Spectral calculus and Lipschitz extension for barycentric metric spaces

TL;DR

This paper computes the metric Markov cotype for barycentric metric spaces, leading to new nonlinear spectral calculus inequalities and a unified approach for Lipschitz extension, including for CAT(0) spaces.

Contribution

It introduces the first understanding of metric Markov cotype in non-Banach barycentric spaces, establishing new inequalities and extension results.

Findings

Computed metric Markov cotype for barycentric metric spaces.

Established new nonlinear spectral calculus inequalities.

Provided new Lipschitz extension results for CAT(0) spaces.

Abstract

The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT(0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.