凸体交线体积的彩色Helly型定理
Colorful Helly-type Theorems for the Volume of Intersections of Convex Bodies
摘要：
我们证明了下面的Helly-type结果。设$\mathcal{C}_1,\dots,\mathcal{C}_{3d}$是$\mathbb{R}^d$中凸体的有限族.假设对于$2d$集合的任何彩色选择，$C_{i_k}\in \mathcal{C}_{i_k}$对于每个$1\leq k\leq 2d$与$1\leq i_1<\dots<i_{2d}\leq 3d$，交集$\bigcap\limits_{k=1}^{2d} C_{i_k}$的体积至少为1。然后，存在$1\leq i \leq 3d$，使得$\bigcap\limits_{C\in \mathcal{C}_i} C$的体积至少为$d^{-O(d^2)}$。
英文摘要：
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 $1\leq k\leq 2d$ with $1\leq i_1<\dots<i_{2d}\leq 3d$, the intersection $\bigcap\limits_{k=1}^{2d} C_{i_k}$ is of volume at least 1. Then there is an $1\leq i \leq 3d$ such that $\bigcap\limits_{C\in \mathcal{C}_i} C$ is of volume at least $d^{-O(d^2)}$.
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