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A note on absolutely minimal extensions in finite metric spaces

Alberto Domínguez Corella, Trí Minh Lê

TL;DR

The paper addresses the existence of absolutely minimal Lipschitz extensions (AMLEs) relative to $2^{X \setminus A}$ in finite metric spaces, revealing a sharp boundary at five points. It provides a constructive five-point counterexample showing nonexistence, while recalling that AMLEs exist for finite spaces with at most four points under Juutinen’s framework, and it notes embedding possibilities into Euclidean spaces. By comparing AMLEs with McShane–Whitney extensions, the work highlights that discrete settings can exhibit nonexistence and non-uniqueness, and that discrete infinity Laplacian solutions may fail to be AMLEs. These results delineate fundamental limits of AMLE theory in finite metric spaces and have implications for discrete analysis on graphs and networks.

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.

A note on absolutely minimal extensions in finite metric spaces

TL;DR

The paper addresses the existence of absolutely minimal Lipschitz extensions (AMLEs) relative to in finite metric spaces, revealing a sharp boundary at five points. It provides a constructive five-point counterexample showing nonexistence, while recalling that AMLEs exist for finite spaces with at most four points under Juutinen’s framework, and it notes embedding possibilities into Euclidean spaces. By comparing AMLEs with McShane–Whitney extensions, the work highlights that discrete settings can exhibit nonexistence and non-uniqueness, and that discrete infinity Laplacian solutions may fail to be AMLEs. These results delineate fundamental limits of AMLE theory in finite metric spaces and have implications for discrete analysis on graphs and networks.

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.
Paper Structure (4 sections, 2 theorems, 33 equations, 1 figure, 2 tables)

This paper contains 4 sections, 2 theorems, 33 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction, existence failure and state of the art
  2. Proofs
  3. Proof of Proposition \ref{['p0']}
  4. Proof of Theorem \ref{['thm0']}

Key Result

Proposition 1.2

If $X$ consists of four points and $A \subset X$ is nonempty, then any function $f:A \to \mathbb R$ admits an AMLE relative to $2^{X \setminus A}$.

Figures (1)

  • Figure 1: An example of a graph with four nodes.

Theorems & Definitions (9)

  • Definition 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5