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

Abstract

Let $\delta >0$ be fixed, let $m=\lceil (1+\delta )n\rceil$, and let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope $G_m=\Gamma (B_1^m)$ we prove $$ \p\left\{d_{\mathrm{BM}}(G_m,B_1^n) \ge c_δn^{5/8}(\log n)^{-1/8}\right\} \ge 1-Ce^{-cn}. $$ Consequently, for all sufficiently large $n$, $$R_1(n)\ge c{n}^{5/8}(\log n{)}^{-1/8}.$$ The proof uses an exact coefficient quantization, a simultaneous Gaussian compression in coefficient quotients, and an inverse-trace estimate for arbitrary inscribed bases.

arXiv commentv2: Substantially strengthened main result. The Gaussian construction now uses only \(m=\lceil(1+δ)n\rceil\) vertices, improves the logarithmic factor in the lower bound to \((\log n)^{-1/8}\), and gives exponentially high probability. The proof has been reorganized and simplified accordingly