On support points and continuous selections

We present a selection theorem concerning support points of convex sets in a Banach space. As a corollary we obtain, for instance, the following result. Denote by BCC(X) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S:BCC(X)→ X such that S(K) is a support point of K for each K ∈ BCC(X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X).