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 $$ \p\left\{d_{\mathrm{BM}}(G_m,B_1^n) \ge c_\delta n^{5/8}(\log n)^{-1/8}\right\} \ge 1-Ce^{-cn}. $$ 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.