The maximum of the 1-measurement of a metric measure space

TL;DR

This paper investigates the structure of the set of distributions of 1-Lipschitz functions on metric measure spaces, focusing on the existence and properties of maximum elements under the Lipschitz order.

Contribution

It provides a necessary condition for the existence of a maximum in the 1-measurement and explores conditions under which such a maximum exists.

Findings

Identifies a necessary condition for the maximum of the 1-measurement.

Analyzes properties of metric measure spaces with a maximum 1-measurement.

Provides examples or criteria for the existence of the maximum.

Abstract

For a metric measure space, we treat the set of distributions of 1-Lipschitz functions, which is called the 1-measurement. On the 1-measurement, we have a partial order relation by the Lipschitz order introduced by Gromov. The aim of this paper is to study the maximum and maximal elements of the 1-measurement with respect to the Lipschitz order. We present a necessary condition of a metric measure space for the existence of the maximum of the 1-measurement. We also consider a metric measure space that has the maximum of its 1-measurement.