Geodesic
On the Structure of the Complements of Chebyshev Sets
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 3, pp. 19-27.

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A set is called a Chebyshev set if it contains a unique best approximation element. We study the structure of the complements of Chebyshev sets, in particular considering the following question: How many connected components can the complement of a Chebyshev set in a finite-dimensional normed or nonsymmetrically normed linear space have? We extend some results from [A. R. Alimov, East J. Approx, 2, No. 2, 215–232 (1996)]. A. L. Brown's characterization of four-dimensional normed linear spaces in which every Chebyshev set is convex is extended to the nonsymmetric setting. A characterization of finite-dimensional spaces that contain a strict sun whose complement has a given number of connected components is established. 
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A. R. Alimov. On the Structure of the Complements of Chebyshev Sets. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 3, pp. 19-27. http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a1/

[1] Alimov A. R., “Chebyshev set's complement”, East. J. Approx., 2:2 (1996), 215–232 | MR | Zbl

[2] Brown A. L., “Chebyshev sets and facial systems of convex sets in finite-dimensional spaces”, Proc. London Math. Soc. (3), 41 (1980), 297–339 | DOI | MR | Zbl

[3] Alimov A. R., “A number of connected components of sun's complement”, East J. Approx., 1:4 (1995), 419–429 | MR | Zbl

[4] Alimov A., “The Number of Connected Components of Chebyshev sets and “suns's” complement”, Proc. A. Ramadze Math. Inst., 117 (1998), 135–136 | MR

[5] Brøndsted A., “Convex sets and Chebyshev sets, II”, Math. Scand., 18 (1966), 5–15 | DOI | MR

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

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

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

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

[10] Tsarkov I. G., “Kompaktnye i slabo kompaktnye chebyshëvskie mnozhestva v lineinykh normirovannykh prostranstvakh”, Trudy MIAN, 189, 1989, 169–184 | MR

[11] Tsarkov I. G., “Ogranichennye chebyshëvskie mnozhestva v konechnomernykh banakhovykh prostranstvakh”, Matem. zametki, 36:1 (1984), 73–87 | MR | Zbl

[12] Brown A. L., “Chebyshev sets and the shapes of convex bodies”, Methods of Functional Analysis in Approximation Theory, Proc. Int. Conf. (Indian Inst. Techn. Bombay, 16–20.XII.1985), Internat. Series of Numerical Math., 76, ed. Micchelli C. A., Birkhäuser, Basel, 1986, 97–121 | MR

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

[14] Deutsch F., “The convexity of Chebyshev sets in Hilbert space”, Topics in polynomials of one and several variables and their applications, Volume dedicated to the memory of P. L. Chebyshev (1821–1894), eds. Rassias Th. M. et al., World Scientific, Singapore, 1993, 143–150 | DOI | MR | Zbl

[15] Berdyshev V. I., “K voprosu o chebyshëvskikh mnozhestvakh”, Dokl. AzSSR, 22:9 (1966), 3–5 | MR | Zbl