On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain 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 from BCC (X) to X such that S(K) is a support point of K for each K in BCC (X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X).