Speaker: Ron Holzman
Title: On convex holes in $d$-dimensional point sets
Abstract: Given a finite set $P \subseteq \mathbb{R}^d$, points
$a_1,a_2,\dotsc,a_{\ell} \in P$ form an $\ell$-hole in $P$ if they are
the vertices of a convex polytope which contains no points of $P$ in its
interior. We construct arbitrarily large point sets in general position
in $\mathbb{R}^d$ having no holes of size $2^{7d}$ or more. This
improves the previously known upper bound of order $d^{d+o(d)}$ due to
Valtr. Our construction uses a certain type of designs, originating from
numerical analysis, known as ordered orthogonal arrays. The bound may be
further improved, to roughly $4^d$, via an approximate version of such
designs. Joint work with Boris Bukh and Ting-Wei Chao.