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Mat. Sb., 2020, Volume 211, Number 8, Pages 132–157 (Mi msb9319)  

Approximative properties of sets and continuous selections

I. G. Tsar'kovab

a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University
b Moscow Center for Fundamental and Applied Mathematics

Abstract: Sets admitting a continuous selection of the operators of best and near-best approximation are studied. Michael's classical continuous selection theorem is extended to the case of a lower semicontinuous metric projection in finite-dimensional spaces (with no a priori convexity conditions on its values). Sufficient conditions on the metric projection implying the solarity of the corresponding set are put forward in finite-dimensional polyhedral spaces. Available results for suns $V$ are employed to establish the existence of continuous selections of the relative (with respect to $V$) Chebyshev near-centre map and of the sets of relative (with respect to $V$) near-Chebyshev points in certain classical spaces.
Bibliography: 30 titles.

Keywords: set-valued mapping, continuous selection, sun, monotone path-connected set, relative Chebyshev centre and point.

Funding Agency Grant Number
Russian Foundation for Basic Research 19-01-00332-а
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 19-01-00332-a).


DOI: https://doi.org/10.4213/sm9319

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English version:
Sbornik: Mathematics, 2020, 211:8, 1190–1211

Bibliographic databases:

UDC: 517.982.256
MSC: Primary 41A65; Secondary 54C65, 41A28, 47A52, 46B20, 54C60
Received: 15.08.2019

Citation: I. G. Tsar'kov, “Approximative properties of sets and continuous selections”, Mat. Sb., 211:8 (2020), 132–157; Sb. Math., 211:8 (2020), 1190–1211

Citation in format AMSBIB
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