Lipschitz clustering in metric spaces

TL;DR

This paper investigates the Lipschitz clustering property in metric spaces, identifying conditions under which it holds or fails, and analyzing its invariance under certain classes of maps.

Contribution

It establishes that uniformly disconnected spaces possess the Lipschitz clustering property and explores how this property behaves under quasihomogeneous and quasisymmetric maps.

Findings

Uniformly disconnected spaces have the Lipschitz clustering property.

Connected spaces may lack this property due to absence of short connecting curves.

The property is invariant under quasihomogeneous maps but not under quasisymmetric maps.

Abstract

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones.