On support points and continuous extensions

Abstract

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 \({\mathcal{BCC}(X)}\) the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping \({S : \mathcal{BCC}(X) \rightarrow X}\) such that S(K) is a support point of K for each \({K \in \mathcal{BCC}(X)}\). Moreover, it is possible to prescribe the values of S on a closed discrete subset of \({\mathcal{BCC}(X)}\).