Essential Dimension and the Flats Spanned by a Point Set


Let P be a finite set of points in ℝd or ℂd.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≤kd−1.

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Correspondence to Ben Lund.

Additional information

Work on this paper was supported by NSF grant CCF-1350572 and by ERC Grant 267165 DISCONV.

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Lund, B. Essential Dimension and the Flats Spanned by a Point Set. Combinatorica 38, 1149–1174 (2018).

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Mathematics Subject Classification (2000)

  • 52C10