Quantitative coarse embeddings of quasi-Banach spaces into a Hilbert space

TL;DR

This paper investigates the coarse embeddability of quasi-Banach spaces into Hilbert spaces, focusing on the Hilbert space compression exponent and its relation to snowflaking transformations.

Contribution

It introduces a method to compute the Hilbert space compression exponent for quasi-Banach spaces and establishes its equivalence to the supremum of snowflaking exponents admitting bi-Lipschitz embeddings.

Findings

Hilbert space compression exponent can be explicitly computed for quasi-Banach spaces.

The compression exponent equals the supremum of snowflaking exponents allowing bi-Lipschitz embedding.

Provides a new characterization linking coarse embeddings and snowflaking transformations.

Abstract

We study how well a quasi-Banach space can be coarsely embedded into a Hilbert space. Given any quasi-Banach space X which coarsely embeds into a Hilbert space, we compute its Hilbert space compression exponent. We also show that the Hilbert space compression exponent of X is equal to the supremum of the amounts of snowflakings of X which admit a bi-Lipschitz embedding into a Hilbert space.