## The geometric structure of Chebyshev sets in $$\ell^\infty(n)$$.(English. Russian original)Zbl 1087.41027

Funct. Anal. Appl. 39, No. 1, 1-8 (2005); translation from Funkts. Anal. Prilozh. 39, No. 1, 1-10 (2005).
Let $$M$$ (resp.$$H$$) be a set (resp. subspace) in $$R^n$$. Some approximative properties of the set $$M\cap N$$ in the subspace $$H$$, where the norm on $$H$$ is induced from $$l^\infty (n)$$, is studied. The results for coordinate and noncoordinate subspaces $$H$$ are completely different. There are two main results in the paper. Theorem 1 describes the approximative properties of intersections of Chebyshev sets, suns, and strict suns in $$l^\infty (n)$$ with coordinate subspaces. Theorem 2 characterizes Chebyshev sets in $$l^\infty (n)$$ in geometric terms.

### MSC:

 41A50 Best approximation, Chebyshev systems

### Keywords:

Chebyshev set; sun; strict sun; best approximation
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### References:

 [1] L. P. Vlasov, ”Approximative properties of sets in normed linear spaces,” Usp. Mat. Nauk, 28, No.6, 3–66 (1973); English transl. Russian Math. Surveys, 28, No. 6, 1–66 (1973). · Zbl 0293.41031 [2] V. A. Koshcheev, ”The connectivity and approximative properties of sets in linear normed spaces,” Mat. Zametki, 17, No.2, 193–204 (1975); English transl. Math. Notes, 17, No. 2, 114–119 (1975). [3] V. A. Koshcheev, ”Connectedness and solar properties of sets in normed linear spaces,” Mat. Zametki, 19, No.2, 267–278 (1976); English transl. Math. Notes, 19, No. 2, 158–164 (1976). [4] A. L. Brown, ”Suns in normed linear spaces which are finite dimensional,” Math. Ann., 279, 81–101 (1987). · Zbl 0607.41027 [5] A. L. Garkavi, ”The theory of best approximation in normed linear spaces,” In: Itogi Nauki i Tekhniki, Matem. Analiz, 1967 [in Russian], VINITI, Moscow, 1969, pp. 75–132; English transl. Progr. Math., Vol. 8: Mathematical Analysis, 1970, pp. 83–150. [6] V. M. Tikhomirov, ”Theory of approximations,” In: Itogi Nauki i Tekhniki, Current Problems in Mathematics, Fundamental Directions [in Russian], Vol. 14, VINITI, Moscow, 1987, pp. 103–260; English transl. Analysis II. Convex analysis and approximation theory. Encycl. Math. Sci., Vol. 14, 1990, pp. 93–243. [7] H. Berens and L. Hetzelt, ”Die Metrische Struktur der Sonnen in n),” Aequat. Math., 27, 274–287 (1984). · Zbl 0544.41025 [8] M. I. Karlov and I. G. Tsar’kov, ”Convexity and connectedness of Chebyshev sets and suns,” Fundam. Prikl. Mat., 3, No.4, 967–978 (1997). · Zbl 0951.41017 [9] A. R. Alimov, ”Geometrical characterization of strict suns in n),” Mat. Zametki, 70, No.1, 3–11 (2001); English transl. Math. Notes, 70, No. 1, 3–10 (2001). · Zbl 1027.46503 [10] A. R. Alimov, ”On the structure of the complement of the Chebyshev sets,” Funkts. Anal. Prilozhen., 35, No.3, 19–27 (2001). · Zbl 1085.35054 [11] V. S. Balagansky and L. P. Vlasov, ”Problem of convexity of Chebyshev sets,” Usp. Mat. Nauk, 51, No.6 (312), 125–188 (1996); English transl. Russian Math. Surveys, 51, No. 6, 1127–1190 (1996). [12] E. V. Oschman, ”Chebyshev sets and continuity of a metric projection,” Izv. Vyssh. Uchebn. Zaved. Mat., 9, 78–82 (1970). [13] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, N.J., 1970. · Zbl 0193.18401 [14] Ch. B. Dunham, ”Characterizability and uniqueness in real Chebyshev approximation,” J. Approx. Theory, 2, 374–383 (1969). · Zbl 0184.08901 [15] D. Braess, ”Geometrical characterizations for nonlinear uniform approximation,” J. Appr. Th. 11, 260–274, (1974). · Zbl 0288.41017 [16] V. M. Tikhomirov, ”A. B. Khodulev (1953–1999),” Matem. Prosv., Ser. 3, 4, 5–7 (2000). [17] G. A. Galperin, ”My friend Andrei Khodulev,” Matem. Prosv., Ser. 3, 4, 8–32 (2000).
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