Geodesic
Existence of Lipschitz selections of the Steiner map
Sbornik. Mathematics, Tome 209 (2018) no. 2, pp. 145-162
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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. 
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B. B. Bednov; P. A. Borodin; K. V. Chesnokova. Existence of Lipschitz selections of the Steiner map. Sbornik. Mathematics, Tome 209 (2018) no. 2, pp. 145-162. http://geodesic.mathdoc.fr/item/SM_2018_209_2_a0/

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