The Convex Hull of Extremal Vector-Valued Continuous Functions

Abstract

Let T be a completely regular space and X a strictly convex n-dimensional real space. We prove that every continuous function from T into the closed unit ball of X can be expressed as an average of eight continuous functions from T into the sphere of X if and only if dim (T) ⩽ n−1, where dim(T) denotes the covering dimension of T. The proof we give can be used to prove the same fact, without hypotheses on T, when X is infinite-dimensional, although in this case it has been proved recently that a better result can be obtained.