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diameter of an empty set
>why diameter of an empty set is minus infinity
diam A = sup{ d(x,y) | x,y in A } is the supremum or least upper
bound of all the distances between points of A. Thus the range
of A is the real numbers, or to be more precise, non-negative
real numbers. Now you look at the definition of supremum.
First it's an upper bound for the nulset. Now every real, or more
precisely every non-negative real is an upper bound. Second it's
the least or smallest upper bound, which in the case of the reals
would be -oo, or in the case of the non-negative reals, would be 0.
Thus as I and others prefer
. . diam nulset = 0
which is intuitively in accordance with the actual range of diam
and not as some others opine
. . diam nulset = -oo
which clashes with the fact that there are no negative diameters.
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