The number of $k$-flats spanned by a set of points

February 23, 2016

In this talk, I will answer a question of Purdy on conditions on a finite point set in $\mathbb{R}^d$ that ensure that it spans more hyperplanes than $(d-2)$-dimensional affine subspaces. In answering this question, I introduce a new measure of the degeneracy of a point set with respect to affine subspaces, and give an asymptotic expression for the number of $k$-dimensional affine subspaces spanned by a point set, for each $0\lt k \lt d$. I will also give a counterexample to an earlier conjecture of Purdy on the question.