Author

O. Friedland

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Preprint Aug 2026

A Log-Free Lower Bound for the Number of Facets of $0/1$-Polytopes

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.

O. Friedland · 2 citations

Preprint Aug 2026

A $5/8$ Lower Bound on the Banach-Mazur Distance to the Cross-Polytope

Let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m = \Gamma(B_1^m)$ be the associated Gaussian Gluskin polytope. In the regime $m = n^3$ we prove that, with probability at least $1-C/n$, $$ d_{\mathrm{BM}}(G_m,B_1^n) \ge c n^{5/8}(\log n)^{-1/4}. $$ This improves the polynomial exponent $4/7$ obtained in the author's preceding work and gives an explicit logarithmic factor. The proof retains the discretization and conditioning/powering framework, but replaces the earlier split into two coefficient regimes by two uniform quotient events. One controls successive directions of the big-coordinate parts; the other compresses the entire small-coordinate cloud near a low-dimensional subspace after every admissible quotient. Suppression, a local Maurey argument, and Gram-Schmidt volume estimates then combine these two forms of control.

O. Friedland · 1 citation