ball.dvi

Flavors of Geometry

MSRI Publications

Volume

, 1997

An Elementary Introduction

to Modern Convex Geometry

KEITH BALL

Contents

Preface

Lecture 1. Basic Notions

Lecture 2. Spherical Sections of the Cube

Lecture 3. Fritz John’s Theorem

Lecture 4. Volume Ratios and Spherical Sections of the Octahedron

Lecture 5. The Brunn–Minkowski Inequality and Its Extensions

Lecture 6. Convolutions and Volume Ratios: The Reverse Isoperimetric

Problem

Lecture 7. The Central Limit Theorem and Large Deviation Inequalities 37
Lecture 8. Concentration of Measure in Geometry

Lecture 9. Dvoretzky’s Theorem

Acknowledgements

References

Index

Preface

These notes are based, somewhat loosely, on three series of lectures given by

myself, J. Lindenstrauss and G. Schechtman, during the Introductory Workshop
in Convex Geometry held at the Mathematical Sciences Research Institute in
Berkeley, early in 1996. A fourth series was given by B. Bollob´

as, on rapid

mixing and random volume algorithms; they are found elsewhere in this book.

The material discussed in these notes is not, for the most part, very new, but

the presentation has been strongly influenced by recent developments: among
other things, it has been possible to simplify many of the arguments in the light
of later discoveries. Instead of giving a comprehensive description of the state of
the art, I have tried to describe two or three of the more important ideas that
have shaped the modern view of convex geometry, and to make them as accessible

KEITH BALL

as possible to a broad audience. In most places, I have adopted an informal style
that I hope retains some of the spontaneity of the original lectures. Needless to
say, my fellow lecturers cannot be held responsible for any shortcomings of this
presentation.

I should mention that there are large areas of research that fall under the

very general name of convex geometry, but that will barely be touched upon in
these notes. The most obvious such area is the classical or “Brunn–Minkowski”
theory, which is well covered in [Schneider 1993]. Another noticeable omission is
the combinatorial theory of polytopes: a standard reference here is [Brøndsted
1983].

Lecture 1. Basic Notions

The topic of these notes is convex geometry. The objects of study are con-

vex bodies: compact, convex subsets of Euclidean spaces, that have nonempty
interior. Convex sets occur naturally in many areas of mathematics: linear pro-
gramming, probability theory, functional analysis, partial differential equations,
information theory, and the geometry of numbers, to name a few.

Although convexity is a simple property to formulate, convex bodies possess

a surprisingly rich structure. There are several themes running through these
notes: perhaps the most obvious one can be summed up in the sentence: “All
convex bodies behave a bit like Euclidean balls.” Before we look at some ways in
which this is true it is a good idea to point out ways in which it definitely is not.
This lecture will be devoted to the introduction of a few basic examples that we
need to keep at the backs of our minds, and one or two well known principles.

The only notational conventions that are worth specifying at this point are

the following. We will use

| · |

to denote the standard Euclidean norm on

. For

a body

, vol(

) will mean the volume measure of the appropriate dimension.

The most fundamental principle in convexity is the

Hahn–Banach separation

theorem

, which guarantees that each convex body is an intersection of half-spaces,

and that at each point of the boundary of a convex body, there is at least one
supporting hyperplane. More generally, if

and

are disjoint, compact, convex

subsets of

, then there is a linear functional

→

for which

(

)

< φ

(

)

whenever

∈

and

∈

The simplest example of a convex body in

is the cube, [

−

. This does

not look much like the Euclidean ball. The largest ball inside the cube has radius
1, while the smallest ball containing it has radius

√

, since the corners of the

cube are this far from the origin. So, as the dimension grows, the cube resembles
a ball less and less.

The second example to which we shall refer is the

-dimensional regular solid

simplex: the convex hull of

+ 1 equally spaced points. For this body, the ratio

of the radii of inscribed and circumscribed balls is

: even worse than for the

cube. The two-dimensional case is shown in Figure 1. In Lecture 3 we shall see

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Figure 1.

Inscribed and circumscribed spheres for an

-simplex.

that these ratios are extremal in a certain well-defined sense.

Solid simplices are particular examples of cones. By a

cone

we just mean

the convex hull of a single point and some convex body of dimension

−

1 (Figure

2). In

, the volume of a cone of height

over a base of (

−

1)-dimensional

volume

Bh/n

The third example, which we shall investigate more closely in Lecture 4, is the

-dimensional “octahedron”, or

cross-polytope:

the convex hull of the 2

points

(

, . . . ,

0), (0

, . . . ,

0), . . . , (0

, . . . ,

1). Since this is the unit ball

of the

norm on

, we shall denote it

. The circumscribing sphere of

has radius 1, the inscribed sphere has radius 1

√

; so, as for the cube, the ratio

√

: see Figure 3, left.

is made up of 2

pieces similar to the piece whose points have nonnegative

coordinates, illustrated in Figure 3, right. This piece is a cone of height 1 over
a base, which is the analogous piece in

−

. By induction, its volume is

−

· · · · ·

1 =

and hence the volume of

is 2

Figure 2.

A cone.

KEITH BALL

, . . . ,

(

, . . . ,

)

Figure 3.

The cross-polytope (left) and one orthant thereof (right).

The final example is the Euclidean ball itself,

∈

≤

We shall need to know the volume of the ball: call it

. We can calculate the

surface “area” of

very easily in terms of

: the argument goes back to the

ancients. We think of the ball as being built of thin cones of height 1: see Figure
4, left. Since the volume of each of these cones is 1

times its base area, the

surface of the ball has area

. The sphere of radius 1, which is the surface of

the ball, we shall denote

−

To calculate

, we use integration in spherical polar coordinates. To specify

a point

we use two coordinates:

, its distance from 0, and

, a point on the

sphere, which specifies the direction of

. The point

plays the role of

−

1 real

coordinates. Clearly, in this representation,

rθ

: see Figure 4, right. We can

Figure 4.

Computing the volume of the Euclidean ball.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

write the integral of a function on

∞

−

(

rθ

) “

dθ

”

−

dr.

(1.1)

The factor

−

appears because the sphere of radius

has area

−

times that

−

. The notation “

dθ

” stands for “area” measure on the sphere: its total

mass is the surface area

. The most important feature of this measure is

its rotational invariance: if

is a subset of the sphere and

is an orthogonal

transformation of

, then

U A

has the same measure as

. Throughout these

lectures we shall normalise integrals like that in (1.1) by pulling out the factor

, and write

∞

−

(

rθ

)

−

dσ

(

)

where

−

is the rotation-invariant measure on

−

of total mass 1. To

find

, we integrate the function

7→

exp

−

both ways. This function is at once invariant under rotations and a product of
functions depending upon separate coordinates; this is what makes the method
work. The integral is

−

∞

−∞

−

√

But this equals

∞

−

dσ dr

∞

−

+ 1

Hence

+ 1

This is extremely small if

is large. From Stirling’s formula we know that

+ 1

∼

√

π e

−

(

+1)

so that

is roughly

πe

To put it another way, the Euclidean ball of

volume

1 has

radius

about

πe

which is pretty big.

KEITH BALL

−

Figure 5.

Comparing the volume of a ball with that of its central slice.

This rather surprising property of the ball in high-dimensional spaces is per-

haps the first hint that our intuition might lead us astray. The next hint is
provided by an answer to the following rather vague question: how is the mass
of the ball distributed? To begin with, let’s estimate the (

−

1)-dimensional

volume of a slice through the centre of the ball of volume 1. The ball has radius

−

(Figure 5). The slice is an (

−

1)-dimensional ball of this radius, so its volume is

−

(

−

By Stirling’s formula again, we find that the slice has volume about

√

when

is large. What are the (

−

1)-dimensional volumes of parallel slices? The

slice at distance

from the centre is an (

−

1)-dimensional ball whose radius is

√

−

(whereas the central slice had radius

), so the volume of the smaller

slice is about

√

−

√

−

(

−

Since

is roughly

πe

), this is about

√

−

πex

(

−

≈

√

exp(

−

πex

)

Thus, if we project the mass distribution of the ball of volume 1 onto a single

direction, we get a distribution that is approximately Gaussian (normal) with
variance 1

πe

). What is remarkable about this is that the variance does not

depend upon

. Despite the fact that the radius of the ball of volume 1 grows

πe

), almost all of this volume stays within a slab of fixed width: for

example, about 96% of the volume lies in the slab

{

∈

−

≤

}

See Figure 6.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

= 2

= 16

= 120

96%

Figure 6.

Balls in various dimensions, and the slab that contains about 96% of

each of them.

So the volume of the ball concentrates close to

any

subspace of dimension

−

1. This would seem to suggest that the volume concentrates near the centre

of the ball, where the subspaces all meet. But, on the contrary, it is easy to see
that, if

is large, most of the volume of the ball lies near its surface. In objects of

high dimension, measure tends to concentrate in places that our low-dimensional
intuition considers small. A considerable extension of this curious phenomenon
will be exploited in Lectures 8 and 9.

To finish this lecture, let’s write down a formula for the volume of a general

body in spherical polar coordinates. Let

be such a body with 0 in its interior,

and for each direction

∈

−

let

(

) be the radius of

in this direction.

Then the volume of

−

(

)

−

ds dσ

−

(

)

dσ

(

)

This tells us a bit about particular bodies. For example, if

is the cube [

−

whose volume is 2

, the radius satisfies

−

(

)

≈

πe

So the “average” radius of the cube is about

πe

This indicates that the volume of the cube tends to lie in its corners, where the
radius is close to

√

, not in the middle of its facets, where the radius is close

to 1. In Lecture 4 we shall see that the reverse happens for

and that this has

a surprising consequence.

KEITH BALL

is (

centrally

)

symmetric

, that is, if

−

∈

whenever

∈

, then

the unit ball of some norm

k · k

{

≤

}

This was already mentioned for the octahedron, which is the unit ball of the

norm

The norm and radius are easily seen to be related by

(

) =

for

∈

−

since

(

) is the largest number

for which

rθ

∈

. Thus, for a general sym-

metric body

with associated norm

k · k

, we have this formula for the volume:

vol(

) =

−

dσ

(

)

Lecture 2. Spherical Sections of the Cube

In the first lecture it was explained that the cube is rather unlike a Euclidean

ball in

: the cube [

−

includes a ball of radius 1 and no more, and is

included in a ball of radius

√

and no less. The cube is a bad approximation

to the Euclidean ball. In this lecture we shall take this point a bit further. A
body like the cube, which is bounded by a finite number of flat facets, is called a

polytope

. Among symmetric polytopes, the cube has the fewest possible facets,

namely 2

. The question we shall address here is this:

is a polytope in

with m facets, how well can

approximate the

Euclidean ball?

Let’s begin by clarifying the notion of approximation. To simplify matters we
shall only consider symmetric bodies. By analogy with the remarks above, we
could define the distance between two convex bodies

and

to be the smallest

for which there is a scaled copy of

inside

and another copy of

times

as large, containing

. However, for most purposes, it will be more convenient

to have an affine-invariant notion of distance: for example we want to regard all
parallelograms as the same. Therefore:

Definition.

The distance

(

K, L

) between symmetric convex bodies

and

is the least positive

for which there is a linear image ˜

such that

⊂

. (See Figure 7.)

Note that this distance is multiplicative, not additive: in order to get a metric (on
the set of linear equivalence classes of symmetric convex bodies) we would need to
take log

instead of

. In particular, if

and

are identical then

(

K, L

) = 1.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Figure 7.

Defining the distance between

and

Our observations of the last lecture show that the distance between the cube
and the Euclidean ball in

at most

√

. It is intuitively clear that it really

√

, i.e., that we cannot find a linear image of the ball that sandwiches the

cube any better than the obvious one. A formal proof will be immediate after
the next lecture.

