Abstract
This paper is concerned with the problem of the existence of Lipschitz selections of the Steiner map , which associates with points of a Banach space the set of their Steiner points. The answer to this problem depends on the geometric properties of the unit sphere of , its dimension, and the number . For general conditions are obtained on the space under which admits no Lipschitz selection. When is finite dimensional it is shown that, if is even, the map has a Lipschitz selection if and only if is a finite polytope; this is not true if is odd. For the (single-valued) map 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.
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