A. R. Alimov, B. B. Bednov, “Monotone path-connectedness of Chebyshev sets in three-dimensional spaces”, Mat. Sb., 2021, Volume 212, Number 5,Pages <nobr>37

This article is cited in 10 papers

Monotone path-connectedness of Chebyshev sets in three-dimensional spaces

A. R. Alimov^abc, B. B. Bednov^ade

^a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
^c Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia
^d Bauman Moscow State Technical University, Moscow, Russia
^e I. M. Sechenov First Moscow State Medical University, Moscow, Russia

Abstract: 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.

UDC: 517.982.256+517.982.252

MSC: 41A65

Received: 09.09.2019 and 16.11.2020

DOI: 10.4213/sm9325