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

Omer Friedland

TL;DR

The work establishes a sharper lower bound on the Banach--Mazur distance from the Gaussian Gluskin polytope to the cross-polytope in the regime $m=n^3$, showing $d_{\rm BM}(G_m,B_1^n) \ge c n^{4/7}(\log n)^{-C}$ w.h.p. by combining a discretization/powering framework with a novel treatment of many small coefficients via Maurey sparsification and a thickening-stable Gaussian measure bound. The analysis splits into a small-$U$ regime, handled via random suppression and Gram--Schmidt control, and a large-$U$ regime, addressed through sparsification inside a quotient space and union bounds. A careful balance of entropy reduction and small-ball estimates yields the $4/7$ exponent, advancing the understanding of optimal distortion to the cross-polytope and moving closer to the known upper bound $n^{5/6}$. The results have implications for the geometry of random quotients of $\ell_1^m$ and related probabilistic nets in high dimensions.

Abstract

Let $Γ$ be an $n\times m$ matrix with independent standard Gaussian entries and let $G_m = Γ(B_1^m)$ 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 $k/\log(nρ)$) 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 $4/7$.

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

TL;DR

The work establishes a sharper lower bound on the Banach--Mazur distance from the Gaussian Gluskin polytope to the cross-polytope in the regime , showing w.h.p. by combining a discretization/powering framework with a novel treatment of many small coefficients via Maurey sparsification and a thickening-stable Gaussian measure bound. The analysis splits into a small- regime, handled via random suppression and Gram--Schmidt control, and a large- regime, addressed through sparsification inside a quotient space and union bounds. A careful balance of entropy reduction and small-ball estimates yields the exponent, advancing the understanding of optimal distortion to the cross-polytope and moving closer to the known upper bound . The results have implications for the geometry of random quotients of and related probabilistic nets in high dimensions.

Abstract

Let be an matrix with independent standard Gaussian entries and let be the associated Gaussian Gluskin polytope (equivalently, a random -dimensional quotient of ). In the regime we prove that, with probability at least , where is the cross-polytope. This improves the previously best-known exponent (up to logarithmic factors) for this Gaussian model; in particular, the same lower bound holds for . 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 to ) 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 .
Paper Structure (34 sections, 32 theorems, 328 equations)

This paper contains 34 sections, 32 theorems, 328 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discretization: a net of sparse coefficient matrices
  4. From Banach--Mazur to a sparse coefficient matrix
  5. Discretization of the coefficient matrix
  6. Exposed indices and the powering reduction
  7. $K/U$ decomposition and a union of cross-polytopes
  8. Global norm control event for $U$-vectors
  9. Few $U$-generators (small-$U$ regime)
  10. A Gram--Schmidt volume bound
  11. Random suppression on the $K$-generators
  12. Block-tail distances for Gaussian images
  13. Suppression implies exponential small Gaussian measure
  14. A global suppression event for the $K$-generators
  15. Small-$U$ regime: a union measure bound
  16. ...and 19 more sections

Key Result

Theorem 1.2

There exist universal constants $c,C>0$ such that for all sufficiently large integers $n$, letting $m = n^3$ and $G_m$ be defined by eq:intro-def-Gm, one has

Theorems & Definitions (76)

  • Definition 1.1: Banach--Mazur distance
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Lemma 2.1: Sparse coefficient representation on $\mathcal{E}_\rho$
  • proof
  • Lemma 2.2: Discretization net and reduction
  • proof
  • Remark 2.3
  • Lemma 2.4: Powering
  • ...and 66 more