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Mat. Sb., 2013, Volume 204, Number 5, Pages 25–44 (Mi msb8127)  

This article is cited in 6 scientific papers (total in 6 papers)

Existence of a Lipschitz selection of the Chebyshev-centre map

Yu. Yu. Druzhinin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper is concerned with the existence of a Lipschitz selection for the operator $T_C$ (the Chebyshev-centre map) that assigns to any bounded subset $M$ of a Banach space $X$ the set $T_C(M)$ of its Chebyshev centres. It is proved that if the unit sphere $S(X)$ of $X$ has an exposed smooth point, then $T_C$ does not have a Lipschitz selection. It is also proved that if $X$ is finite dimensional the operator $T_C$ has a Lipschitz selection if and only if $X$ is polyhedral. The operator $T_C$ is also shown to have a Lipschitz selection in the space $\mathbf c_0(K)$ and $\mathbf c$-spaces.
Bibliography: 4 titles.

Keywords: Chebyshev centre, Lipschitz selection, metric projection.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00952a


DOI: https://doi.org/10.4213/sm8127

Full text: PDF file (555 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2013, 204:5, 641–660

Bibliographic databases:

UDC: 517.982.256
MSC: 41A65
Received: 03.04.2012 and 26.11.2012

Citation: Yu. Yu. Druzhinin, “Existence of a Lipschitz selection of the Chebyshev-centre map”, Mat. Sb., 204:5 (2013), 25–44; Sb. Math., 204:5 (2013), 641–660

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. X. Shi, G. Mao, B. D. O. Anderson, Z. Yang, J. Chen, “Robust localization using range measurements with unknown and bounded errors”, IEEE Trans. Wirel. Commun., 16:6 (2017), 4065–4078  crossref  isi  scopus
    2. M. V. Balashov, “Inscribed balls and their centers”, Comput. Math. Math. Phys., 57:12 (2017), 1899–1907  mathnet  crossref  crossref  isi  elib
    3. B. B. Bednov, P. A. Borodin, K. V. Chesnokova, “Existence of Lipschitz selections of the Steiner map”, Sb. Math., 209:2 (2018), 145–162  mathnet  crossref  crossref  adsnasa  isi  elib
    4. Yu. Yu. Druzhinin, “On Selections from the Best $n$-Nets”, Math. Notes, 104:5 (2018), 678–682  mathnet  crossref  crossref  isi  elib
    5. I. G. Tsarkov, “Ustoichivost otnositelnogo chebyshëvskogo proektora v poliedralnykh prostranstvakh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 235–245  mathnet  crossref  elib
    6. 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
  • Математический сборник Sbornik: Mathematics (from 1967)
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