The main result of this lecture will imply that, if a polytope is to have small

distance from the Euclidean ball, it must have very many facets: exponentially
many in the dimension

Theorem 2.1.

Let

be a

(

symmetric

)

polytope in

with

(

K, B

) =

Then

has at least

)

facets

On the other hand

for each

there is a polytope

with

facets whose distance from the ball is at most

The arguments in this lecture, including the result just stated, go back to the
early days of packing and covering problems. A classical reference for the subject
is [Rogers 1964].

Before we embark upon a proof of Theorem 2.1, let’s look at a reformulation

that will motivate several more sophisticated results later on.

A symmetric

convex body in

with

pairs of facets can be realised as an

-dimensional

slice (through the centre) of the cube in

. This is because such a body is

the intersection of

slabs in

, each of the form

{

x, v

i| ≤

}

, for some

nonzero vector

. This is shown in Figure 8.

Thus

is the set

{

x, v

i| ≤

1 for 1

≤

}

, for some sequence (

)

vectors in

. The linear map

7→

(

x, v

, . . . ,

x, v

)

embeds

as a subspace

, and the intersection of

with the cube

[

−

is the set of points

for which

| ≤

1 for each coordinate

. So

this intersection is the image of

under

. Conversely, any

-dimensional slice

of [

−

is a body with at most

pairs of faces. Thus, the result we are

aiming at can be formulated as follows:

The cube in

has almost spherical sections whose dimension

is roughly

log

and not more.

KEITH BALL

Figure 8.

Any symmetric polytope is a section of a cube.

In Lecture 9 we shall see that all symmetric

-dimensional convex bodies have

almost spherical sections of dimension about log

. As one might expect, this is

a great deal more difficult to prove for general bodies than just for the cube.

For the proof of Theorem 2.1, let’s forget the symmetry assumption again and

just ask for a polytope

{

x, v

i ≤

1 for 1

≤

}

with

facets for which

⊂

What do these inclusions say about the vectors (

)? The first implies that each

has length at most 1, since, if not,

would be a vector in

but not

. The second inclusion says that if

does not belong to

then

does

not belong to

: that is, if

> d

, there is an

for which

x, v

1. This is

equivalent to the assertion that for every unit vector

there is an

for which

θ, v

i ≥

Thus our problem is to look for as few vectors as possible,

, v

, . . . , v

, of

length at most 1, with the property that for every unit vector

there is some

with

θ, v

i ≥

. It is clear that we cannot do better than to look for vectors

of length exactly 1: that is, that we may assume that all facets of our polytope
touch the ball. Henceforth we shall discuss only such vectors.

For a fixed unit vector

and

∈

1), the set

(

ε, v

) of

∈

−

for which

θ, v

i ≥

is called a

spherical cap

(or just a

cap

); when we want to be precise,

we will call it the

cap about

. (Note that

does not refer to the radius!) See

Figure 9, left.

We want every

∈

−

to belong to at least one of the (1

)-caps determined

by the (

). So our task is to estimate the number of caps of a given size needed

to cover the entire sphere. The principal tool for doing this will be upper and
lower estimates for the area of a spherical cap. As in the last lecture, we shall

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Figure 9.

Left:

-cap

(

ε, v

)

about

. Right: cap of radius

about

measure this area as a proportion of the sphere: that is, we shall use

−

our measure. Clearly, if we show that each cap occupies only a small proportion
of the sphere, we can conclude that we need plenty of caps to cover the sphere.
What is slightly more surprising is that once we have shown that spherical caps
are not

too

small, we will also be able to deduce that we

can

cover the sphere

without using too many.

In order to state the estimates for the areas of caps, it will sometimes be

convenient to measure the size of a cap in terms of its radius, instead of using
the

measure. The cap of radius

about

∈

−

| ≤

as illustrated in Figure 9, right. (In defining the radius of a cap in this way we
are implicitly adopting a particular metric on the sphere: the metric induced by
the usual Euclidean norm on

.) The two estimates we shall use are given in

the following lemmas.

Lemma 2.2 (Upper bound for spherical caps).

For

≤

ε <

the cap

(

ε, u

)

−

has measure at most

−

nε

Lemma 2.3 (Lower bound for spherical caps).

For

≤

a cap of

radius

−

has measure at least

(

−

We can now prove Theorem 2.1.

Proof.

Lemma 2.2 implies the first assertion of Theorem 2.1 immediately. If

is a polytope in

with

facets and if

⊂

, we can find

caps

(

, v

) covering

−

. Each covers at most

−

)

of the sphere, so

≥

exp

To get the second assertion of Theorem 2.1 from Lemma 2.3 we need a little

more argument. It suffices to find

= 4

points

, v

, . . . , v

on the sphere so

that the caps of radius 1 centred at these points cover the sphere: see Figure 10.

Such a set of points is called a 1

-net

for the sphere: every point of the sphere is

KEITH BALL

Figure 10.

The

-cap has radius 1.

within distance 1 of some

. Now, if we choose a set of points on the sphere any

two of which are at

least

distance 1 apart, this set cannot have too many points.

(Such a set is called 1

-separated

.) The reason is that the caps of radius

about

these points will be disjoint, so the sum of their areas will be at most 1. A cap of
radius

has area at least

, so the number

of these caps satisfies

≤

This does the job, because a

maximal

1-separated set is automatically a 1-net:

if we can’t add any further points that are at least distance 1 from all the points
we have got, it can only be because every point of the sphere is

within

distance 1

of at least one of our chosen points. So the sphere has a 1-net consisting of only
4

points, which is what we wanted to show.

Lemmas 2.2 and 2.3 are routine calculations that can be done in many ways.
We leave Lemma 2.3 to the dedicated reader. Lemma 2.2, which will be quoted
throughout Lectures 8 and 9, is closely related to the Gaussian decay of the
volume of the ball described in the last lecture. At least for smallish

(which is

the interesting range) it can be proved as follows.

Proof.

The proportion of

−

belonging to the cap

(

ε, u

) equals the pro-

portion of the solid ball that lies in the “spherical cone” illustrated in Figure 11.

√

−

Figure 11.

Estimating the area of a cap.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

As is also illustrated, this spherical cone is contained in a ball of radius

√

−

(if

≤

√

2), so the ratio of its volume to that of the ball is at most

−

≤

−

nε

In Lecture 8 we shall quote the upper estimate for areas of caps repeatedly. We
shall in fact be using yet a third way to measure caps that differs very slightly
from the

(

ε, u

) description. The reader can easily check that the preceding

argument yields the same estimate

−

nε

for this other description.

Lecture 3. Fritz John’s Theorem

In the first lecture we saw that the cube and the cross-polytope lie at distance

at most

√

from the Euclidean ball in

, and that for the simplex, the distance

is at most

. It is intuitively clear that these estimates cannot be improved. In

this lecture we shall describe a strong sense in which this is as bad as things get.
The theorem we shall describe was proved by Fritz John [1948].

John considered ellipsoids inside convex bodies. If (

)

is an orthonormal

basis of

and (

) are positive numbers, the ellipsoid

(

x, e

≤

)

has volume

. John showed that each convex body contains a unique

ellipsoid of largest volume and, more importantly, he

characterised

it. He showed

that if

is a symmetric convex body in

and

is its maximal ellipsoid then

⊂

√

Hence, after an affine transformation (one taking

) we can arrange that

⊂

√

A nonsymmetric

may require

, like the simplex, rather than

√

John’s characterisation of the maximal ellipsoid is best expressed after an

affine transformation that takes the maximal ellipsoid to

. The theorem states

that

is the maximal ellipsoid in

if a certain condition holds—roughly, that

there be plenty of points of contact between the boundary of

and that of the

ball. See Figure 12.

Theorem 3.1 (John’s Theorem).

Each convex body

contains an unique

ellipsoid of maximal volume

This ellipsoid is

if and only if the following

conditions are satisfied

⊂

and

(

for some

)

there are Euclidean unit

vectors

(

)

on the boundary of

and positive numbers

(

)

satisfying

= 0

(3.1)

and

x, u

for each

∈

(3.2)

KEITH BALL

Figure 12.

The maximal ellipsoid touches the boundary at many points.

According to the theorem the points at which the sphere touches

∂K

can be

given a mass distribution whose centre of mass is the origin and whose inertia
tensor is the identity matrix. Let’s see where these conditions come from. The
first condition, (3.1), guarantees that the (

) do not all lie “on one side of the

sphere”. If they did, we could move the ball away from these contact points and
blow it up a bit to obtain a larger ball in

. See Figure 13.

The second condition, (3.2), shows that the (

) behave rather like an or-

thonormal basis in that we can resolve the Euclidean norm as a (weighted) sum
of squares of inner products. Condition (3.2) is equivalent to the statement that

x, u

for all

This guarantees that the (

) do not all lie close to a proper subspace of

. If

they did, we could shrink the ball a little in this subspace and expand it in an
orthogonal direction, to obtain a larger ellipsoid inside

. See Figure 14.

Condition (3.2) is often written in matrix (or operator) notation as

⊗

(3.3)

where

is the identity map on

and, for any unit vector

⊗

is the

rank-one orthogonal projection onto the span of

, that is, the map

7→ h

x, u

The trace of such an orthogonal projection is 1. By equating the traces of the

Figure 13.

An ellipsoid where all contacts are on one side can grow.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Figure 14.

An ellipsoid (solid circle) whose contact points are all near one plane

can grow.

matrices in the preceding equation, we obtain

In the case of a

symmetric

convex body, condition (3.1) is redundant, since

we can take any sequence (

) of contact points satisfying condition (3.2) and

replace each

by the pair

each with half the weight of the original.

Let’s look at a few concrete examples. The simplest is the cube. For this

body the maximal ellipsoid is

, as one would expect. The contact points are

the standard basis vectors (

, e

, . . . , e

) of

and their negatives, and they

satisfy

⊗

That is, one can take all the weights

equal to 1 in (3.2). See Figure 15.

The simplest nonsymmetric example is the simplex. In general, there is no

natural way to place a regular simplex in

, so there is no natural description of

the contact points. Usually the best way to talk about an

-dimensional simplex

is to realise it in

: for example as the convex hull of the standard basis

Figure 15.

The maximal ellipsoid for the cube.

KEITH BALL

Figure 16.

is contained in the convex body determined by the hyperplanes

tangent to the maximal ellipsoid at the contact points.

vectors in

. We leave it as an exercise for the reader to come up with a nice

description of the contact points.

One may get a bit more of a feel for the second condition in John’s Theorem by

interpreting it as a rigidity condition. A sequence of unit vectors (

) satisfying

the condition (for some sequence (

)) has the property that if

is a linear map

of determinant 1, not all the images

T u

can have Euclidean norm less than 1.

John’s characterisation immediately implies the inclusion mentioned earlier:

is symmetric and

is its maximal ellipsoid then

⊂

√

. To check this

we may assume

. At each contact point

, the convex bodies

and

have the same supporting hyperplane. For

, the supporting hyperplane at

any point

is perpendicular to

. Thus if

∈

we have

x, u

i ≤

1 for each

and we conclude that

is a subset of the convex body

{

∈

x, u

i ≤

1 for 1

≤

}

(3.4)

An example of this is shown in Figure 16.

In the symmetric case, the same argument shows that for each

∈

we have

x, u

i| ≤

1 for each

. Hence, for

∈

x, u

≤

| ≤

√

, which is exactly the statement

⊂

√

n B

. We leave as a slightly

trickier exercise the estimate

| ≤

in the nonsymmetric case.

Proof of John’s Theorem.

