CONVEX HULL DEVIATION AND CONTRACTIBILITY

We study the Hausdorff distance between a set and its convex hull. Let X be a Banach space, define the CHD-constant of the space X as the supremum of this distance over all subsets of the unit ball in X. In the case of finite dimensional Banach spaces we obtain the exact upper bound of the CHD-constant depending on the dimension of the space. We give an upper bound for the CHD-constant in Lp spaces. We prove that the CHD-constant is not greater than the maximum of Lipschitz constants of metric projection operators onto hyperplanes. This implies that for a Hilbert space the CHD-constant equals 1. We prove a characterization of Hilbert spaces and study the contractibility of proximally smooth sets in a uniformly convex and uniformly smooth Banach space.