On Lipschitz selections of affine-set valued mappings

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, p) into this family. Let F be such a mapping satisfying the following condition: for every subset M' subset of M consisting of at most 2(k+1) points, the restriction F/(M'), of F to M' has a selection f(M') (i.e. f(M')(x) epsilon F(x) for all x epsilon M') satisfying the Lipschitz condition parallel to f(M')(x)- f(M'), (y) parallel to less than or equal to p(x, y), x, y epsilon M'. Then F has a Lipschitz selection f : M --> H such that parallel tof(x) - f(y) parallel to less than or equal to gamma rho (x, y), x, y epsilon M where gamma = gamma (k) is a constant depending only on k. (The upper bound of the number of points in 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.