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)(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) ® XS:(X)X such that S(K) is a support point of K for each K Î BCC(X)K(X). Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X)(X).