### 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.

Language | English (US) |
---|---|

Pages | 673-681 |

Number of pages | 9 |

Journal | Canadian Mathematical Bulletin |

Volume | 59 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Lipschitz retractions in Hadamard spaces via gradient flow semigroups.** / Bačák, Miroslav; Kovalev, Leonid Vladimirovich.

Research output: Contribution to journal › Article

*Canadian Mathematical Bulletin*, vol. 59, no. 4, pp. 673-681. https://doi.org/10.4153/CMB-2016-033-3

}

TY - JOUR

T1 - Lipschitz retractions in Hadamard spaces via gradient flow semigroups

AU - Bačák, Miroslav

AU - Kovalev, Leonid Vladimirovich

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - Finite subset space

KW - Gradient flow

KW - Hadamard space

KW - Lie-Trotter-Kato formula

KW - Lipschitz retraction

UR - http://www.scopus.com/inward/record.url?scp=84992129536&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992129536&partnerID=8YFLogxK

U2 - 10.4153/CMB-2016-033-3

DO - 10.4153/CMB-2016-033-3

M3 - Article

VL - 59

SP - 673

EP - 681

JO - Canadian Mathematical Bulletin

T2 - Canadian Mathematical Bulletin

JF - Canadian Mathematical Bulletin

SN - 0008-4395

IS - 4

ER -