Farthest Points of Bounded Closed Sets

Farthest points of bounded closed subsets of a real Hilbert space are considered. If the set at a farthest point satisfies the support condition of strong convexity, then the corresponding max-projection operator is shown to be Lipschitz continuous as a function of a point of the space; if the set is convex, then the max-projection operator is shown to be Lipschitz continuous as a function of the set with respect to the Pliś metric. The relation of the Lipschitz constant with the constant in the support condition of strong convexity is discussed. An algorithm of iterative approximation to a farthest point is given.