Existence of Lipschitz selections of the Steiner map

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.
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