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

Let $δ>0$ be fixed, let $m = \lceil(1+δ)n\rceil$, and let $Γ$ be an $n\times m$ matrix with independent standard Gaussian entries. For the Gaussian Gluskin polytope $G_m = Γ(B_1^m)$ 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,