The proof is in two parts, the harder of which

is to show that if

ellipsoid of largest volume, then we can find an

appropriate system of weights on the contact points. The easier part is to show
that if such a system of weights exists, then

is the

unique

ellipsoid of maximal

volume. We shall describe the proof only in the symmetric case, since the added
complications in the general case add little to the ideas.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

We begin with the easier part. Suppose there are unit vectors (

) in

∂K

and

numbers (

) satisfying

⊗

Let

x, e

≤

be an ellipsoid in

, for some orthonormal basis (

) and positive

. We want

to show that

≤

and that the product is equal to 1 only if

= 1 for all

Since

E ⊂

we have that for each

the hyperplane

{

x, u

= 1

}

does not

cut

. This implies that each

belongs to the

polar ellipsoid

y, e

≤

(The reader unfamiliar with duality is invited to check this.) So, for each

, e

≤

Hence

, e

≤

But the left side of the equality is just

, because, by condition (3.2), we

have

, e

= 1

for each

. Finally, the fact that the geometric mean does not exceed the arith-

metic mean (the AM/GM inequality) implies that

≤

and there is equality in the first of these inequalities only if all

are equal to 1.

We now move to the harder direction. Suppose

is an ellipsoid of largest

volume in

. We want to show that there is a sequence of contact points (

)

and positive weights (

) with

⊗

We already know that, if this is possible, we must have

= 1

KEITH BALL

So our aim is to show that the matrix

can be written as a convex combina-

tion of (a finite number of) matrices of the form

⊗

, where each

is a contact

point. Since the space of matrices is finite-dimensional, the problem is simply
to show that

belongs to the convex hull of the set of all such rank-one

matrices,

{

⊗

is a contact point

}

We shall aim to get a contradiction by showing that if

not

, we can

perturb the unit ball slightly to get a new ellipsoid in

of larger volume than

the unit ball.

Suppose that

is not in

. Apply the separation theorem in the space of

matrices to get a linear functional

(on this space) with the property that

< φ

(

⊗

)

(3.5)

for each contact point

. Observe that

can be represented by an

matrix

= (

), so that, for any matrix

= (

(

) =

Since all the matrices

⊗

and

are symmetric, we may assume the same

for

. Moreover, since these matrices all have the same trace, namely 1, the

inequality

(

)

< φ

(

⊗

) will remain unchanged if we add a constant to

each diagonal entry of

. So we may assume that the trace of

is 0: but this

says precisely that

(

) = 0.

Hence, unless the identity has the representation we want, we have found a

symmetric matrix

with zero trace for which

(

⊗

)

for every contact point

. We shall use this

to build a bigger ellipsoid inside

Now, for each vector

, the expression

(

⊗

)

is just the number

. For sufficiently small

δ >

0, the set

∈

(

δH

)

≤

is an ellipsoid and as

tends to 0 these ellipsoids approach

. If

is one of

the original contact points, then

(

δH

)

= 1 +

δu

Hu >

does not belong to

. Since the boundary of

is compact (and the function

7→

is continuous)

will not contain any other point of

∂K

as long as

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

is sufficiently small. Thus, for such

, the ellipsoid

is strictly inside

and

some slightly expanded ellipsoid is inside

It remains to check that each

has volume at least that of

. If we denote

by (

) the eigenvalues of the symmetric matrix

δH

, the volume of

, so the problem is to show that, for each

, we have

≤

1. What

we know is that

is the trace of

δH

, which is

, since the trace of

0. So the AM/GM inequality again gives

≤

as required.

There is an analogue of John’s Theorem that characterises the unique ellipsoid
of minimal volume containing a given convex body. (The characterisation is
almost identical, guaranteeing a resolution of the identity in terms of contact
points of the body and the Euclidean sphere.) This minimal volume ellipsoid
theorem can be deduced directly from John’s Theorem by duality. It follows
that, for example, the ellipsoid of minimal volume containing the cube [

−

is the obvious one: the ball of radius

√

. It has been mentioned several times

without proof that the distance of the cube from the Euclidean ball in

exactly

√

We can now see this easily: the ellipsoid of minimal volume outside

the cube has volume

√

times that of the ellipsoid of maximal volume inside

the cube. So we cannot sandwich the cube between homothetic ellipsoids unless
the outer one is at least

√

times the inner one.

We shall be using John’s Theorem several times in the remaining lectures. At

this point it is worth mentioning important extensions of the result. We can view
John’s Theorem as a description of those linear maps from Euclidean space to a
normed space (whose unit ball is

) that have largest determinant, subject to the

condition that they have norm at most 1: that is, that they map the Euclidean
ball into

. There are many other norms that can be put on the space of linear

maps. John’s Theorem is the starting point for a general theory that builds
ellipsoids related to convex bodies by maximising determinants subject to other
constraints on linear maps. This theory played a crucial role in the development
of convex geometry over the last 15 years. This development is described in
detail in [Tomczak-Jaegermann 1988].

Lecture 4. Volume Ratios and Spherical Sections of the

Octahedron

In the second lecture we saw that the

-dimensional cube has almost spherical

sections of dimension about log

but not more. In this lecture we will examine

the corresponding question for the

-dimensional cross-polytope

. In itself,

this body is as far from the Euclidean ball as is the cube in

: its distance from

the ball, in the sense described in Lecture 2 is

√

. Remarkably, however, it has

KEITH BALL

almost spherical sections whose dimension is about

. We shall deduce this from

what is perhaps an even more surprising statement concerning intersections of
bodies. Recall that

contains the Euclidean ball of radius

√

. If

is an

orthogonal transformation of

then

U B

also contains this ball and hence so

does the intersection

∩

U B

. But, whereas

does not lie in any Euclidean

ball of radius less than 1, we have the following theorem [Kaˇsin 1977]:

Theorem 4.1.

For each

there is an orthogonal transformation

for which

the intersection

∩

U B

is contained in the Euclidean ball of radius

√

(

and contains the ball of radius

√

(The constant 32 can easily be improved: the important point is that it is inde-
pendent of the dimension

.) The theorem states that the intersection of just two

copies of the

-dimensional octahedron may be approximately spherical. Notice

that if we tried to approximate the Euclidean ball by intersecting rotated copies
of the cube, we would need exponentially many in the dimension, because the
cube has only 2

facets and our approximation needs exponentially many facets.

In contrast, the octahedron has a much larger number of facets, 2

: but of course

we need to do a lot more than just count facets in order to prove Theorem 4.1.
Before going any further we should perhaps remark that the cube has a property
that corresponds to Theorem 4.1. If

is the cube and

is the same orthogonal

transformation as in the theorem, the convex hull

conv(

∪

U Q

)

is at distance at most 32 from the Euclidean ball.

In spirit, the argument we shall use to establish Theorem 4.1 is Kaˇsin’s original

one but, following [Szarek 1978], we isolate the main ingredient and we organise
the proof along the lines of [Pisier 1989]. Some motivation may be helpful. The
ellipsoid of maximal volume inside

is the Euclidean ball of radius

√

. (See

Figure 3.) There are 2

points of contact between this ball and the boundary of

: namely, the points of the form

, . . . ,

one in the middle of each facet of

. The vertices,

(

, . . . ,

are the points of

furthest from the origin. We are looking for a rotation

U B

whose facets chop off the spikes of

(or vice versa). So we want to know that

the points of

at distance about 1

√

from the origin are fairly typical, while

those at distance 1 are atypical.

For a unit vector

∈

−

, let

(

) be the radius of

in the direction

(

) =

−

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

In the first lecture it was explained that the volume of

can be written

−

(

)

dσ

and that it is equal to 2

!. Hence

−

(

)

dσ

≤

√

Since the average of

(

)

is at most 2

√

, the value of

(

) cannot often be

much more than 2

√

. This feature of

is captured in the following definition

of Szarek.

Definition.

Let

be a convex body in

. The

volume ratio

vr(

) =

vol(

)

vol(

)

where

is the ellipsoid of maximal volume in

The preceding discussion shows that vr(

)

≤

2 for all

. Contrast this with the

cube in

, whose volume ratio is about

√

2. The only property of

that

we shall use to prove Kaˇsin’s Theorem is that its volume ratio is at most 2. For
convenience, we scale everything up by a factor of

√

and prove the following.

Theorem

4.2.

Let

be a symmetric convex body in

that contains the

Euclidean unit ball

and for which

vol(

)

vol(

)

Then there is an orthogonal transformation

for which

∩

U K

⊂

Proof.

It is convenient to work with the norm on

whose unit ball is

. Let

k · k

denote this norm and

| · |

the standard Euclidean norm. Notice that, since

⊂

, we have

k ≤ |

for all

∈

The radius of the body

∩

U K

in a given direction is the minimum of the

radii of

and

U K

in that direction. So the norm corresponding to the body

∩

U K

is the

maximum

of the norms corresponding to

and

U K

. We need

to find an orthogonal transformation

with the property that

max (

U θ

)

≥

for every

∈

−

. Since the maximum of two numbers is at least their average,

it will suffice to find

with

U θ

≥

for all

θ.

KEITH BALL

For each

∈

write

(

) for the average

(

U x

). One sees imme-

diately that

is a norm (that is, it satisfies the triangle inequality) and that

(

)

≤ |

for every

, since

preserves the Euclidean norm. We shall show in

a moment that there is a

for which

−

(

)

dσ

≤

(4.1)

This says that

(

) is large on average: we want to deduce that it is large

everywhere.

Let

be a point of the sphere and write

(

) =

, for 0

< t

≤

1. The crucial

point will be that, if

is close to

, then

(

) cannot be much more than

. To

be precise, if

−

| ≤

then

(

)

≤

(

) +

(

−

)

≤

−

| ≤

Hence,

(

) is at most 2

for every

in a spherical cap of radius

about

From Lemma 2.3 we know that this spherical cap has measure at least

−

≥

So 1

(

)

is at least 1

)

on a set of measure at least (

. Therefore

−

(

)

dσ

≥

)

By (4.1), the integral is at most

, so

≥

). Thus our arbitrary point

satisfies

(

)

≥

It remains to find

satisfying (4.1). Now, for any

, we have

(

)

U θ

≥ k

U θ

k k

so it will suffice to find a

for which

−

U θ

dσ

≤

(4.2)

The hypothesis on the volume of

can be written in terms of the norm as

−

dσ

The group of orthogonal transformations carries an invariant probability mea-
sure. This means that we can average a function over the group in a natural
way. In particular, if

is a function on the sphere and

is some point on the

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

sphere, the average over orthogonal

of the value

(

U θ

) is just the average of

on the sphere: averaging over

mimics averaging over the sphere:

ave

(

U θ

) =

−

(

)

dσ

(

)

Hence,

ave

−

U θ

dσ

(

) =

−

ave

U θ

dσ

(

)

−

dσ

(

)

dσ

(

)

−

dσ

(

)

Since the average over all

of the integral is at most

, there is at least one

for which the integral is at most

. This is exactly inequality (4.2).

The choice of

in the preceding proof is a random one. The proof does not

in any way tell us how to find an explicit

for which the integral is small. In

the case of a general body

, this is hardly surprising, since we are assuming

nothing about how the volume of

is distributed. But, in view of the earlier

remarks about facets of

chopping off spikes of

U B

, it is tempting to think

that for the particular body

we might be able to write down an appropriate

explicitly. In two dimensions the best choice of

is obvious: we rotate the

diamond through 45

◦

and after intersection we have a regular octagon as shown

in Figure 17.

The most natural way to try to generalise this to higher dimensions is to look

for a

such that each vertex of

U B

is exactly aligned through the centre of a

facet of

: that is, for each standard basis vector

U e

is a multiple of

one of the vectors (

, . . . ,

). (The multiple is

√

since

U e

has length 1.)

Thus we are looking for an

orthogonal matrix

each of whose entries is

∩

Figure 17.

The best choice for

in two dimensions is a

◦

rotation.

