EXTREME-POINTS AND RETRACTIONS IN BANACH-SPACES

For T a completely regular topological space and X a strictly convex B anach space, we study the extremal structure of the unit ball of the s pace C(T, X) of continuous and bounded functions from T into X. We sho w that when dim X is an even integer then every point in the unit ball of C(T, X) can be expressed as the average of three extreme points if , and only if, dim T < dim X, where dim T is the covering dimension of T. We also prove that, if X is infinite-dimensional the aforementione d representation of the points in the unit ball of C(T, X) is always p ossible without restrictions on the topological space T. Finally, we d educe from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a repr esentation as the mean of three retractions of the unit ball onto the unit sphere.