Barycentric selectors and a Steiner-type point of a convex body in a Banach space

Let F be a mapping from a metric space (M,ρ) 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 f : M→X satisfying f(x)∈F(x),x∈M. 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 Km(X) 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 SX(m) : Km(X)→X 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.