Geodesic
A~connected compact locally Chebyshev set in a~finite-dimensional space is a~Chebyshev set
Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 455-465

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Let $X$ be a Banach space. A set $M\subset X$ is a Chebyshev set if, for each $x\in X$, there exists a unique best approximation to $x$ in $M$. A set $M$ is locally Chebyshev if, for any point $x\in M$, there exists a Chebyshev set $F_x\subset M$ such that some neighbourhood of $x$ in $M$ lies in $F_x$. It is shown that each connected compact locally Chebyshev set in a finite-dimensional normed space is a Chebyshev set. Bibliography: 11 titles.
Keywords: Chebyshev set, metric projection, Chebyshev layer, covering
Mots-clés : homotopy. 
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     author = {K. S. Shklyaev},
     title = {A~connected compact locally {Chebyshev} set in a~finite-dimensional space is {a~Chebyshev} set},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_3_a4/}
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K. S. Shklyaev. A~connected compact locally Chebyshev set in a~finite-dimensional space is a~Chebyshev set. Sbornik. Mathematics, Tome 211 (2020) no. 3, pp. 455-465. http://geodesic.mathdoc.fr/item/SM_2020_211_3_a4/