A remark on intersection of convex sets

Let X be a real Banach space. Let {G(gamma): gamma is an element of Gamma} be a family of closed, convex subsets of X. We show that either the intersection boolean AND(gammais an element ofGamma) (G(gamma))(epsilon) of the epsilon-neighborhood of the sets G(gamma) is bounded for each epsilon>0, or it is unbounded for each epsilon>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces. (C) 2003 Elsevier Inc. All rights reserved.