色单纯深度猜想

摘要：

给定点或颜色的$d+1$集合，$S_1,\ldots,S_{d+1}$在$\mathbb R^d$中，彩色单形是一个集合$T\subseteq\bigcup_{i=1}^{d+1}S_i$，使得对于所有$i\in\{1,\ldots,d+1\}$，$|T\cap S_i|\leq 1$。彩色Carathéodory定理指出，如果$\mathbf 0$在每个$S_i$的凸包中，则存在一个彩色单形$T$，其凸包中包含$\mathbf 0$。Deza，Huang，Stephen，and Terlaky（Colourful simplicial depth，Discrete Comput. Geom.，35，597- 604（2006））证明，当对所有$i\in\{1,\ldots,d+1\}$求$|S_i|=d+1$时，总是至少有$d^2+1$个在其凸包中包含$\mathbf 0$的彩色单形。我们证明了这个猜想通过组合的方法。

英文摘要：

Given $d+1$ sets of points, or colours, $S_1,\ldots,S_{d+1}$ in $\mathbb R^d$, a colourful simplex is a set $T\subseteq\bigcup_{i=1}^{d+1}S_i$ such that $|T\cap S_i|\leq 1$, for all $i\in\{1,\ldots,d+1\}$. The colourful Carathéodory theorem states that, if $\mathbf 0$ is in the convex hull of each $S_i$, then there exists a colourful simplex $T$ containing $\mathbf 0$ in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597--604 (2006)) conjectured that, when $|S_i|=d+1$ for all $i\in\{1,\ldots,d+1\}$, there are always at least $d^2+1$ colourful simplices containing $\mathbf 0$ in their convex hulls. We prove this conjecture via a combinatorial approach.

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