KEITH BALL

√

. Such matrices, apart from the factor

√

, are called

Hadamard matrices

In what dimensions do they exist? In dimensions 1 and 2 there are the obvious

(1)

and

√

−

√

It is not too hard to show that in larger dimensions a Hadamard matrix cannot
exist unless the dimension is a multiple of 4. It is an open problem to determine
whether they exist in all dimensions that are multiples of 4. They are known to
exist, for example, if the dimension is a power of 2: these examples are known
as the Walsh matrices.

In spite of this, it seems extremely unlikely that one might prove Kaˇsin’s

Theorem using Hadamard matrices. The Walsh matrices certainly do not give
anything smaller than

−

; pretty miserable compared with

−

. There

are some good reasons, related to Ramsey theory, for believing that one cannot
expect to find genuinely explicit matrices of any kind that would give the right
estimates.

Let’s return to the question with which we opened the lecture and see how

Theorem 4.1 yields almost spherical sections of octahedra. We shall show that,
for each

, the 2

-dimensional octahedron has an

-dimensional slice which

is within distance 32 of the (

-dimensional) Euclidean ball. By applying the

argument of the theorem to

, we obtain an

orthogonal matrix

such

that

U x

≥

√

for every

∈

, where

k · k

denotes the

norm. Now consider the map

→

with matrix

. For each

∈

, the norm of

T x

U x

≥

√

On the other hand, the Euclidean norm of

T x

U x

√

So, if

belongs to the image

, then, setting

T x

≥

√

By the Cauchy–Schwarz inequality, we have

≤

√

, so the slice of

by the subspace

has distance at most 32 from

, as we wished to show.

A good deal of work has been done on embedding of other subspaces of

into

-spaces of low dimension, and more generally subspaces of

into low-

dimensional

, for 1

< p <

2. The techniques used come from probability

theory:

-stable random variables, bootstrapping of probabilities and deviation

estimates. We shall be looking at applications of the latter in Lectures 8 and 9.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

The major references are [Johnson and Schechtman 1982; Bourgain et al. 1989;
Talagrand 1990].

Volume ratios have been studied in considerable depth. They have been found

to be closely related to the so-called cotype-2 property of normed spaces: this re-
lationship is dealt with comprehensively in [Pisier 1989]. In particular, Bourgain
and Milman [1987] showed that a bound for the cotype-2 constant of a space
implies a bound for the volume ratio of its unit ball. This demonstrated, among
other things, that there is a uniform bound for the volume ratios of slices of
octahedra of all dimensions. A sharp version of this result was proved in [Ball
1991]: namely, that for each

has largest volume ratio among the balls of

-dimensional subspaces of

. The proof uses techniques that will be discussed

in Lecture 6.

This is a good point to mention a result of Milman [1985] that looks super-

ficially like the results of this lecture but lies a good deal deeper. We remarked
that while we can almost get a sphere by intersecting two copies of

, this is

very far from possible with two cubes. Conversely, we can get an almost spher-
ical convex hull of two cubes but not of two copies of

. The

QS-Theorem

(an abbreviation for “quotient of a subspace”) states that if we combine the two
operations, intersection and convex hull, we can get a sphere no matter what
body we start with.

Theorem 4.3 (QS-Theorem).

There is a constant

(

independent of ev-

erything

)

such that

for any symmetric convex body

of any dimension

there

are linear maps

and

and an ellipsoid

with the following property

= conv(

∪

then

E ⊂

∩

⊂

Lecture 5. The Brunn–Minkowski Inequality

and Its Extensions

In this lecture we shall introduce one of the most fundamental principles in

convex geometry: the Brunn–Minkowski inequality. In order to motivate it, let’s
begin with a simple observation concerning convex sets in the plane. Let

⊂

be such a set and consider its slices by a family of parallel lines, for example those
parallel to the

-axis. If the line

meets

, call the length of the slice

(

The graph of

is obtained by shaking

down onto the

-axis like a deck of

cards (of different lengths). This is shown in Figure 18. It is easy to see that the
function

is concave on its support. Towards the end of the last century, Brunn

investigated what happens if we try something similar in higher dimensions.

Figure 19 shows an example in three dimensions. The central, hexagonal, slice

has larger volume than the triangular slices at the ends: each triangular slice
can be decomposed into four smaller triangles, while the hexagon is a union of
six such triangles. So our first guess might be that the slice area is a concave

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Figure 20.

The area of a cone’s section increases with

Theorem 5.1 (Brunn).

Let

be a convex body in

let

be a unit vector

and for each

let

be the hyperplane

{

∈

x, u

}

Then the function

7→

vol (

∩

)

(

−

is concave on its support

One consequence of this is that if

is centrally symmetric, the largest slice

perpendicular to a given direction is the central slice (since an even concave
function is largest at 0). This is the situation in Figure 19.

Brunn’s Theorem was turned from an elegant curiosity into a powerful tool by

Minkowski. His reformulation works in the following way. Consider three parallel
slices of a convex body in

at positions

and

, where

= (1

−

)

λt

for

some

∈

1). This is shown in Figure 21.

Call the slices

, and

and think of them as subsets of

−

. If

∈

and

∈

, the point (1

−

)

λy

belongs to

: to see this, join the points

(

r, x

) and (

t, y

) in

and observe that the resulting line segment crosses

(

−

)

λy

). So

includes a new set

−

)

λA

{

−

)

λy

∈

, y

∈

}

(

t, y

)

(

s, x

)

Figure 21.

The section

contains the weighted average of

and

KEITH BALL

(This way of using the addition in

to define an addition of sets is called

Minkowski addition

.) Brunn’s Theorem says that the volumes of the three sets

, and

−

satisfy

vol (

)

(

−

≥

−

) vol (

)

(

−

vol (

)

(

−

The Brunn–Minkowski inequality makes explicit the fact that all we really know
about

is that it includes the Minkowski combination of

and

. Since we

have now eliminated the role of the ambient space

, it is natural to rewrite

the inequality with

in place of

−

Theorem 5.2 (Brunn–Minkowski inequality).

and

are nonempty

compact subsets of

then

vol ((1

−

)

λB

)

≥

−

) vol (

)

vol (

)

(The hypothesis that

and

be nonempty corresponds in Brunn’s Theorem

to the restriction of a function to its support.) It should be remarked that the
inequality is stated for general compact sets, whereas the early proofs gave the
result only for convex sets. The first complete proof for the general case seems
to be in [L

ıusternik 1935].

To get a feel for the advantages of Minkowski’s formulation, let’s see how it

implies the classical isoperimetric inequality in

Theorem 5.3 (Isoperimetric inequality).

Among bodies of a given volume

Euclidean balls have least surface area

Proof.

Let

be a compact set in

whose volume is equal to that of

, the

Euclidean ball of radius 1. The surface “area” of

can be written

vol(

∂C

) = lim

→

vol (

εB

)

−

vol (

)

as shown in Figure 22. By the Brunn–Minkowski inequality,

vol (

εB

)

≥

vol (

)

vol (

)

εB

Figure 22.

Expressing the area as a limit of volume increments.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Hence

vol (

εB

)

≥

vol (

)

vol (

)

≥

vol (

) +

nε

vol (

)

(

−

vol (

)

vol(

∂C

)

≥

vol(

)

(

−

vol(

)

Since

and

have the same volume, this shows that vol(

∂C

)

≥

vol(

and the latter equals vol(

∂B

), as we saw in Lecture 1.

This relationship between the Brunn–Minkowski inequality and the isoperimetric
inequality will be explored in a more general context in Lecture 8.

The Brunn–Minkowski inequality has an alternative version that is formally

weaker. The AM/GM inequality shows that, for

in (0

1),

−

) vol(

)

vol(

)

≥

vol(

)

−

)

vol(

)

λ/n

So the Brunn–Minkowski inequality implies that, for compact sets

and

∈

1),

vol((1

−

)

λB

)

≥

vol(

)

−

vol(

)

(5.1)

Although this multiplicative Brunn–Minkowski inequality is weaker than the
Brunn–Minkowski inequality for

particular

, and

, if one knows (5.1) for

all

, and

one can easily deduce the Brunn–Minkowski inequality for all

, and

. This deduction will be left for the reader.

Inequality (5.1) has certain advantages over the Brunn–Minkowski inequality.

(i) We no longer need to stipulate that

and

be nonempty, which makes the

inequality easier to use.

(ii) The dimension

has disappeared.

(iii) As we shall see, the multiplicative inequality lends itself to a particularly

simple proof because it has a generalisation from sets to functions.

Before we describe the functional Brunn–Minkowski inequality let’s just remark
that the multiplicative Brunn–Minkowski inequality can be reinterpreted back in
the setting of Brunn’s Theorem: if

7→

(

) is a function obtained by scanning

a convex body with parallel hyperplanes, then log

is a concave function (with

the usual convention regarding

−∞

In order to move toward a functional generalisation of the multiplicative

Brunn–Minkowski inequality let’s reinterpret inequality (5.1) in terms of the
characteristic functions of the sets involved. Let

, and

denote the char-

acteristic functions of

, and (1

−

)

λB

respectively; so, for example,

(

) = 1 if

∈

and 0 otherwise. The volumes of

, and (1

−

)

λB

are the integrals

, and

. The Brunn–Minkowski inequality says

that

≥

−

KEITH BALL

But what is the relationship between

, and

that guarantees its truth? If

(

) = 1 and

(

) = 1 then

∈

and

∈

, so

−

)

λy

∈

−

)

λB,

and hence

((1

−

)

λy

) = 1. This certainly ensures that

((1

−

)

λy

)

≥

(

)

−

(

)

for any

and

This inequality for the three functions at least has a homogeneity that matches
the desired inequality for the integrals.

In a series of papers, Pr´

ekopa and

Leindler proved that this homogeneity is enough.

Theorem 5.4 (The Pr´

ekopa–Leindler inequality).

and

are

nonnegative measurable functions on

∈

and for all

and

((1

−

)

λy

)

≥

(

)

−

(

)

(5.2)

then

≥

−

It is perhaps helpful to notice that the Pr´

ekopa–Leindler inequality looks like

H¨

older’s inequality, backwards. If

and

were given and we set

(

) =

(

)

−

(

)

(for each

), then H¨

older’s inequality says that

≤

−

(H¨

older’s inequality is often written with 1

instead of 1

−

, 1

instead of

, and

replaced by

.) The difference between Pr´

ekopa–Leindler and

H¨

older is that, in the former, the value

(

) may be much larger since it is a

supremum over many pairs (

x, y

) satisfying

= (1

−

)

λy

rather than just

the pair (

z, z

Though it generalises the Brunn–Minkowski inequality, the Pr´

ekopa–Leindler

inequality is a good deal simpler to prove, once properly formulated. The argu-
ment we shall use seems to have appeared first in [Brascamp and Lieb 1976b].
The crucial point is that the passage from sets to functions allows us to prove
the inequality by induction on the dimension, using only the one-dimensional
case. We pay the small price of having to do a bit extra for this case.

Proof of the Pr´

ekopa–Leindler inequality.

We start by checking the

one-dimensional Brunn–Minkowski inequality. Suppose

and

are nonempty

measurable subsets of the line. Using

| · |

to denote length, we want to show that

−

)

λB

| ≥

−

)

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

We may assume that

and

are compact and we may shift them so that

the right-hand end of

and the left-hand end of

are both at 0. The set

−

)

λB

now includes the essentially disjoint sets (1

−

)

and

λB

, so its

length is at least the sum of the lengths of these sets.

Now suppose we have nonnegative integrable functions

, and

on the

line, satisfying condition (5.2). We may assume that

and

are bounded. Since

the inequality to be proved has the same homogeneity as the hypothesis (5.2),
we may also assume that

and

are normalised so that sup

= sup

= 1.

