Radii of Euclidean sections of $\ell_p$-balls

TL;DR

This paper analyzes the radius of Euclidean sections of the unit ball of $\ell_1^N$ (the cross-polytope) by bounding the scaling constant $\lambda$ that compares a $d$-dimensional section to a $d$-dimensional Euclidean ball. The main results give a sharp upper bound $\lambda \le \sqrt{2/\pi}\sqrt{N}\,\sqrt{d}/\mu_d$ (with $\mu_d=\mathbb{E}\|G^{(d)}\|_2$) for sections satisfying $\lambda\|x\|_2\le\|x\|_1$, and a lower bound $\lambda'\ge\sqrt{d}$ for sections with $\|x\|_1\le\lambda'\|x\|_2$, showing that the inner radius scales as $d^{-1/2}$ and the outer radius as $N^{-1/2}$ up to constants. The bounds are proved using Gaussian-averaged arguments and $2$-absolutely summing norms, and the upper bound is shown to be essentially tight, with saturations in several explicit settings. The authors extend the results to the complex case and to $p\in[1,2]$, introducing generalized chi-means $\mu_{d,p}$ and providing corresponding bounds, while noting that a full analogue of the lower bound for $p>1$ remains open. The paper also surveys related questions for other normed and non-commutative spaces (notably Schatten spaces $S_p^N$) and discusses algorithmic aspects and open problems, linking Dvoretzky-type phenomena to applications in theoretical computer science and quantum information.

Abstract

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = Ω(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to some norm, much better lower bounds on $d$ have been obtained, with most research focusing on such lower bounds and on the degree of approximation of the section by a $d$-dimensional Euclidean ball. In this note we concentrate on another parameter, namely the radius of the approximating ball. We focus on the case of the unit ball of the space $\ell_1^N$ (the so-called cross-polytope), which is relevant to various questions of interest in theoretical computer science. We will also survey other instances where similar questions for other normed spaces (most often $\ell_p$-spaces or their non-commutative analogues) were found relevant to problems in various areas of mathematics and its applications, and state some open problems. Finally, in view of the computer science ramifications, we will comment on the algorithmic aspects of finding nearly spherical sections.

Theorems & Definitions (11)

• Theorem 2.1
• Remark 2.2
• proof : Proof of Theorem \ref{['thm:bound-on-radius']} (i)
• proof : Proof of Theorem \ref{['thm:bound-on-radius']} (ii)
• Lemma 2.3
• proof : Proof of Lemma \ref{['lemma:norm12']}
• Remark 2.4
• Theorem 3.1
• Remark 3.2
• proof : Proof of Theorem \ref{['thm:general-p-complex-case']}