A remark on intersection of convex sets

Let X be a real Banach space. Let {Gγ : γ ∈ Γ } be a family of closed, convex subsets of X. We show that either the intersection γ ∈Γ (Gγ ) of the -neighborhood of the sets Gγ is bounded for each >0, or it is unbounded for each >0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces.  2003 Elsevier Inc. All rights reserved.