The observable diameter of metric measure spaces and the existence of points of positive measures

TL;DR

This paper establishes a fundamental link between the observable diameter of metric measure spaces and the existence of points with positive measure, providing a new characterization within Gromov's theory.

Contribution

It proves that the partial diameter or observable diameter is zero if and only if a point with positive measure exists, clarifying a key aspect of metric measure space invariants.

Findings

Observable diameter equals zero iff a point with positive measure exists

Characterizes the relationship between measure concentration and point measures

Provides a new criterion for measure distribution in metric spaces

Abstract

A metric measure space is a metric space with a Borel measure. In Gromov's theory of metric measure spaces, there are important invariants called the partial diameter and the observable diameter. We obtain the result that the partial diameter or the observable diameter equals zero if and only if there exists a point that has positive measure.