0/1多面体的面数的无对数下界
A Log-Free Lower Bound for the Number of Facets of $0/1$-Polytopes
- Sorbonne Université（索邦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对0/1多面体，移除了前人下界中的对数因子，通过随机符号多面体与拉德马赫速率体的比较等方法，得到了更优的面数下界。
AI中文摘要：
设g(n)表示n维欧氏空间中满维0/1多面体的最大面数。我们证明存在绝对常数c>0和n₀，使得当n≥n₀时，g(n)≥(cn)^(n/2)。这移除了Gatzouras、Giannopoulos和Markoulakis给出的下界(cn/log n)^(n/2)中的对数因子。证明将随机符号多面体与两个被固定水平间隙分隔的拉德马赫速率体进行比较：缺失内部体的面在外部体的平坦区域上具有均匀小的足迹；进入内部体的面会形成空的缓冲离散帽。对于浅穿透，似然片定位缩小了相关范围的熵，并允许条件ε-网论证；对于深穿透，全局离散化即可满足要求。
英文摘要：
Let $g(n)$ denote the largest number of facets of a full-dimensional $0/1$-polytope in $\R^n$. We prove that there are absolute constants $c>0$ and $n_0$ such that $$ g(n)\ge (cn)^{n/2}\quad(n\ge n_0). $$ This removes the logarithmic factor from the lower bound $\bigl(cn/\log n\bigr)^{n/2}$ of Gatzouras, Giannopoulos, and Markoulakis. The proof compares a random sign polytope with two Rademacher rate bodies separated by a fixed level gap. Facets missing the inner body have uniformly small footprints on a flat patch of the outer body. A facet entering the inner body forces an empty buffered discrete cap. For shallow penetration, a likelihood-slab localization reduces the relevant range entropy and permits a conditional $\varepsilon$-net argument; for deep penetration, a global discretization suffices.
