A remark on intersection of convex sets

AbstractLet 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)ϵ of the ϵ-neighborhood of the sets Gγ is bounded for each ϵ>0, or it is unbounded for each ϵ>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces