Geodesic
Three-Dimensional Spaces Where All Bounded Chebyshev Sets Are Monotone Path Connected
Matematičeskie zametki, Tome 114 (2023) no. 3, pp. 323-338.

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In a three-dimensional normed space $X$, any bounded Chebyshev set is monotone path connected if and only if one of the following two conditions holds: (1) the set of extreme points of the sphere in the dual space is dense in this sphere; (2) $X=Y\oplus_\infty \mathbb R$ (i.e., the unit sphere of $X$ is a cylinder).
Keywords: Chebyshev set, monotone path connected set, bounded Chebyshev set. 
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B. B. Bednov. Three-Dimensional Spaces Where All Bounded Chebyshev Sets Are Monotone Path Connected. Matematičeskie zametki, Tome 114 (2023) no. 3, pp. 323-338. http://geodesic.mathdoc.fr/item/MZM_2023_114_3_a0/

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