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

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 $s\leq αn$ and $i_1,\ldots, i_s\in I$ such that $|P_{i_1}\cap\cdots\cap P_{i_s}|\leq (Cn)^n\,|P|,$ where $α, C>0$ 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.

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