Colorful Helly-type Theorems for the Volume of Intersections of Convex Bodies

We prove the following Helly-type result. Let ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_{3d}$ be finite families of convex bodies in ${\mathbb{R}}^d$. Assume that for any colorful selection of $2d$ sets, $C_{i_k}\in {\mathcal{C}}_{i_k}$ for each $1\le k\le 2d$ with $1\leq i_1<\dots<i_{2d}\leq 3d$,theintersection$\bigcap\limits_{k=1}^{2d} C_{i_k}$isofvolumeatleast1.Thenthereisan$1\leq i \leq 3d$suchthat$\bigcap\limits_{C\in $\mathcal{C}$_i} C$isofvolumeatleast$d^{-O(d^2)}$.