Barycentric selectors and a Steiner-type point of a convex body in a Banach space |
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Authors: | P. Shvartsman |
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Affiliation: | Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, Israel |
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Abstract: | Let F be a mapping from a metric space into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping satisfying . The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector SX(m) defined on the family of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set. |
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Keywords: | Set-valued mapping Lipschitz selection Helly-type theorems Steiner point Barycenter |
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