Lipschitz retractions in Hadamard spaces via gradient flow semigroups

TL;DR

This paper proves the existence of Lipschitz retractions from finite subset spaces to smaller ones in Hadamard spaces, extending previous results known for Hilbert spaces and addressing an open question.

Contribution

It establishes the existence of Lipschitz retractions in Hadamard spaces, generalizing known results from Hilbert spaces and solving an open problem.

Findings

Lipschitz retractions exist in Hadamard spaces for finite subset spaces.

The result extends the class of metric spaces where such retractions are known.

Addresses an open question posed by Kovalev.

Abstract

Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r: X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive.