Multiple sets exponential concentration and higher order eigenvalues

TL;DR

This paper introduces a new measure of concentration involving multiple sets and demonstrates how higher order eigenvalues of the metric Laplacian enhance concentration inequalities, with applications to Riemannian manifolds.

Contribution

It establishes a novel link between higher order eigenvalues and improved measure concentration, extending classical inequalities to multiple sets.

Findings

Higher order eigenvalues lead to exponential concentration improvements.

The new concentration notion applies to generic metric measure spaces.

Results recover and extend classical eigenvalue estimates on Riemannian manifolds.

Abstract

On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of k distinct sets. We show that the k-th eigenvalues of the metric Laplacian gives exponential improved concentration with k sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of Chung, Grigory'an and Yau [11].