The Intrinsic Spread of a Configuration in R d

How "spread out" is a finite set of points in general position in real affine d-space? On the line, a natural measure would be the ratio between the greatest distance and the smallest; this is invariant under affine transformations, so it depends only on the "affine shape" of the configuration. If we use the same definition in higher dimensions, the property of being invariant under affine transformations is lost: we can stretch in one direction but not in another, and the ratio will change. The same thing happens if we use the maximum ratio of distances from points to hyperplanes spanned by other points. Thus it seems most natural to use the following definition, which is affinely invariant and also generalizes the "correct" definition in the one-dimensional case:

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