On the volume bound in the Dvoretzky–Rogers lemma

Ferenc Fodor, Márton Naszódi and Tamás Zarnócz

Vol. 301 (2019), No. 1, 89–99

DOI: 10.2140/pjm.2019.301.89

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. Pełczyński 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 ${ℝ}^d$, 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 Pełczyński and Szarek, and provide a lower bound for the probability that the volume of such a random parallelotope is large.

Keywords

isotropic vectors, John's theorem, Dvoretzky–Rogers lemma, decomposition of the identity, volume

Mathematical Subject Classification 2010

Primary: 52A22, 52A38, 52A40, 52B11

Milestones

Received: 10 April 2018

Revised: 19 December 2018

Accepted: 19 December 2018

Published: 16 September 2019