A note on absolutely minimal extensions in finite metric spaces

TL;DR

This paper investigates the existence of absolutely minimal Lipschitz extensions in finite metric spaces, showing they always exist for up to four points but may fail in five-point spaces.

Contribution

The paper establishes the boundary at five points where absolutely minimal Lipschitz extensions can fail to exist in finite metric spaces.

Findings

AMLE existence guaranteed for spaces with up to four points

AMLE may not exist in five-point metric spaces

Provides insight into the structure of finite metric spaces regarding Lipschitz extensions

Abstract

Absolutely minimal Lipschitz extensions (AMLEs) are known to exist in many infinite metric settings, but the finite case is less settled. In metric spaces with at most four points, every function on a nonempty subset admits an AMLE in the sense that the Lipschitz constant cannot be further reduced on sets that are disjoint from the prescribed domain. We show that in five-point spaces such extensions may fail to exist.