Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Friedland, Omer
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203751602