Geodesic
Monotone path-connectedness of Chebyshev sets in three-dimensional spaces
Sbornik. Mathematics, Tome 212 (2021) no. 5, pp. 636-654

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We characterize the three-dimensional Banach spaces in which any Chebyshev set is monotone path-connected. Namely, we show that in a three-dimensional space $X$ each Chebyshev set is monotone path-connected if and only if one of the following two conditions is satisfied: any exposed point of the unit sphere of $X$ is a smooth point or $X=Y\oplus_\infty \mathbb R$ (that is, the unit sphere of $X$ is a cylinder). Bibliography: 17 titles.
Keywords: Chebyshev set, sun, monotone path-connected set, cylindrical norm. 
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     author = {A. R. Alimov and B. B. Bednov},
     title = {Monotone path-connectedness of {Chebyshev} sets in three-dimensional spaces},
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A. R. Alimov; B. B. Bednov. Monotone path-connectedness of Chebyshev sets in three-dimensional spaces. Sbornik. Mathematics, Tome 212 (2021) no. 5, pp. 636-654. http://geodesic.mathdoc.fr/item/SM_2021_212_5_a1/