By Fubini’s Theorem, we can write the integrals of

and

as integrals of the

lengths of their level sets:

(

)

(

≥

)

dt,

and similarly for

. If

(

)

≥

and

(

)

≥

then

((1

−

)

λy

)

≥

. So we

have the inclusion

(

≥

)

⊃

−

)(

≥

) +

(

≥

)

For 0

≤

t <

1 the sets on the right are nonempty so the one-dimensional Brunn–

Minkowski inequality shows that

(

≥

)

| ≥

−

)

(

≥

)

(

≥

)

Integrating this inequality from 0 to 1 we get

≥

−

)

and the latter expression is at least

−

by the AM/GM inequality. This does the one-dimensional case.

The induction that takes us into higher dimensions is quite straightforward, so

we shall just sketch the argument for sets in

, rather than functions. Suppose

and

are two such sets and, for convenience, write

= (1

−

)

λB.

Choose a unit vector

and, as before, let

be the hyperplane

{

∈

x, u

}

perpendicular to

at “position”

. Let

denote the slice

∩

and similarly

for

and

, and regard these as subsets of

−

. If

and

are real numbers,

and if

= (1

−

)

λt

, the slice

includes (1

−

)

λB

. (This is reminiscent

KEITH BALL

of the earlier argument relating Brunn’s Theorem to Minkowski’s reformulation.)
By the inductive hypothesis in

−

vol(

)

≥

vol(

)

−

vol(

)

Let

, and

be the functions on the line, given by

(

) = vol(

)

(

) = vol(

)

(

) = vol(

)

Then, for

, and

as above,

(

)

≥

(

)

−

(

)

By the one-dimensional Pr´

ekopa–Leindler inequality,

≥

−

But this is exactly the statement vol(

)

≥

vol(

)

−

vol(

)

, so the inductive

step is complete.

The proof illustrates clearly why the Pr´

ekopa–Leindler inequality makes things

go smoothly. Although we only carried out the induction for

sets

, we required

the one-dimensional result for the

functions

we get by scanning sets in

To close this lecture we remark that the Brunn–Minkowski inequality has nu-

merous extensions and variations, not only in convex geometry, but in combina-
torics and information theory as well. One of the most surprising and delightful
is a theorem of Busemann [1949].

Theorem 5.5 (Busemann).

Let

be a symmetric convex body in

and

for each unit vector

let

(

)

be the volume of the slice of

by the subspace

orthogonal to

Then the body whose radius in each direction

(

)

is itself

convex

The Brunn–Minkowski inequality is the starting point for a highly developed
classical theory of convex geometry. We shall barely touch upon the theory in
these notes. A comprehensive reference is the recent book [Schneider 1993].

Lecture 6. Convolutions and Volume Ratios:

The Reverse Isoperimetric Problem

In the last lecture we saw how to deduce the classical isoperimetric inequality

from the Brunn–Minkowski inequality. In this lecture we will answer the

reverse question. This has to be phrased a bit carefully, since there is no upper
limit to the surface area of a body of given volume, even if we restrict attention
to convex bodies. (Consider a very thin pancake.) For this reason it is natural to
consider affine equivalence classes of convex bodies, and the question becomes:
given a convex body, how small can we make its surface area by applying an

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

affine (or linear) transformation that preserves volume? The answer is provided
by the following theorem from [Ball 1991].

Theorem 6.1.

Let

be a convex body and

a regular solid simplex in

Then there is an affine image of

whose volume is the same as that of

and

whose surface area is no larger than that of

Thus, modulo affine transformations, simplices have the largest surface area
among convex bodies of a given volume. If

is assumed to be centrally sym-

metric then the estimate can be strengthened: the cube is extremal among sym-
metric bodies. A detailed proof of Theorem 6.1 would be too long for these
notes. We shall instead describe how the symmetric case is proved, since this is
considerably easier but illustrates the most important ideas.

Theorem 6.1 and the symmetric analogue are both deduced from volume-ratio

estimates. In the latter case the statement is that among symmetric convex
bodies, the cube has largest volume ratio. Let’s see why this solves the reverse
isoperimetric problem. If

is any cube, the surface area and volume of

are

related by

vol(

∂Q

) = 2

vol(

)

(

−

We wish to show that any other convex body

has an affine image ˜

for which

vol(

∂

)

≤

vol( ˜

)

(

−

Choose ˜

so that its maximal volume ellipsoid is

, the Euclidean ball of

radius 1. The volume of ˜

is then at most 2

, since this is the volume of the

cube whose maximal ellipsoid is

. As in the previous lecture,

vol(

∂

) = lim

→

vol( ˜

εB

)

−

vol( ˜

)

Since ˜

⊃

, the second expression is at most

lim

→

vol( ˜

)

−

vol( ˜

)

= vol( ˜

) lim

→

(1 +

)

−

vol( ˜

) =

vol( ˜

)

vol( ˜

)

(

−

≤

vol( ˜

)

(

−

which is exactly what we wanted.

The rest of this lecture will thus be devoted to explaining the proof of the

volume-ratio estimate:

Theorem 6.2.

Among symmetric convex bodies the cube has largest volume

ratio

As one might expect, the proof of Theorem 6.2 makes use of John’s Theorem
from Lecture 3. The problem is to show that, if

is a convex body whose

maximal ellipsoid is

, then vol(

)

≤

. As we saw, it is a consequence of

KEITH BALL

John’s theorem that if

is the maximal ellipsoid in

, there is a sequence (

)

of unit vectors and a sequence (

) of positive numbers for which

⊗

and for which

⊂

{

x, u

i| ≤

1 for 1

≤

}

We shall show that this

has volume at most 2

. The principal tool will be a

sharp inequality for norms of generalised convolutions. Before stating this let’s
explain some standard terms from harmonic analysis.

and

→

are bounded, integrable functions, we define the

convolu-

tion

∗

and

∗

(

) =

(

)

(

−

)

dy.

Convolutions crop up in many areas of mathematics and physics, and a good
deal is known about how they behave. One of the most fundamental inequalities
for convolutions is Young’s inequality: If

∈

, and

= 1 +

then

∗

≤ k

(Here

k · k

means the

norm on

, and so on.) Once we have Young’s in-

equality, we can give a meaning to convolutions of functions that are not both
integrable and bounded, provided that they lie in the correct

spaces. Young’s

inequality holds for convolution on any locally compact group, for example the
circle. On compact groups it is sharp: there is equality for constant functions.
But on

, where constant functions are not integrable, the inequality can be

improved (for most values of

and

). It was shown by Beckner [1975] and

Brascamp and Lieb [1976a] that the correct constant in Young’s inequality is
attained if

and

are appropriate Gaussian densities: that is, for some positive

and

(

) =

−

and

(

) =

−

. (The appropriate choices of

and

the value of the best constant for each

and

will not be stated here. Later we

shall see that they can be avoided.)

How are convolutions related to convex bodies? To answer this question we

need to rewrite Young’s inequality slightly. If 1

= 1, the

norm

∗

can be realised as

(

∗

)(

)

(

)

for some function

with

= 1. So the inequality says that, if 1

2, then

Z Z

(

)

(

−

)

(

)

dy dx

≤ k

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

We may rewrite the inequality again with

(

−

) in place of

(

), since this

doesn’t affect

Z Z

(

)

(

−

)

(

−

)

dy dx

≤ k

(6.1)

This can be written in a more symmetric form via the map from

into

that

takes (

x, y

) to (

y, x

−

) =: (

u, v, w

). The range of this map is the subspace

{

(

u, v, w

) :

= 0

}

Apart from a factor coming from the Jacobian of this map, the integral can be
written

(

)

(

)

(

)

where the integral is with respect to two-dimensional measure on the subspace

. So Young’s inequality and its sharp forms estimate the integral of a product

function on

over a subspace. What is the simplest product function? If

and

are each the characteristic function of the interval [

−

1], the function

given by

(

u, v, w

) =

(

)

(

)

(

)

is the characteristic function of the cube [

−

⊂

. The integral of

over

a subspace of

is thus the area of a slice of the cube: the area of a certain

convex body. So there is some hope that we might use a convolution inequality
to estimate volumes.

Brascamp and Lieb proved rather more than the sharp form of Young’s in-

equality stated earlier. They considered not just two-dimensional subspaces of

but

-dimensional subspaces of

. It will be more convenient to state their

result using expressions analogous to those in (6.1) rather than using integrals
over subspaces. Notice that the integral

Z Z

(

)

(

−

)

(

−

)

dy dx

can be written

x, v

dx,

where

= (0

1),

= (1

−

1) and

= (

−

0) are vectors in

. The theorem

of Brascamp and Lieb is the following.

Theorem 6.3.

(

)

are vectors in

and

(

)

are positive numbers satis-

fying

and if

(

)

are nonnegative measurable functions on the line

then

(

x, v

)

KEITH BALL

is “maximised” when the

(

)

are appropriate Gaussian densities

(

) =

−

where the

depend upon

the

and the

The word maximised is in quotation marks since there are degenerate cases for
which the maximum is not attained. The value of the maximum is not easily
computed since the

are the solutions of nonlinear equations in the

and

. This apparently unpleasant problem evaporates in the context of convex

geometry: the inequality has a normalised form, introduced in [Ball 1990], which
fits perfectly with John’s Theorem.

Theorem 6.4.

(

)

are unit vectors in

and

(

)

are positive numbers

for which

⊗

and if

(

)

are nonnegative measurable functions

then

(

x, u

)

≤

Y Z

In this reformulation of the theorem, the

play the role of 1

: the Fritz John

condition ensures that

as required, and miraculously guarantees that

the correct constant in the inequality is 1 (as written). The functions

have

been replaced by

, since this ensures that equality occurs if the

are identical

Gaussian densities. It may be helpful to see why this is so. If

(

) =

−

for all

, then

(

x, u

)

= exp

−

x, u

−|

−

so the integral is

−

Y Z

−

Y Z

Armed with Theorem 6.4, let’s now prove Theorem 6.2.

Proof of the volume-ratio estimate.

Recall that our aim is to show that,

for

and

as usual, the body

{

x, u

i| ≤

1 for 1

≤

}

has volume at most 2

. For each

let

be the characteristic function of the

interval [

−

1] in

. Then the function

7→

x, u

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

is exactly the characteristic function of

. Integrating and applying Theorem

6.4 we have

vol(

)

≤

Y Z

= 2

The theorems of Brascamp and Lieb and Beckner have been greatly extended
over the last twenty years. The main result in Beckner’s paper solved the old
problem of determining the norm of the Fourier transform between

spaces.

There are many classical inequalities in harmonic analysis for which the best
constants are now known. The paper [Lieb 1990] contains some of the most
up-to-date discoveries and gives a survey of the history of these developments.

The methods described here have many other applications to convex geometry.

There is also a reverse form of the Brascamp–Lieb inequality appropriate for
analysing, for example, the ratio of the volume of a body to that of the minimal
ellipsoid containing it.

Lecture 7. The Central Limit Theorem

and Large Deviation Inequalities

The material in this short lecture is not really convex geometry, but is intended

to provide a context for what follows. For the sake of readers who may not be
familiar with probability theory, we also include a few words about independent
random variables.

To begin with, a

probability measure

on a set Ω is just a measure of total

mass

(Ω) = 1. Real-valued functions on Ω are called random variables and

the integral of such a function

: Ω

→

, its mean, is written

and called

the expectation of

. The variance of

(

−

)

. It is customary to

suppress the reference to Ω when writing the measures of sets defined by random
variables. Thus

(

{

∈

Ω :

(

)

}

)

is written

(

X <

1): the probability that

is less than 1.

