Infinity modulus and the essential metric

TL;DR

This paper explores the concept of infinity modulus in metric spaces, establishing its link to shortest path lengths, and introduces the essential metric that aligns with previously studied metrics.

Contribution

It extends the notion of infinity modulus to general metric spaces and defines the essential metric, connecting it to existing literature.

Findings

Infinity modulus relates to shortest path lengths in metric spaces.

The essential metric recovers a previously known metric by De Cecco and Palmieri.

The formulation accounts for exceptional families in metric measure spaces.

Abstract

We study $\infty$-modulus on general metric spaces and establish its relation to shortest lengths of paths. This connection was already known for modulus on graphs, but the formulation in metric measure spaces requires more attention to exceptional families. We use this to define a metric that we call the essential metric, and show how this recovers a metric that had already been advanced in the literature by De Cecco and Palmieri.