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. 
@article{MZM_2023_114_3_a0,
     author = {B. B. Bednov},
     title = {Three-Dimensional {Spaces} {Where} {All} {Bounded} {Chebyshev} {Sets} {Are} {Monotone} {Path} {Connected}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {323--338},
     publisher = {mathdoc},
     volume = {114},
     number = {3},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_114_3_a0/}
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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/