Concentration of product spaces

TL;DR

This paper explores how the concentration property of metric measure spaces behaves under taking their product, providing a complete answer to a question posed in Gromov's book.

Contribution

It offers a full characterization of when the product of two concentrating sequences also concentrates, resolving a previously partial understanding.

Findings

Established conditions for product spaces to concentrate

Extended Gromov's partial results to a complete theorem

Clarified the relationship between concentration and product operations

Abstract

We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov's book. We obtain a complete answer for this question.