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Mat. Sb., 2018, Volume 209, Number 2, Pages 3–21 (Mi msb8800)  

This article is cited in 2 scientific papers (total in 2 papers)

Existence of Lipschitz selections of the Steiner map

B. B. Bednov, P. A. Borodin, K. V. Chesnokova

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper is concerned with the problem of the existence of Lipschitz selections of the Steiner map $\mathrm{St}_n$, which associates with $n$ points of a Banach space $X$ the set of their Steiner points. The answer to this problem depends on the geometric properties of the unit sphere $S(X)$ of $X$, its dimension, and the number $n$. For $n\ge 4$ general conditions are obtained on the space $X$ under which $\mathrm{St}_n$ admits no Lipschitz selection. When $X$ is finite dimensional it is shown that, if $n\ge 4$ is even, the map $\mathrm{St}_n$ has a Lipschitz selection if and only if $S(X)$ is a finite polytope; this is not true if $n\ge 3$ is odd. For $n=3$ the (single-valued) map $\mathrm{St}_3$ is shown to be Lipschitz continuous in any smooth strictly-convex two-dimensional space; this ceases to be true in three-dimensional spaces.
Bibliography: 21 titles.

Keywords: Banach space, Steiner point, Lipschitz selection, linearity coefficient.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-08335-а
18-01-00333-а
Dynasty Foundation
Ministry of Education and Science of the Russian Federation НШ 6222.2018.1
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant nos. 15-01-08335-а and 18-01-00333-а) and the programme of the President of the Russian Federation for the state support of leading scientific schools (grant no. НШ-6222.2018.1). Borodin's work was also supported by the Dmitry Zimin Dynasty Foundation.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8800

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English version:
Sbornik: Mathematics, 2018, 209:2, 145–162

Bibliographic databases:

UDC: 517.982.256+517.988.38
MSC: 41A65, 52A40
Received: 20.08.2016 and 08.03.2017

Citation: B. B. Bednov, P. A. Borodin, K. V. Chesnokova, “Existence of Lipschitz selections of the Steiner map”, Mat. Sb., 209:2 (2018), 3–21; Sb. Math., 209:2 (2018), 145–162

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. R. Alimov, I. G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Russian Math. Surveys, 74:5 (2019), 775–849  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. B. B. Bednov, “The set of geometric medians for four-element subsets in Lindenstrauss spaces”, Moscow University Mathematics Bulletin, 74:6 (2019), 215–220  mathnet  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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