AI 中文总结
本文针对不同体积子集问题及其对偶版本，构造点集改进了H_{a,d}(n)的上界，还改进了D_d(n)的上界并给出D_2(n)、D_3(n)的首个非平凡下界。
AI 中文摘要
对于满足2≤a≤d+1的参数，最大整数H_{a,d}(n)是多少？该整数满足：R^d中任意n个点构成的集合，若不存在a个点位于同一个(a-2)-flat上，则必包含一个大小为H_{a,d}(n)的子集，该子集所确定的(a-1)-维单形具有两两不同的(a-1)-维体积。我们针对a和d的若干种情形，构造了n个点的集合，改进了H_{a,d}(n)的已知最佳上界。我们还研究了该问题的对偶版本：设D_d(n)为最大数，使得对于R^d中任意处于一般位置的n个超平面排列，总能找到一个大小为D_d(n)的超平面子集，该子集所确定的所有d-维单形具有两两不同的d-维体积。我们改进了D_d(n)的当前已知上界，并给出了D_2(n)和D_3(n)的首个非平凡下界。
英文摘要
For $2 \leq a \leq d+1$, what is the largest integer $H_{a,d} (n)$ such that every set of $n$ points in $\mathbb{R}^d$ with no $a$ points on a common $(a-2)$-flat contains a subset of $H_{a,d} (n)$ points whose determined $(a-1)$-dimensional simplices have pairwise distinct $(a-1)$-dimensional volumes? We construct $n$-point sets that improve the best known upper bounds for $H_{a,d}(n)$ in several cases of $a$ and $d$. We also study a dual version of the problem. Let $D_d(n)$ the maximum number such that for any arrangement of $n$ hyperplanes in general position in $\mathbb{R}^d$, we can always find a subset of $D_d(n)$ hyperplanes for which all the $d$-dimensional simplices that they define have distinct $d$-dimensional volumes. We improve the current known upper bound for $D_d(n)$ and give the first nontrivial lower bound for $D_2(n)$ and $D_3 (n)$.
Comments9 pages
