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

Abstract

Let $Γ$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m = Γ(B_1^m)$ 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.

Disclosure

“e about the relation between the two arguments. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this manuscript, the author used OpenAI’s ChatGPT for language, exposition, organization, and critical review. The mathematical argument, including the two-event proof architecture and the bound n5/8 (log n)−3/8 , was obtained independently by the author before this use. During a later AI”

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2026-08-23T05:21:58.575015Z

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