Lipschitz retractions in Hadamard spaces via gradient flow semigroups

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 $n\ge 2.$ 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.

References in corpus (3)

• Lipschitz retraction of finite subsets of Hilbert spaces
• A new proof of the Lie-Trotter-Kato formula in Hadamard spaces
• Symmetric products of the line: embeddings and retractions

Cited by in corpus (2)

• Lipschitz clustering in metric spaces
• On Lipschitz Retraction of Finite Subsets of Normed Spaces