Two crucial, and closely related, ideas distinguish probability theory from

general measure theory. The first is independence. Two random variables

and

are said to be independent if, for any functions

and

(

)

(

) =

(

)

(

)

Independence can always be viewed in a canonical way. Let (Ω

, µ

) be a product

space (Ω

Ω

, µ

⊗

), where

and

are probabilities. Suppose

and

are random variables on Ω for which the value

(

, ω

) depends only upon

while

(

, ω

) depends only upon

. Then any integral (that converges

appropriately)

(

)

(

) =

(

))

(

))

dµ

⊗

(

s, t

)

KEITH BALL

Ω

Figure 23.

Independence and product spaces

can be written as the product of integrals

(

))

dµ

(

)

(

))

dµ

(

) =

(

)

(

)

by Fubini’s Theorem. Putting it another way, on each line

{

(

, t

) :

∈

Ω

}

is fixed, while

exhibits its full range of behaviour in the correct proportions.

This is illustrated in Figure 23.

In a similar way, a sequence

, X

, . . . , X

of independent random variables

arises if each variable is defined on the product space Ω

Ω

. . .

Ω

and

depends only upon the

-th coordinate.

The second crucial idea, which we will not discuss in any depth, is the use

of many different

-fields on the same space. The simplest example has already

been touched upon. The product space Ω

Ω

carries two

-fields, much smaller

than the product field, which it inherits from Ω

and Ω

respectively. If

and

are the

-fields on Ω

and Ω

, the sets of the form

Ω

⊂

Ω

for

∈ F

form a

-field on Ω

Ω

; let’s call it ˜

. Similarly,

{

Ω

∈ F

}

“Typical” members of these

-fields are shown in Figure 24.

Figure 24.

Members of the “small”

-fields

and

Ω

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

One of the most beautiful and significant principles in mathematics is the cen-

tral limit theorem: any random quantity that arises as the sum of many small
independent contributions is distributed very much like a Gaussian random vari-
able. The most familiar example is coin tossing. We use a coin whose decoration
is a bit austere: it has +1 on one side and

−

1 on the other. Let

, ε

, . . . , ε

be the outcomes of

independent tosses. Thus the

are independent random

variables, each of which takes the values +1 and

−

1 each with probability

(Such random variables are said to have a

Bernoulli distribution

.) Then the

normalised sum

√

belongs to an interval

of the real line with probability very close to

√

−

dt.

The normalisation 1

√

, ensures that the variance of

is 1: so there is some

hope that the

will all be similarly distributed.

The standard proof of the central limit theorem shows that much more is true.

Any sum of the form

with real coefficients

will have a roughly Gaussian distribution as long as each

is fairly small compared with

. Some such smallness condition is clearly

needed since if

= 1

and

· · ·

= 0

the sum is just

, which is not much like a Gaussian. However, in many in-

stances, what one really wants is not that the sum is distributed like a Gaussian,
but merely that the sum cannot be large (or far from average) much more often
than an appropriate Gaussian variable. The example above clearly satisfies such
a condition:

never deviates from its mean, 0, by more than 1.

The following inequality provides a deviation estimate for any sequence of

coefficients. In keeping with the custom among functional analysts, I shall refer
to the inequality as Bernstein’s inequality. (It is not related to the Bernstein
inequality for polynomials on the circle.) However, probabilists know the result
as Hoeffding’s inequality, and the earliest reference known to me is [Hoeffding
1963]. A stronger and more general result goes by the name of the Azuma–
Hoeffding inequality; see [Williams 1991], for example.

Theorem 7.1 (Bernstein’s inequality).

, ε

, . . . , ε

are independent

Bernoulli random variables and if

, a

, . . . , a

satisfy

= 1,

then for each

positive

we have

Prob

> t

≤

−

KEITH BALL

This estimate compares well with the probability of finding a standard Gaussian
outside the interval [

−

t, t

√

∞

−

ds.

The method by which Bernstein’s inequality is proved has become an industry
standard.

Proof.

We start by showing that, for each real

≤

(7.1)

The idea will then be that

cannot be large too often, since, whenever it

is large, its exponential is enormous.

To prove (7.1), we write

λa

and use independence to deduce that this equals

λa

For each

the expectation is

λa

−

λa

= cosh

λa

Now, cosh

≤

for any real

, so, for each

λa

≤

Hence

≤

since

= 1.

To pass from (7.1) to a probability estimate, we use the inequality variously

known as Markov’s or Chebyshev’s inequality: if

is a nonnegative random

variable and

is positive, then

Prob(

≥

)

≤

(because the integral includes a bit where a function whose value is at least

integrated over a set of measure Prob(

≥

)).

Suppose

≥

0. Whenever

≥

, we will have

≥

. Hence

Prob

≥

≤

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

by (7.1). So

Prob

≥

≤

−

and in a similar way we get

Prob

≤ −

≤

−

Putting these inequalities together we get

Prob

≥

≤

−

In the next lecture we shall see that deviation estimates that look like Bernstein’s
inequality hold for a wide class of functions in several geometric settings. For
the moment let’s just remark that an estimate similar to Bernstein’s inequality,

Prob

≥

≤

−

(7.2)

holds for

= 1, if the

1 valued random variables

are replaced by in-

dependent random variables

each of which is uniformly distributed on the

interval

−

. This already has a more geometric flavour, since for these (

)

the vector (

, X

, . . . , X

) is distributed according to Lebesgue measure on

the cube

−

⊂

. If

= 1, then

is the distance of the point

(

, X

, . . . , X

) from the subspace of

orthogonal to (

, a

, . . . , a

). So (7.2)

says that most of the mass of the cube lies close to any subspace of

, which is

reminiscent of the situation for the Euclidean ball described in Lecture 1.

Lecture 8. Concentration of Measure in Geometry

The aim of this lecture is to describe geometric analogues of Bernstein’s de-

viation inequality. These geometric deviation estimates are closely related to
isoperimetric inequalities. The phenomenon of which they form a part was in-
troduced into the field by V. Milman: its development, especially by Milman
himself, led to a new, probabilistic, understanding of the structure of convex
bodies in high dimensions. The phenomenon was aptly named the concentration
of measure.

We explained in Lecture 5 how the Brunn–Minkowski inequality implies the

classical isoperimetric inequality in

: among bodies of a given volume, the

Euclidean balls have least surface area. There are many other situations where
isoperimetric inequalities are known; two of them will be described below. First
let’s recall that the argument from the Brunn–Minkowski inequality shows more
than the isoperimetric inequality.

Let

be a compact subset of

. For each point

, let

(

x, A

) be the

distance from

(

x, A

) = min

−

∈

}

KEITH BALL

Figure 25.

-neighbourhood.

For each positive

, the Minkowski sum

εB

is exactly the set of points

whose distance from

is at most

. Let’s denote such an

-neighbourhood

;

see Figure 25.

The Brunn–Minkowski inequality shows that, if

is an Euclidean ball of the

same volume as

, we have

vol(

)

≥

vol(

)

for any

ε >

This formulation of the isoperimetric inequality makes much clearer the fact that
it relates the measure and the metric on

. If we blow up a set in

using the

metric, we increase the measure by at least as much as we would for a ball.

This idea of comparing the volumes of a set and its neighbourhoods makes

sense in any space that has both a measure and a metric, regardless of whether
there is an analogue of Minkowski addition. For any metric space (Ω

, d

) equipped

with a Borel measure

, and any positive

and

, it makes sense to ask: For

which sets

of measure

do the blow-ups

have smallest measure? This

general isoperimetric problem has been solved in a variety of different situations.
We shall consider two closely related geometric examples. In each case the mea-
sure

will be a probability measure: as we shall see, in this case, isoperimetric

inequalities may have totally unexpected consequences.

In the first example, Ω will be the sphere

−

, equipped with either

the geodesic distance or, more simply, the Euclidean distance inherited from

as shown in Figure 26. (This is also called the

chordal metric;

it was used

in Lecture 2 when we discussed spherical caps of given radii.) The measure
will be

−

, the rotation-invariant probability on

−

. The solutions of

the isoperimetric problem on the sphere are known exactly: they are spherical
caps (Figure 26, right) or, equivalently, they are balls in the metric on

−

Thus, if a subset

of the sphere has the same measure as a cap of radius

, its

neighbourhood

has measure at least that of a cap of radius

This statement is a good deal more difficult to prove than the classical isoperi-

metric inequality on

: it was discovered by P. L´

evy, quite some time after the

isoperimetric inequality in

. At first sight, the statement looks innocuous

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

(

x, y

)

Figure 26.

The Euclidean metric on the sphere. A spherical cap (right) is a ball

for this metric.

enough (despite its difficulty): but it has a startling consequence.

Suppose

, so that

has the measure of a hemisphere

. Then, for each positive

the set

has measure at least that of the set

, illustrated in Figure 27. The

complement of

is a spherical cap that, as we saw in Lecture 2, has measure

about

−

nε

. Hence

(

)

≥

−

nε

, so almost the entire sphere lies within

distance

, even though there may be points rather far from

. The measure

and the metric on the sphere “don’t match”: the mass of

concentrates very

close to any set of measure

. This is clearly related to the situation described

in Lecture 1, in which we found most of the mass of the ball concentrated near
each hyperplane: but now the phenomenon occurs for any set of measure

The phenomenon just described becomes even more striking when reinter-

preted in terms of Lipschitz functions. Suppose

−

→

is a function on

the sphere that is 1-Lipschitz: that is, for any pair of points

and

on the

sphere,

(

)

−

(

)

| ≤ |

−

There is at least one number

, the median of

, for which both the sets (

≤

)

and (

≥

) have measure at least

. If a point

has distance at most

from

Figure 27.

-neighbourhood of a hemisphere.

KEITH BALL

(

≤

), then (since

is 1-Lipschitz)

(

)

≤

ε.

By the isoperimetric inequality all but a tiny fraction of the points on the sphere
have this property:

(

f > M

)

≤

−

nε

Similarly,

is larger than

−

on all but a fraction of the sphere. Putting

these statements together we get

(

−

> ε

)

≤

−

nε

So, although

may vary by as much as 2 between a point of the sphere and

its opposite, the function is nearly equal to

on almost the entire sphere:

practically constant.

In the case of the sphere we thus have the following pair of properties.

(i) If

⊂

Ω with

(

) =

then

(

)

≥

−

nε

(ii) If

: Ω

→

is 1-Lipschitz there is a number

for which

(

−

> ε

)

≤

−

nε

Each of these statements may be called an approximate isoperimetric inequal-
ity. We have seen how the second can be deduced from the first. The reverse
implication also holds (apart from the precise constants involved). (To see why,
apply the second property to the function given by

(

) =

(

x, A

).)

In many applications, exact solutions of the isoperimetric problem are not as

important as deviation estimates of the kind we are discussing. In some cases
where the exact solutions are known, the two properties above are a good deal
easier to prove than the solutions: and in a great many situations, an exact
isoperimetric inequality is not known, but the two properties are.

The for-

mal similarity between property 2 and Bernstein’s inequality of the last lecture
is readily apparent. There are ways to make this similarity much more than
merely formal: there are deviation inequalities that have implications for Lips-
chitz functions and imply Bernstein’s inequality, but we shall not go into them
here.

In our second example, the space Ω will be

equipped with the ordinary

Euclidean distance.

The measure will be the standard Gaussian probability

measure on

with density

(

) = (2

)

−

−|

The solutions of the isoperimetric problem in Gauss space were found by Borell
[1975]. They are half-spaces. So, in particular, if

⊂

and

(

) =

, then

(

) is at least as large as

(

), where

is the half-space

{

∈

≤

}

and so

{

≤

}

: see Figure 28.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

Figure 28.

-neighbourhood of a half-space.

The complement of

has measure

√

∞

−

≤

−

Hence,

(

)

≥

−

Since

does not appear in the exponent, this looks much weaker than the state-

ment for the sphere, but we shall see that the two are more or less equivalent.

Borell proved his inequality by using the isoperimetric inequality on the

sphere. A more direct proof of a deviation estimate like the one just derived
was found by Maurey and Pisier, and their argument gives a slightly stronger,
Sobolev-type inequality [Pisier 1989, Chapter 4]. We too shall aim directly for
a deviation estimate, but a little background to the proof may be useful.

There was an enormous growth in understanding of approximate isoperimetric

inequalities during the late 1980s, associated most especially with the name of
Talagrand. The reader whose interest has been piqued should certainly look at
Talagrand’s gorgeous articles [1988; 1991a], in which he describes an approach to
deviation inequalities in product spaces that involves astonishingly few structural
hypotheses. In a somewhat different vein (but prompted by his earlier work),
Talagrand [1991b] also found a general principle, strengthening the approximate
isoperimetric inequality in Gauss space. A simplification of this argument was
found by Maurey [1991]. The upshot is that a deviation inequality for Gauss
space can be proved with an extremely short argument that fits naturally into
these notes.

Theorem 8.1 (Approximate isoperimetric inequality for Gauss space).

Let

⊂

be measurable and let

be the standard Gaussian measure on

Then

(

x,A

)

dµ

≤

(

)

Consequently

(

) =

(

)

≥

−

KEITH BALL

Proof.

We shall deduce the first assertion directly from the Pr´

ekopa–Leindler

inequality (with

) of Lecture 5. To this end, define functions

, and

, as follows:

(

) =

(

x,A

)

(

)

(

) =

(

)

(

)

(

) =

(

)

where

is the Gaussian density. The assertion to be proved is that

(

x,A

)

dµ

(

)

≤

which translates directly into the inequality

≤

By the Pr´

ekopa–Leindler inequality it is enough to check that, for any

and

(

)

(

)

≤

It suffices to check this for

∈

, since otherwise

(

) = 0. But, in this case,

(

x, A

)

≤ |

−

. Hence

)

(

)

(

) =

(

x,A

)

−

≤

exp

−

= exp

−

exp

−

= (2

)

which is what we need.

To deduce the second assertion from the first, we use Markov’s inequality, very

much as in the proof of Bernstein’s inequality of the last lecture. If

(

) =

then

(

x,A

)

dµ

≤

The integral is at least

µ d

(

x, A

)

≥

µ d

(

x, A

)

≥

≤

−

and the assertion follows.

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

It was mentioned earlier that the Gaussian deviation estimate above is essentially
equivalent to the concentration of measure on

−

. This equivalence depends

upon the fact that the Gaussian measure in

is concentrated in a spherical shell

of thickness approximately 1, and radius approximately

√

. (Recall that the

Euclidean ball of volume 1 has radius approximately

√

.) This concentration is

easily checked by direct computation using integration in spherical polars: but
the inequality we just proved will do the job instead. There is an Euclidean
ball of some radius

whose Gaussian measure is

. According to the theorem

above, Gaussian measure concentrates near the boundary of this ball. It is not
hard to check that

is about

√

. This makes it quite easy to show that the

deviation estimate for Gaussian measure guarantees a deviation estimate on the
sphere of radius

√

with a decay rate of about

−

. If everything is scaled

down by a factor of

√

, onto the sphere of radius 1, we get a deviation estimate

that decays like

−

nε

and

now appears in the exponent. The details are left

to the reader.

The reader will probably have noticed that these estimates for Gauss space

and the sphere are not quite as strong as those advertised earlier, because in
each case the exponent is

. . . ε

. . .

instead of

. . . ε

. . .

. In some applications,

the sharper results are important, but for our purposes the difference will be
irrelevant. It was pointed out to me by Talagrand that one can get as close
as one wishes to the correct exponent

. . . ε

. . .

by using the Pr´

ekopa–Leindler

inequality with

close to 1 instead of

and applying it to slightly different

and

For the purposes of the next lecture we shall assume an estimate of

−

even though we proved a weaker estimate.

Lecture 9. Dvoretzky’s Theorem

Although this is the ninth lecture, its subject, Dvoretzky’s Theorem, was re-

ally the start of the modern theory of convex geometry in high dimensions. The
phrase “Dvoretzky’s Theorem” has become a generic term for statements to the
effect that high-dimensional bodies have almost ellipsoidal slices. Dvoretzky’s
original proof shows that any symmetric convex body in

has almost ellip-

soidal sections of dimension about

√

log

. A few years after the publication

of Dvoretzky’s work, Milman [Milman 1971] found a very different proof, based
upon the concentration of measure, which gave slices of dimension log

. As we

saw in Lecture 2 this is the best one can get in general. Milman’s argument gives
the following.

Theorem 9.1.

There is a positive number

such that

for every

ε >

and

every natural number

every symmetric convex body of dimension

has a slice

of dimension

≥

cε

log(1 +

−

)

log

KEITH BALL

that is within distance

1 +

of the

-dimensional Euclidean ball

There have been many other proofs of similar results over the years. A par-
ticularly elegant one [Gordon 1985] gives the estimate

≥

cε

log

(removing

the logarithmic factor in

−

), and this estimate is essentially best possible. We

chose to describe Milman’s proof because it is conceptually easier to motivate
and because the concentration of measure has many other uses. A few years ago,
Schechtman found a way to eliminate the log factor within this approach, but
we shall not introduce this subtlety here. We shall also not make any effort to
be precise about the dependence upon

With the material of Lecture 8 at our disposal, the plan of proof of Theorem

9.1 is easy to describe. We start with a symmetric convex body and we consider
a linear image

whose maximal volume ellipsoid is the Euclidean ball. For this

we will try to find almost spherical sections, rather than merely ellipsoidal

ones. Let

k · k

be the norm on

whose unit ball is

. We are looking for a

-dimensional space

with the property that the function

7→ k

is almost constant on the Euclidean sphere of

∩

−

. Since

contains

, we have

k ≤ |

for all

∈

, so for any

and

−

k − k

k| ≤ k

−

k ≤ |

−

Thus

k · k

is a Lipschitz function on the sphere in

, (indeed on all of

). (We

used the same idea in Lecture 4.) From Lecture 8 we conclude that the value of

is almost constant on a very large proportion of

−

: it is almost equal to

its average

−

dσ,

on most of

−

We now choose our

-dimensional subspace at random. (The exact way to do

this will be described below.) We can view this as a random embedding

→

For any particular unit vector

∈

, there is a very high probability that its

image

T ψ

will have norm

T ψ

close to

. This means that even if we select

quite a number of vectors

, ψ

, . . . , ψ

−

we can guarantee that there

will be some choice of

for which

all

the norms

T ψ

will be close to

. We

will thus have managed to pin down the radius of our slice in many different
directions. If we are careful to distribute these directions well over the sphere in

, we may hope that the radius will be almost constant on the entire sphere.

For these purposes, “well distributed” will mean that all points of the sphere in

are close to one of our chosen directions. As in Lecture 2 we say that a set

{

, ψ

, . . . , ψ

}

−

is a

-net

for the sphere if every point of

−

is within

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

(Euclidean) distance

of at least one

. The arguments in Lecture 2 show that

−

has a

-net with no more than

elements. The following lemma states that, indeed, pinning down the norm on
a very fine net, pins it down everywhere.

Lemma 9.2.

Let

k · k

be a norm on

and suppose that for each point

some

-net on

−

we have

−

)

≤ k

k ≤

(1 +

)

for some

γ >

Then

for

every

∈

−

)

−

≤ k

k ≤

(1 +

)

−

Proof.

Clearly the value of

plays no real role here so assume it is 1. We

start with the upper bound. Let

be the maximum possible ratio

for

nonzero

and let

be a point of

−

with

. Choose

in the

-net

with

−

| ≤

. Then

−

k ≤

−

| ≤

Cδ

, so

k ≤ k

−

k ≤

(1 +

) +

Cδ.

Hence

≤

(1 +

)

−

To get the lower bound, pick some

in the sphere and some

in the

-net

with

−

| ≤

. Then

−

)

≤ k

k ≤ k

−

k ≤ k

(1 +

)

−

| ≤ k

(1 +

)

−

Hence

k ≥

−

(1 +

)

−

)

−

According to the lemma, our approach will give us a slice that is within distance

1 +

−

of the Euclidean ball (provided we satisfy the hypotheses), and this distance can
be made as close as we wish to 1 if

and

are small enough.

We are now in a position to prove the basic estimate.

Theorem 9.3.

Let

be a symmetric convex body in

whose ellipsoid of

maximal volume is

and put

−

dσ

KEITH BALL

as above

Then

has almost spherical slices whose dimension is of the order of

Proof.

Choose

and

small enough to give the desired accuracy, in accordance

with the lemma.

Since the function

7→ k

is Lipschitz (with constant 1) on

−

, we know

from Lecture 8 that, for any

≥

k −

> t

≤

−

In particular,

k −

> M γ

≤

−

)

≤ k

k ≤

(1 +

)

on all but a proportion 2

−

of the sphere.

Let

be a

-net on the sphere in

with at most (4

/δ

)

elements. Choose

a random embedding of

: more precisely, fix a particular copy of

and consider its images under orthogonal transformations

a random subspace with respect to the invariant probability on the group of
orthogonal transformations. For each fixed

in the sphere of

, its images

U ψ

, are uniformly distributed on the sphere in

. So for each

∈ A

, the

inequality

−

)

≤ k

U ψ

k ≤

(1 +

)

holds for

outside a set of measure at most 2

−

. So there will be at

least one

for which this inequality holds for

all

, as long as the sum of

the probabilities of the bad sets is at most 1. This is guaranteed if

−

This inequality is satisfied by

of the order of

2 log(4

/δ

)

Theorem 9.3 guarantees the existence of spherical slices of

of large dimension,

provided the average

−

dσ

is not too small. Notice that we certainly have

≤

1 since

k ≤ |

for all

In order to get Theorem 9.1 from Theorem 9.3 we need to get a lower estimate
for

of the order of

√

log

√

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

This is where we must use the fact that

is the

maximal

volume ellipsoid in

. We saw in Lecture 3 that in this situation

⊂

√

, so

k ≥ |

√

for

all

, and hence

≥

√

But this estimate is useless, since it would not give slices of dimension bigger
than 1. It is vital that we use the more detailed information provided by John’s
Theorem.

Before we explain how this works, let’s look at our favourite examples. For

specific norms it is usually much easier to compute the mean

by writing it as

an integral with respect to Gaussian measure on

. As in Lecture 8 let

the standard Gaussian measure on

, with density

)

−

−|

By using polar coordinates we can write

−

dσ

Γ(

√

2Γ((

+ 1)

dµ

(

)

√

dµ

(

)

The simplest norm for which to calculate is the

norm. Since the body we

consider is supposed to have

as its maximal ellipsoid we must use

√

, for

which the corresponding norm is

√

Since the integral of this sum is just

times the integral of any one coordinate

it is easy to check that

√

dµ

(

) =

So for the scaled copies of

, we have

bounded below by a fixed number, and

Theorem 9.3 guarantees almost spherical sections of dimension proportional to

. This was first proved, using exactly the method described here, in [Figiel et al.

1977], which had a tremendous influence on subsequent developments. Notice
that this result and Kaˇsin’s Theorem from Lecture 4 are very much in the same
spirit, but neither implies the other. The method used here does not achieve
dimensions as high as

2 even if we are prepared to allow quite a large distance

from the Euclidean ball. On the other hand, the volume-ratio argument does
not give sections that are very close to Euclidean: the volume ratio is the closest
one gets this way. Some time after Kaˇsin’s article appeared, the gap between
these results was completely filled by Garnaev and Gluskin [Garnaev and Gluskin
1984]. An analogous gap in the general setting of Theorem 9.1, namely that the
existing proofs could not give a dimension larger than some fixed multiple of
log

, was recently filled by Milman and Schechtman.

KEITH BALL

What about the cube? This body

has

as its maximal ellipsoid, so our job

is to estimate

√

max

dµ

(

)

At first sight this looks much more complicated than the calculation for

since we cannot simplify the integral of a maximum. But, instead of estimating
the mean of the function max

, we can estimate its median (and from Lecture

8 we know that they are not far apart). So let

be the number for which

(max

| ≤

) =

(max

| ≥

) =

From the second identity we get

√

max

dµ

(

)

≥

√

We estimate

from the first identity. It says that the cube [

−

R, R

]

has Gauss-

ian measure

. But the cube is a “product” so

([

−

R, R

]

) =

√

−

In order for this to be equal to

we need the expression

√

−

to be about 1

−

(log 2)

/n.

Since the expression approaches 1 roughly like

−

we get an estimate for

of the order of

√

log

. From Theorem 9.3 we then

recover the simple result of Lecture 2 that the cube has almost spherical sections
of dimension about log

There are many other bodies and classes of bodies for which

can be ef-

ficiently estimated. For example, the correct order of the largest dimension of
Euclidean slice of the

balls, was also computed in the paper [Figiel et al. 1977]

mentioned earlier.

We would like to know that for a

general

body with maximal ellipsoid

have

dµ

(

)

≥

(constant)

log

(9.1)

just as we do for the cube. The usual proof of this goes via the Dvoretzky–Rogers
Lemma, which can be proved using John’s Theorem. This is done for example in
[Pisier 1989]. Roughly speaking, the Dvoretzky–Rogers Lemma builds something
like a cube around

, at least in a subspace of dimension about

, to which we

then apply the result for the cube. However, I cannot resist mentioning that the
methods of Lecture 6, involving sharp convolution inequalities, can be used to

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

show that among all symmetric bodies

with maximal ellipsoid

the cube

is precisely the one for which the integral in (9.1) is smallest. This is proved by
showing that for each

, the Gaussian measure of

is at most that of [

−

r, r

]

The details are left to the reader.

This last lecture has described work that dates back to the seventies. Although

some of the material in earlier lectures is more recent (and some is much older),
I have really only scratched the surface of what has been done in the last twenty
years. The book of Pisier to which I have referred several times gives a more
comprehensive account of many of the developments. I hope that readers of
these notes may feel motivated to discover more.

Acknowledgements

I would like to thank Silvio Levy for his help in the preparation of these notes,

and one of the workshop participants, John Mount, for proofreading the notes
and suggesting several improvements. Finally, a very big thank you to my wife
Sachiko Kusukawa for her great patience and constant love.

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KEITH BALL

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Index

affine transformation, 8, 13, 32
almost, see approximate
AM/GM inequality, 17, 19, 29, 31
approximate

ball, 8, 20

intersection, 20
section, 8, 9, 19, 24, 49

ellipsoid, see approximate ball
isoperimetric inequality, 44, 45

area, see also volume
arithmetic mean, see AM/GM inequality
average radius, 7
Azuma, Kazuoki, 39
Azuma–Hoeffding inequality, 39

, 3

, 4

ball, see also approximate ball

and cube, 2, 8, 9, 19
and octahedron, 3
and simplex, 2
central section, 6
circumscribed, 2, 3, 8
distribution of volume, 6
Euclidean, 4
for arbitrary norm, 8
inscribed, 2, 3, 8
near polytope, 9
section, 6
similarities to convex body, 2
volume, 4

Ball, Keith M., 25, 33, 36
Beckner, William, 34, 37
Bernoulli distribution, 39
Bernstein’s inequality, 39, 41, 44, 46
Bernstein, Sergei N., 39
Bollob´

as, B´

ela, 1

bootstrapping of probabilities, 24
Borell, Christer, 44
bound, see inequality
Bourgain, Jean, 25
Brascamp, Herm Jan, 30, 34, 35, 37
Brascamp–Lieb reverse inequality, 37
Brunn’s Theorem, 26, 28, 29
Brunn, Hermann, 25, 26
Brunn–Minkowski

inequality, 25, 28, 32, 41

functional, 29
multiplicative, 29

theory, 2

Brøndsted, Arne, 2
Busemann’s Theorem, 32
Busemann, Herbert, 32

cap, 10, 22

radius, 11
volume, 10, 11

Cauchy–Schwarz inequality, 24
central

limit theorem, 37, 39
section, 6

centrally symmetric, see symmetric
characteristic function, 29
Chebyshev’s inequality, 40
chordal metric, 42
circumscribed ball, 2, 8
classical convex geometry, 2
coin toss, 39
combinatorial

theory of polytopes, 2

combinatorics, 32
concave function, 25, 27, 29
concentration, see also distribution

of measure, 7, 41, 47

KEITH BALL

cone, 3, 26

spherical, 12
volume, 3

contact point, 13
convex

body

definition, 2
ellipsoids inside, 13
similarities to ball, 2
symmetric, 8

hull, 2, 3

convolution

generalised, 34
inequalities, 52

cotype-2 property, 25
cross-polytope, 3, 8

and ball, 13
volume, 3

ratio, 19, 21

cube, 2, 7, 33, 52, 53

and ball, 8, 9, 13, 19
distribution of volume, 41
maximal ellipsoid, 15
sections of, 8
volume, 7

distribution, 7
ratio, 19, 21

deck of cards, 25
density, see distribution
determinant maximization, 19
deviation estimate, 24, 39

geometric, 41
in Gauss space, 45

differential equations, 2
dimension-independent constant, 6, 20,

25, 47

distance

between convex bodies, 8

distribution, see also concentration, see

also volume distribution

Gaussian, 6, 12
normal, 6
of volume, 12

duality, 17, 19
Dvoretzky’s Theorem, 47
Dvoretzky, Aryeh, 47, 52
Dvoretzky–Rogers Lemma, 52

ellipsoid, 13

maximal, see maximal ellipsoid

minimal, 37
polar, 17

Euclidean

metric, 11

on sphere, 42

norm, 2, 14

expectation, 37

Figiel, Tadeusz, 51, 52
Fourier transform, 37
Fubini’s Theorem, 31, 38
functional, 2, 18

analysis, 2
Brunn–Minkowski inequality, 29

Garnaev, Andre˘ı, 51
Gaussian

distribution, 6, 12, 34, 36, 39, 44, 46, 47,

51, 53

measure, see Gaussian distribution

generalized convolution, 34
geometric

mean, see AM/GM inequality

geometry

of numbers, 2

Gluskin, E. D., 51
Gordon, Yehoram, 48

H¨

older inequality, 30

Hadamard matrix, 24
Hahn–Banach separation theorem, 2
harmonic

analysis, 34, 37

Hoeffding, Wassily, 39
homogeneity, 30

identity

resolution of, 14, 19

independence

from

dimension,

see

dimension-independent constant

independent variables, 37
inequality

AM/GM, 17, 19, 29, 31
Bernstein’s, 39, 41, 44, 46
Brunn–Minkowski, 25, 28, 32, 41

functional, 29
multiplicative, 29

Cauchy–Schwarz, 24
Chebyshev’s, 40
convolution, 52
deviation, see deviation estimate
for cap volume, 11
for norms of convolutions, 34

AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY

H¨

older, 30

isoperimetric, 28, 32, 41, 44, 45
Markov’s, 40, 46
Pr´

ekopa–Leindler, 30, 46, 47

Young’s, 34

inertia tensor, 14
information theory, 2, 32
inscribed ball, 2, 8
intersection

of convex bodies, 20

invariant measure on

(

), 22, 50

isoperimetric

inequality, 28, 32, 41

approximate, 44, 45
in Gauss space, 45
on sphere, 42

problem

in Gauss space, 44

John’s Theorem, 13, 33, 36, 51, 52

extensions, 19
proof of, 16

John, Fritz, 13
Johnson, William B., 25

Kaˇ

sin’s Theorem, 20, 51

Kaˇ

sin, B. S., 20

Kusukawa, Sachiko, 53

norm, 34

norm, 3, 8, 24, 51

L´

evy, Paul, 42

large deviation inequalities, see deviation

estimate

Leindler, L´

aszl´

o, 30

Levy, Silvio, 53
Lieb, Elliott H., 30, 34, 35, 37
Lindenstrauss, Joram, 1, 25, 51, 52
linear

programming, 2

Lipschitz function, 43, 50
Lyusternik, Lazar A., 28

Markov’s inequality, 40, 46
mass, see also volume

distribution on contact points, 14

Maurey, Bernard, 45
maximal

ellipsoid, 13, 21, 34, 48, 49, 51, 53

for cube, 15

mean, see AM/GM inequality
metric

and measure, 42
on space of convex bodies, 8
on sphere, 11

Milman, Vitali D., 25, 41, 47, 51, 52
minimal ellipsoid, 19, 37
Minkowski

addition, 28, 42

Minkowski, Hermann, 27
Mount, John, 53
MSRI, 1
multiplicative

Brunn–Minkowski inequality, 29

(

), 22

net, 11, 48
nonsymmetric convex body, 13, 15
norm

, 3, 8, 24, 51

and radius, 8
Euclidean, 2, 14, 21
of convolution, 34
with given ball, 8, 21, 48

normal distribution, 6
normalisation of integral, 5, 39

octahedron, see cross-polytope
omissions, 2
orthant, 3
orthogonal

group, 22, 50
projection, 14

orthonormal basis, 13, 14

partial differential equations, 2
Pisier, Gilles, 20, 45, 52
polar

coordinates, 4, 7, 47, 51
ellipsoid, 17

polytope, 8

as section of cube, 9
combinatorial theory of –s, 2
near ball, 9

Pr´

ekopa, Andr´

as, 30

Pr´

ekopa–Leindler inequality, 30, 46, 47

probability

bootstrapping of, 24
measure, 37, 42
that. . . , 37
theory, 2, 24, 37

projection

orthogonal, 14

QS-Theorem, 25

KEITH BALL

quotient of a subspace, 25

(

), 7

radius

and norm, 8
and volume for ball, 5
average, 7
in a direction, 7, 32
of body in a direction, 21

Ramsey theory, 24
random

subspace, 48, 50
variable, 37

ratio

between volumes, see volume ratio

resolution of the identity, 14, 19
reverse

Brascamp–Lieb inequality, 37

rigidity condition, 16
Rogers, Claude Ambrose, 9, 52

−

, 4

Schechtman, Gideon, 1, 25, 48, 51
Schneider, Rolf, 2, 32
section

ellipsoidal, see approximate ball
length of, 25
of ball, 6
of cube, 8, 9
parallel, 27
spherical, see approximate ball

sections

almost spherical, 50

1-separated set, 12
separation

theorem, 18

shaking down a set, 25

, 5

-field, 38

simplex, 15, 33

regular, 2

slice, see section
sphere, see also ball

cover by caps, 10
measure on, 5
volume, 5

spherical

cap, see cap
cone, 12
coordinates, see polar coordinates
section, see approximate ball

Stirling’s formula, 5, 6
supporting hyperplane, 2, 16
symmetric body, 8, 15, 25, 33, 47, 49
Szarek, Stanis law Jerzy, 20, 21

Talagrand, Michel, 25, 45
tensor product, 14
Tomczak-Jaegermann, Nicole, 19

vol, 2
volume

distribution

of ball, 6
of cube, 7, 41

of ball, 4
of cone, 3
of cross-polytope, 3
of ellipsoid, 13
of symmetric body, 8
ratio, 19, 21, 25, 33, 36, 51

vr, 21

Walsh matrix, 24
weighted average, 14, 27
Williams, David, 39

Young’s inequality, 34

Keith Ball

Department of Mathematics

University College

University of London

London

United Kingdom

kmb@math.ucl.ac.uk