Lipschitz retractions in Hadamard spaces via gradient flow semigroups

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Abstract

Let X(n), for n ∈ 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: X(n) → X(n - 1) for n ≥ 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.

Original languageEnglish (US)
Pages (from-to)673-681
Number of pages9
JournalCanadian Mathematical Bulletin
Volume59
Issue number4
DOIs
StatePublished - Dec 1 2016

Keywords

  • Finite subset space
  • Gradient flow
  • Hadamard space
  • Lie-Trotter-Kato formula
  • Lipschitz retraction

ASJC Scopus subject areas

  • Mathematics(all)

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