Quantitative Helly-Type Theorem for the Diameter of Convex Sets

Abstract

We provide a new quantitative version of Helly’s theorem: there exists an absolute constant $\alpha >1$ with the following property. If $\{P_i: i\in I\}$ is a finite family of convex bodies in ${\mathbb {R}}^n$ with ${\mathrm{int}} (\bigcap _{i\in I}P_i )\ne \emptyset$, then there exist $z\in {\mathbb {R}}^n$, $s\leqslant \alpha n$ and $i_1,\ldots i_s\in I$ such that

$$\begin{aligned} z+P_{i_1}\cap \cdots \cap P_{i_s}\subseteq cn^{3/2}\Big (z+\bigcap _{i\in I}P_i\Big ), \end{aligned}$$

where $c>0$ is an absolute constant. This directly gives a version of the “quantitative” diameter theorem of Bárány, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound $O(n^{3/2})$ can be improved to $O(\sqrt{n})$.