Geodesic
The Geometric Structure of Chebyshev Sets in $\ell^\infty(n)$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 1-10.

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A subset $M$ of a normed linear space $X$ is called a Chebyshev set if each $x\in X$ has a unique nearest point in $M$. We characterize Chebyshev sets in $\ell^\infty(n)$ in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in $\ell^\infty(n)$ by coordinate subspaces.
Keywords: Chebyshev set, sun, strict sun, best approximation. 
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A. R. Alimov. The Geometric Structure of Chebyshev Sets in $\ell^\infty(n)$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 1-10. http://geodesic.mathdoc.fr/item/FAA_2005_39_1_a0/

[1] Vlasov L. P., “Approksimativnye svoistva mnozhestv v lineinykh normirovannykh prostranstvakh”, UMN, 28:6 (1973), 3–66 | MR | Zbl

[2] Koscheev V. A., “Svyaznost i approksimativnye svoistva mnozhestv v lineinykh normirovannykh prostranstvakh”, Matem. zametki, 17:2 (1975), 193–204 | MR | Zbl

[3] Koscheev V. A., “Svyaznost i solnechnye svoistva mnozhestv v lineinykh normirovannykh prostranstvakh”, Matem. zametki, 19:2 (1976), 267–278 | MR | Zbl

[4] Brown A. L., “Suns in normed linear spaces which are finite dimensional”, Math. Ann., 279 (1987), 81–101 | DOI | MR

[5] Garkavi A. L., “Teoriya nailuchshego priblizheniya v lineinykh normirovannykh prostranstvakh”, Itogi nauki i tekhniki. Matematicheskii analiz, VINITI, Moskva, 1967, 83–150

[6] Tikhomirov V. M., “Teoriya priblizhenii”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 14, 1987, 103–260

[7] Berens H., Hetzelt L., “Die Metrische Struktur der Sonnen in $\ell^\infty(n)$”, Aequat. Math., 27 (1984), 274–287 | DOI | MR | Zbl

[8] Karlov M. I., Tsarkov I. G., “Vypuklost i svyaznost chebyshëvskikh mnozhestv i solnts”, Fundam. prikl. matem., 3:4 (1997), 967–978 | MR | Zbl

[9] Alimov A. R., “Geometricheskaya kharakterizatsiya strogikh solnts v $\ell^\infty(n)$”, Matem. zametki, 70:1 (2001), 3–11 | DOI | MR | Zbl

[10] Alimov A. R., “O strukture dopolneniya k chebyshëvskim mnozhestvam”, Funkts. analiz i ego pril., 35:3 (2001), 19–27 | DOI | MR | Zbl

[11] Balaganskii V. S., Vlasov L. P., “Problema vypuklosti chebyshëvskikh mnozhestv”, UMN, 51:6 (1996), 125–188 | DOI | MR | Zbl

[12] Oshman E. V., “Chebyshëvskie mnozhestva i nepreryvnost metricheskoi proektsii”, Izv. VUZov, Matematika, 1970, no. 9, 78–82 | MR | Zbl

[13] Rokafellar R., Vypuklyi analiz, Mir, M., 1973

[14] Dunham Ch. B., “Characterizability and uniqueness in real Chebyshev approximation”, J. Approx. Theory, 2 (1969), 374–383 | DOI | MR | Zbl

[15] Braess D., “Geometrical characterizations for nonlinear uniform approximation”, J. Approx. Theory, 11 (1974), 260–274 | DOI | MR | Zbl

[16] Tikhomirov V. M., “A. B. Khodulëv (1953–1999)”, Matem. prosveschenie, ser. 3, 2000, no. 4, 5–7 | MR

[17] Galperin G. A., “Moi drug Andrei Khodulëv”, Matem. prosveschenie, ser. 3, 2000, no. 4, 8–32