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Three-Dimensional Spaces Where All Bounded Chebyshev Sets Are Monotone Path Connected

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Abstract

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).

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Acknowledgments

The author is grateful to P. A. Borodin and I. G. Tsar’kov for attention and valuable comments.

Funding

This work was financially supported by the Russian Science Foundation, project 22-21-00415, https://rscf.ru/en/project/22-21-00415/.

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Authors and Affiliations

  1. I. M. Sechenov First Moscow State Medical University, Moscow, 119992, Russia

    B. B. Bednov

  2. Bauman Moscow State Technical University, Moscow, 105005, Russia

    B. B. Bednov

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  1. B. B. Bednov
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Correspondence to B. B. Bednov.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 323–338 https://doi.org/10.4213/mzm13569.

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Bednov, B.B. Three-Dimensional Spaces Where All Bounded Chebyshev Sets Are Monotone Path Connected. Math Notes 114, 283–295 (2023). https://doi.org/10.1134/S0001434623090018

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