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Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups

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