On Lipschitz selections of affine-set valued mappings |
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Authors: | P Shvartsman |
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Institution: | (1) Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan, 610066, P. R. China;(2) Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P. R. China |
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Abstract: | We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M,r) ({\cal M},\rho) into this family.¶Let F be such a mapping satisfying the following condition: for every subset M¢ ì M {\cal M'} \subset {\cal M} consisting of at most 2k+1 points, the restriction F|M¢ F|_{\cal M'} of F to M¢ {\cal M'} has a selection fM¢ (i.e. fM¢(x) ? F(x) for all x ? M¢) f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) satisfying the Lipschitz condition ||fM¢(x) - fM¢(y)|| £ r(x,y ), x,y ? M¢ \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} . Then F has a Lipschitz selection f : M ? H f : {\cal M} \to H such that ||f(x) - f(y) || £ gr(x,y ), x,y ? M \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} where g = g(k) \gamma = \gamma(k) is a constant depending only on k. (The upper bound of the number of points in M¢ {\cal M'} , 2k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees. |
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