Essential dimension and the flats spanned by a point set

Abstract

Let $P$ be a finite set of points in ${\mathbb{R}}^d$ or ${\mathbb{C}}^d$. We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by $P$ is at least the number of $(d-2)$-flats spanned by $P$. In answering this question, we define a new measure of the degeneracy of a point set with respect to affine subspaces, termed the "essential dimension". We use the essential dimension to give an asymptotic expression for the number of $k$-flats spanned by $P$, for $1\le k\le d-1$.

23 pages

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