On the volume bound in the Dvoretzky--Rogers lemma

TL;DR

This paper links the volume bounds in the Dvoretzky--Rogers lemma to the expected volume of random parallelotopes, providing probabilistic insights and improvements over previous bounds.

Contribution

It extends Pivovarov's expectation result to broader measures and connects it to the Dvoretzky--Rogers lemma, offering a probabilistic proof of existing bounds.

Findings

Volume bound equals the expectation of squared volume of random parallelotopes.

Probabilistic proof of Pelczyński and Szarek's improvement.

Lower bound for the probability of large volume in random parallelotopes.

Abstract

The classical Dvoretzky--Rogers lemma provides a deterministic algorithm by which, from any set of isotropic vectors in Euclidean $d$-space, one can select a subset of $d$ vectors whose determinant is not too small. Subsequently, Pelczy\'nski and Szarek improved this lower bound by a factor depending on the dimension and the number of vectors. Pivovarov, on the other hand, determined the expectation of the square of the volume of parallelotopes spanned by $d$ independent random vectors in $\mathbb{R}^d$, each one chosen according to an isotropic measure. We extend Pivovarov's result to a class of more general probability measures, which yields that the volume bound in the Dvoretzky--Rogers lemma is, in fact, equal to the expectation of the squared volume of random parallelotopes spanned by isotropic vectors. This allows us to give a probabilistic proof of the improvement of Pelczy\'nski and Szarek, and provide a lower bound for the probability that the volume of such a random parallelotope is large.