Brascamp-Lieb inequality and quantitative versions of Helly's theorem

TL;DR

This paper develops new quantitative versions of Helly's theorem using advanced inequalities and decompositions, providing bounds on intersections of half-spaces with applications to convex geometry.

Contribution

It introduces novel quantitative bounds for Helly's theorem by combining Brascamp-Lieb inequalities, approximate John's decompositions, and reverse isoperimetric techniques.

Findings

Existence of small subfamilies with controlled intersection volume

Improved bounds compared to previous results

Application of Brascamp-Lieb inequality in convex geometry

Abstract

We provide a number of new quantitative versions of Helly's theorem. For example, we show that for every family $\{P_i:i\in I\}$ of closed half-spaces $$P_i=\{x\in {\mathbb R}^n:\langle x,w_i\rangle \leq 1\}$$ in ${\mathbb R}^n$ such that $P=\bigcap_{i\in I}P_i$ has positive volume, there exist $s\leq \alpha n$ and $i_1,\ldots, i_s\in I$ such that $$|P_{i_1}\cap\cdots\cap P_{i_s}|\leq (Cn)^n\,|P|,$$ where $\alpha, C>0$ are absolute constants. These results complement and improve previous work of B\'ar\'any-Katchalski-Pach and Nasz\'odi. Our method combines the work of Srivastava on approximate John's decompositions with few vectors, a new estimate on the corresponding constant in the Brascamp-Lieb inequality and an appropriate variant of Ball's proof of the reverse isoperimetric inequality.