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

Abstract

Let $\Gamma$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m=\Gamma (B_1^m)$ be the associated Gaussian Gluskin polytope (equivalently, a random $n$-dimensional quotient of ${\ell }_1^m$). In the regime $m=n^3$ we prove that, with probability at least $1-2/n$, $$d_{\mathrm{BM}}(G_m,B_1^n)\ge c{n}^{4/7}(\log n{)}^{-C},$$ where $B_1^n = \conv\{\pm e_1,\dots,\pm e_n\}$ is the cross-polytope. This improves the previously best-known exponent $5/9$ (up to logarithmic factors) for this Gaussian model; in particular, the same lower bound holds for ${\sup }_K{d}_{\mathrm{BM}}(K,B_1^n)$. 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 $k/\log (n\rho )$) 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 $4/7$.