Spaces of small metric cotype

TL;DR

This paper explores the properties of metric cotype in metric spaces, showing that bi-Lipschitz ultrametric spaces have infinite metric cotype and examining invariance under certain transformations.

Contribution

It demonstrates that bi-Lipschitz ultrametric spaces have infinite metric cotype and investigates invariance properties of metric cotype under snowflaking and Gromov-Hausdorff limits.

Findings

Ultrametric spaces have infinite metric cotype 1.

Metric cotype inequalities are invariant under snowflaking.

Partial converse of the main result established.

Abstract

Naor and Mendel's metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz equivalent to an ultrametric space has infinimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov-Hausdorff limits, and use these facts to establish a partial converse of the main result.