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Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups

Published online by Cambridge University Press:  20 November 2018

Miroslav Bačák  and
Leonid V. Kovalev
Miroslav Bačák
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04 103 Leipzig, Germany e-mail: bacak@mis.mpg.de
Leonid V. Kovalev
Affiliation:
Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA e-mail: lvkovale@syr.edu
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Abstract

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Let $X\left( n \right)$, for $n\,\in \,\mathbb{N}$, be the set of all subsets of a metric space $\left( x,\,d \right)$ of cardinality at most $n$. The set $X\left( n \right)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r:\,X\left( n \right)\,\to \,X\left( n\,-\,1 \right)$ for $n\,\ge \,2$. It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if $X$ is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when $X$ is a Hadamard space. In this paper we answer the question in the positive.

Keywords

53C2347H2054E4058D07finite subset spacegradient flowHadamard spaceLie–Trotter–Kato formulaLipschitz retraction
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 673 - 681
Copyright
Copyright © Canadian Mathematical Society 2016

References

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