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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

[1] A. A. Flerov, “Locally Chebyshev sets on the plane”, Math. Notes, 97:1 (2015), 136–142 | DOI | DOI | MR | Zbl

[2] A. R. Alimov, “On approximative properties of locally Chebyshev sets”, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44:1 (2018), 36–42 | MR | Zbl

[3] A. R. Alimov, I. G. Tsar'kov, “Connectedness and solarity in problems of best and near-best approximation”, Russian Math. Surveys, 71:1 (2016), 1–77 | DOI | DOI | MR | Zbl

[4] A. A. Flerov, Izbrannye geometricheskie svoistva mnozhestv s konechnoznachnoi metricheskoi proektsiei, Diss. ... kand. fiz.-matem. nauk, MGU, M., 2016, 68 pp.

[5] L. P. Vlasov, “Approximative properties of sets in normed linear spaces”, Russian Math. Surveys, 28:6 (1973), 1–66 | DOI | MR | Zbl

[6] A. R. Alimov, M. I. Karlov, “Sets with external Chebyshev layer”, Math. Notes, 69:2 (2001), 269–273 | DOI | DOI | MR | Zbl

[7] A. Czarnecki, M. Kulczycki, W. Lubawski, “On the connectedness of boundary and complement for domains”, Ann. Polon. Math., 103:2 (2011), 189–191 | DOI | MR | Zbl

[8] A. T. Fomenko, D. B. Fuks, Kurs gomotopicheskoi topologii, Nauka, M., 1989, 496 pp. | MR | Zbl

[9] V. V. Prasolov, Elements of homology theory, Grad. Stud. Math., 81, Amer. Math. Soc., Providence, RI, x+418 pp. | DOI | MR | Zbl

[10] A. D. Aleksandrov, N. Yu. Netsvetaev, Geometriya, 2-e izd., BKhV-Peterburg, SPb., 2010, 612 pp. | MR | Zbl

[11] M. Brown, “A proof of the generalized Schoenflies theorem”, Bull. Amer. Math. Soc., 66:2 (1960), 74–76 | DOI | MR | Zbl