Lipschitz means and mixers on metric spaces

TL;DR

This paper explores Lipschitz continuous versions of the mean and median in metric spaces, linking them to key geometric properties like bounded turning and quasisymmetric parameterization.

Contribution

It introduces Lipschitz means and medians in metric spaces and connects them to fundamental geometric concepts, expanding the understanding of metric space topology.

Findings

Lipschitz means and medians relate to bounded turning property.

They are connected to quasisymmetric parameterization.

The work bridges topological measures with geometric properties.

Abstract

The standard arithmetic measures of center, the mean and median, have natural topological counterparts which have been widely used in continuum theory. In the context of metric spaces it is natural to consider the Lipschitz continuous versions of the mean and median. We show that they are related to familiar concepts of the geometry of metric spaces: the bounded turning property, the existence of quasisymmetric parameterization, and others.