Embeddability of snowflaked metrics with applications to the nonlinear geometry of the spaces $L_p$ and $\ell_{p}$ for $0<p<\infty$

TL;DR

This paper explores the metric embeddability of $L_p$ and $\,ell_p$ spaces for all positive p, providing a comprehensive roadmap for Lipschitz embeddings and examining their nonlinear geometric structures.

Contribution

It offers a complete classification of Lipschitz embeddability between $L_p$ and $\ell_p$ spaces, including their snowflaked versions, and investigates the complex subspace and nonlinear geometries within.

Findings

Complete Lipschitz embeddability roadmap for $L_p$ and $\ell_p$ spaces.

Identification of fundamental open problems related to weaker embeddings.

Insights into the dissimilarities between subspace structures and nonlinear geometries.

Abstract

We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range $0<p<\infty$ from a metric viewpoint and give a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with both their ad-hoc distances and their snowflakings. Through connections with weaker forms of embeddings that lead to basic (yet fundamental) open problems, we also set the challenging goal of understanding the dissimilarities between the well-known subspace structure and the different nonlinear geometries that coexist inside $L_{p}$ and $\ell_{p}$.