An improved lower bound on the Banach--Mazur distance to the cross-polytope

Let $\mathrm{\Gamma }$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m=\mathrm{\Gamma }(B_1^m)$ be the associated Gaussian Gluskin polytope (equivalently, a random $n$-dimensional quotient of ${\mathrm{\ell }}_1^m$). In the regime $m=n^3$ we prove that, with probability at least $1-2\mathrm{/}n$, $$d_{\mathrm{B}\mathrm{M}}(G_m,B_1^n)\ge c{n}^{4\mathrm{/}7}(\log n{)}^{-C},$$ where $B_1^n=\text{conv}\{\pm e_1,\dots ,\pm e_n\}$ is the cross-polytope. This improves the previously best-known exponent $5\mathrm{/}9$ (up to logarithmic factors) for this Gaussian model; in particular, the same lower bound holds for ${\sup }_K{d}_{\mathrm{B}\mathrm{M}}(K,B_1^n)$. The main new ingredient is a conditioning-compatible treatment of the regime of many small-coefficients''. After passing to a suitable Gaussian quotient, we apply a Maurey-type sparsification that reduces the relevant entropy (in effect shrinking the support size from $k$ to $k/\log(nρ)$) at the cost of a Euclidean thickening. We control this enlargement via a Gaussian measure bound stable under Euclidean thickening. In the complementary regime of few small-coefficients’', we give a streamlined argument avoiding the global tilting step in earlier work. Together these ingredients rebalance entropy and small-ball estimates and yield the exponent $4\mathrm{/}7$.