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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2011 Volume 75, Issue 5, Pages 19–46 (Mi im4280)

This article is cited in 12 papers

On the convexity of $N$-Chebyshev sets

P. A. Borodin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We define $N$-Chebyshev sets in a Banach space $X$ for every positive integer $N$ (when $N=1$, these are ordinary Chebyshev sets) and study conditions that guarantee their convexity. In particular, we prove that all $N$-Chebyshev sets are convex when $N$ is even and $X$ is uniformly convex or $N\geqslant 3$ is odd and $X$ is smooth uniformly convex.

Keywords: Chebyshev set, convexity problem.

UDC: 517.982.256

MSC: 46B20, 41A50, 41A65

Received: 29.12.2009
Revised: 03.06.2010

DOI: 10.4213/im4280


 English version:
Izvestiya: Mathematics, 2011, 75:5, 889–914

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© Steklov Math. Inst. of RAS, 2025