Regular Random Sections of Convex Bodies and the Random Quotient-of-Subspace Theorem

TL;DR

This paper introduces a new convex body position that ensures regular geometric estimates for random sections and quotients of subspaces, extending Pisier's work and improving probabilistic Euclidean approximation results.

Contribution

It establishes a novel convex body position guaranteeing regular estimates for random sections and quotients, with a new topological proof approach.

Findings

Existence of a new convex body position with regular geometric properties

High-probability estimates for random sections matching deterministic bounds

Improved probabilistic Euclidean approximation for quotients of subspaces

Abstract

It was shown by G. Pisier that any finite-dimensional normed space admits an $\alpha$-regular $M$-position, guaranteeing not only regular entropy estimates but moreover regular estimates on the diameters of minimal sections of its unit-ball and its dual. We revisit Pisier's argument and show the existence of a \emph{different} position, which guarantees the same estimates for \emph{randomly sampled} sections \emph{with high-probability}. As an application, we obtain a \emph{random} version of V. Milman's Quotient-of-Subspace Theorem, asserting that in the above position, \emph{typical} quotients of subspaces are isomorphic to Euclidean, with a distance estimate which matches the best-known deterministic one (and beating all prior estimates which hold with high-probability). Our main novel ingredient is a new position of convex bodies, whose existence we establish by using topological arguments and a fixed-point theorem.