0/1多面体的几乎阶乘多个面
Almost factorial many facets for 0/1-polytopes
浏览论文内容
中文总结 AI 辅助
针对福卡达与齐格勒提出的0/1多面体最大面数渐近行为问题，本文通过超单纯形与排列多面体的组合学，构造出n≥10时面数至少为(n−⌈2log₂(n)⌉−1)!的0/1多面体，大幅改进了下界并确定了log g(n)的渐近误差范围。
中文摘要 AI 辅助
福卡达（Fukuda，1995）与齐格勒（Ziegler，2000）提出了一个长期存在的问题：n维0/1多面体最多可拥有的面数g(n)的渐近行为。巴拉尼（Bárány）与波尔（Pór，2001）通过概率方法得出了一个重要结果，证明g(n)至少随n呈超指数增长。在本文中，我们提出了一种视角的重大转变，由此证明对于每个n≥10，存在一个0/1多面体，其面数至少为(n−⌈2log₂(n)⌉−1)!，这显著改进了当前已知的g(n)下界。此外，结合已知的上界，我们的构造确定了log g(n)的渐近行为，其误差为O((log n)²)。本文采用的方法是初等且完全确定性的，证明的核心思想来自超单纯形与排列多面体的组合学。
英文摘要
A long-standing question posed by Fukuda (1995) and Ziegler (2000) inquires about the asymptotic behavior of $g(n)$, the maximum number of facets that an $n$-dimensional $0/1$-polytope can have. A remarkable result by Bárány and Pór (2001) via probabilistic methods established that $g(n)$ is at least superexponential in $n$. In this paper, we propose a drastic change of perspective, which leads us to show that for each $n\geq 10$ there exists a $0/1$-polytope having at least $(n-\lceil 2\log_2 (n)\rceil - 1)!$ facets. This provides a significant improvement over the currently known lower bounds for $g(n)$. Furthermore, when combined with known upper bounds, our construction establishes the asymptotic behavior of $\log g(n)$ up to an error of $O((\log n)^2)$. The methods employed throughout this paper are elementary and fully deterministic. The underlying ideas in our proof stem from the combinatorics of hypersimplices and permutohedra.
