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

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 $$P_i=\{x\in {\mathbb{R}}^n:⟨x,w_i⟩\le 1\}$$ in ${\mathbb{R}}^n$ such that $P={\bigcap }_{i\in I}{P}_i$ has positive volume, there exist $s\le \alpha n$ and $i_1,\dots ,i_s\in I$ such that $$|P_{i_1}\cap \cdots \cap P_{i_s}|\le (Cn{)}^n|P|,$$ where $\alpha ,C>0$ are absolute constants. These results complement and improve previous work of Bárány-Katchalski-Pach and Naszódi. 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.