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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 4, Pages 235–245 (Mi timm1590)

Stability of the relative Chebyshev projection in polyhedral spaces

I. G. Tsar'kov

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper is concerned with structural and stability properties of the set of Chebyshev centers of a set. Given a nonempty bounded subset $M$ of a metric space $(X,\varrho)$, the quantity $\operatorname{diam} M =\sup_{x,y\in M}\varrho(x,y)$ is called the diameter of $M$, and $r_M:=r(M):=\inf\{a\geqslant 0, x\in X \mid M\subset B(x,a)\}$, the Chebyshev radius of $M$. A point $x_0\in X$ for which $M\subset B(x_0,r(M))$ is called a Chebyshev center of $M$. The concept of a Chebyshev center and related stability, existence and uniqueness problems are important in various branches of mathematics. We study the structure of the set of Chebyshev centers and the stability of the Chebyshev projection (the Chebyshev center map). In the space $X=C(Q)$, where $Q$ is a normal topological space, we describe the structure of the Chebyshev center of sets with a unique Chebyshev center. The Chebyshev projection is the mapping associating with a nonempty bounded set the set of all its Chebyshev centers. Given a nonempty bounded set $M$ of a space $X$ and a nonempty set $Y\subset X$, the relative Chebyshev radius is defined as $r_Y(M)=\inf_{y\in Y} r(y,M)$, where $r(x,M):=\inf\{r\ge 0\mid M\subset B(x,r)\}=\sup_{y\in M}\|x-y\|$. The set of relative Chebyshev centers is defined as $\mathrm{Z}_Y(M):=\{y\in Y\mid r(y,M)=r_Y(M)\}$. The mapping $M\mapsto \mathrm{Z}_Y(M)$ is called the relative Chebyshev projection (with respect to the set $Y$). Stability properties of the relative Chebyshev projection in finite-dimensional polyhedral spaces are studied. In particular, in a finite-dimensional polyhedral space, the projection $\mathrm{Z}_Y( \cdot )$, where $Y$ is a subspace, is shown to be globally Lipschitz continuous.

Keywords: Chebyshev center, Chebyshev projection, stability.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00295 Ministry of Education and Science of the Russian Federation ÍØ-6222.2018.1 This work was supported by the Russian Foundation for Basic Research (project no. 16-01-00295) and by the RF President's Grant for State Support of Leading Scientific Schools (project no. NSh-6222.2018.1).

DOI: https://doi.org/10.21538/0134-4889-2018-24-4-235-245

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Bibliographic databases:

UDC: 517.982.256
MSC: 41A65
Revised: 14.11.2018
Accepted:19.11.2018

Citation: I. G. Tsar'kov, “Stability of the relative Chebyshev projection in polyhedral spaces”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 235–245

Citation in format AMSBIB
\Bibitem{Tsa18} \by I.~G.~Tsar'kov \paper Stability of the relative Chebyshev projection in polyhedral spaces \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2018 \vol 24 \issue 4 \pages 235--245 \mathnet{http://mi.mathnet.ru/timm1590} \crossref{https://doi.org/10.21538/0134-4889-2018-24-4-235-245} \elib{https://elibrary.ru/item.asp?id=36517714} 

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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
2. I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Sb. Math., 211:8 (2020), 1190–1211
3. I. G. Tsar'kov, “Properties of Monotone Connected Sets”, Math. Notes, 109:5 (2021), 819–827
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