RUS  ENG
Full version
JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2018 Volume 209, Number 2, Pages 3–21 (Mi sm8800)

This article is cited in 3 papers

Existence of Lipschitz selections of the Steiner map

B. B. Bednov, P. A. Borodin, K. V. Chesnokova

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper is concerned with the problem of the existence of Lipschitz selections of the Steiner map $\mathrm{St}_n$, which associates with $n$ points of a Banach space $X$ the set of their Steiner points. The answer to this problem depends on the geometric properties of the unit sphere $S(X)$ of $X$, its dimension, and the number $n$. For $n\geqslant 4$ general conditions are obtained on the space $X$ under which $\mathrm{St}_n$ admits no Lipschitz selection. When $X$ is finite dimensional it is shown that, if $n\geqslant 4$ is even, the map $\mathrm{St}_n$ has a Lipschitz selection if and only if $S(X)$ is a finite polytope; this is not true if $n\geqslant 3$ is odd. For $n=3$ the (single-valued) map $\mathrm{St}_3$ is shown to be Lipschitz continuous in any smooth strictly-convex two-dimensional space; this ceases to be true in three-dimensional spaces.
Bibliography: 21 titles.

Keywords: Banach space, Steiner point, Lipschitz selection, linearity coefficient.

UDC: 517.982.256+517.988.38

MSC: 41A65, 52A40

Received: 20.08.2016 and 08.03.2017

DOI: 10.4213/sm8800


 English version:
Sbornik: Mathematics, 2018, 209:2, 145–162

Bibliographic databases:


© Steklov Math. Inst. of RAS, 2025