Geodesic
On the convexity of $N$-Chebyshev sets
Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 889-914

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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. 
@article{IM2_2011_75_5_a1,
     author = {P. A. Borodin},
     title = {On the convexity of $N${-Chebyshev} sets},
     journal = {Izvestiya. Mathematics },
     pages = {889--914},
     publisher = {mathdoc},
     volume = {75},
     number = {5},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a1/}
}
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P. A. Borodin. On the convexity of $N$-Chebyshev sets. Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 889-914. http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a1/