Linear extension operators between spaces of Lipschitz maps and optimal transport

TL;DR

This paper introduces the concept of K-random projections, a weaker form of Lipschitz retracts, and demonstrates their role in linear extension operators for Lipschitz maps, linking to optimal transport and approximation properties.

Contribution

It defines K-random projections related to Kantorovich-Rubinstein distance and establishes their equivalence with the existence of certain linear extension operators and approximation properties.

Findings

K-random projections enable linear extension of Lipschitz maps.

Existence of K-random projections is equivalent to weak* continuous operators.

Characterizes metric spaces with bounded approximation property in free spaces.

Abstract

Motivated by the notion of K-gentle partition of unity introduced in [12] and the notion of K-Lipschitz retract studied in [17], we study a weaker notion related to the Kantorovich-Rubinstein transport distance, that we call K-random projection. We show that K-random projections can still be used to provide linear extension operators for Lipschitz maps. We also prove that the existence of these random projections is necessary and sufficient for the existence of weak* continuous operators. Finally we use this notion to characterize the metric spaces (X, d) such that the free space F(X) has the bounded approximation propriety.