Lipschitz extensions to finitely many points

TL;DR

This paper studies Lipschitz map extensions in quasi-metric spaces, establishing bounds on Lipschitz constants when extending to finitely many points, with specific results for Hilbert and Banach spaces, and discusses applications to metric transforms.

Contribution

It provides new bounds for Lipschitz extensions to finitely many points in quasi-metric, Hilbert, and Banach spaces, advancing the theory of Lipschitz map extensions.

Findings

Extension bounds depend on the number of added points

Bound is linear for general quasi-metric spaces

Bound is square root for Hilbert to Banach space extensions

Abstract

We consider Lipschitz maps with values in quasi-metric spaces and extend such maps to finitely many points. We prove that in this context every 1-Lipschitz map admits an extension such that its Lipschitz constant is bounded from above by the number of added points plus one. Moreover, we prove that if the source space is a Hilbert space and the target space is a Banach space, then there exists an extension such that its Lipschitz constant is bounded from above by the square root of the total of added points plus one. We discuss applications to metric transforms.