The Convex Hull of Extremal Vector‐Valued Continuous Functions

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.