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Published:2016-06-28
Printed: Dec 2016
• Miroslav Bačák,
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04 103 Leipzig, Germany
• Leonid V. Kovalev,
Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
 Format: LaTeX MathJax PDF

## 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 $r\colon X(n)\to X(n-1)$ 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.
 Keywords: finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retraction
 MSC Classifications: 53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces 47H20 - Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07] 54E40 - Special maps on metric spaces 58D07 - Groups and semigroups of nonlinear operators [See also 17B65, 47H20]

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