Essential dimension and the flats spanned by a point set

TL;DR

This paper introduces the concept of essential dimension to analyze the structure of point sets in Euclidean and complex spaces, providing bounds on the number of flats spanned and addressing a question by Purdy.

Contribution

It defines the new measure of essential dimension and uses it to derive asymptotic formulas for the number of flats spanned by a point set, answering Purdy's question.

Findings

Established a lower bound on the number of hyperplanes versus (d-2)-flats.

Introduced the essential dimension as a measure of degeneracy.

Provided asymptotic estimates for the number of k-flats spanned.

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 $1 \leq k \leq d-1$.