Lipschitz Retractions in Hadamard Spaces via Gradient Flow Semigroups

Bačák, Miroslav ; Kovalev, Leonid V.

Canadian mathematical bulletin, Tome 59 (2016) no. 4, pp. 673-681

Résumé

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.

Détail

DOI : 10.4153/CMB-2016-033-3

Mots-clés : 53C23, 47H20, 54E40, 58D07, finite subset space, gradient flow, Hadamard space, Lie–Trotter–Kato formula, Lipschitz retraction