Absolute Lipschitz extendability and linear projection constants

TL;DR

This paper investigates the absolute Lipschitz extendability constant of finite metric spaces, linking it to projection constants of Lipschitz-free spaces, and provides specific bounds for small cases.

Contribution

It establishes a method to determine absolute extendability constants via relative projection constants and demonstrates how to compute these using linear programming.

Findings

ae(3)=4/3

ae(4)≥(5+4√2)/7

Linear programming can be used to compute projection constants

Abstract

We prove that the absolute extendability constant of a finite metric space may be determined by computing relative projection constants of certain Lipschitz-free spaces. As an application, we show that $\mbox{ae}(3)=4/3$ and $\mbox{ae}(4)\geq (5+4\sqrt{2})/7$. Moreover, we discuss how to compute relative projection constants by solving linear programming problems.