On Lipschitz selections of affine-set valued mappings

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 $({\cal M},\rho)$ into this family.¶Let F be such a mapping satisfying the following condition: for every subset ${\cal M'} \subset {\cal M}$ consisting of at most 2^k+1 points, the restriction $F|_{\cal M'}$ of F to ${\cal M'}$ has a selection $f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'})$ satisfying the Lipschitz condition $\parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'}$. Then F has a Lipschitz selection $f : {\cal M} \to H$ such that $\parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M}$ where $\gamma = \gamma(k)$ is a constant depending only on k. (The upper bound of the number of points in ${\cal M'}$, 2^k+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.