Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups

Canadian mathematical bulletin

Leonid V. Kovalev¹  Miroslav Bačák² 

[1] Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA;Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04 103 Leipzig, Germany

关键词: finite subset space;    gradient flow;    Hadamard space;    Lie-Trotter-Kato formula;    Lipschitz retraction;   

DOI  :  10.4153/CMB-2016-033-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $X(n),$ for $ninmathbb{N},$ 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 $rcolon X(n)o X(n-1)$ for $nge2.$ 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 Hadamardspace. In the present paper we answer this question in the positive.

【 授权许可】

Unknown