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

## Ben Lund, Rutgers University

## 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.