The Intrinsic Spread of a Configuration in R d
Jacob Eli Goodman, Richard M. Pollack, Bernd Sturmfels · Journal of the American Mathematical Society · 1990
How spread out is a finite set of points in general position in real d-space? On the line, a natural measure would be the ratio between the greatest distance and the smallest; this is invariant under 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 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: