Gaussian polytopes with large Banach-Mazur distance to the cross-polytope

Abstract

Let $B_1^n$ be the standard cross-polytope in ${\mathbb{R}}^n$, let $g_1,\dots ,g_m$ be independent standard Gaussian vectors in ${\mathbb{R}}^n$, and set $G_m=\mathrm{conv}\pm g_1,\dots ,\pm g_m$. For $m=n^3$ it is proved that $$\mathbb{P}\left\{d_{\mathrm{BM}}(G_m,B_1^n)⩾c{n}^{5/8}(\ln n{)}^{-5/8}\right\}⩾1-\frac{2}{n}$$ for a suitable absolute constant $c>0$. This independently improves the polynomial exponent $4/7$ in Friedland's preceding work. Independent concurrent work of Friedland, which appeared after completion of the present manuscript, obtains the same polynomial exponent with the stronger logarithmic factor $(\ln n{)}^{-1/4}$ by a different argument. The proof uses Friedland's discretization and conditioning argument together with the $K/U$ decomposition. A selected family of $K$ vectors is suppressed and the remaining $K$ vectors are quotiented out. In the resulting quotient simultaneous bounds are proved for every top-dimensional exterior product formed from the suppressed $K$ vectors and the $U$ vectors. A Dvoretzky-Rogers selection after L"owner normalization converts these determinant estimates into a bound for the minimum volume ellipsoid of the whole projected polytope and Maurey's empirical method then gives the required Gaussian measure estimate.

arXiv commentv2: Added discussion of independent concurrent work of Friedland (arXiv:2608.17743), including the chronology and a comparison of the two proofs. The main theorem and proof are unchanged