Extreme points and retractions in Banach spaces

ForT a completely regular topological space andX a strictly convex Banach space, we study the extremal structure of the unit ball of the spaceC(T,X) of continuous and bounded functions fromT intoX. We show that when dimX is an even integer then every point in the unit ball ofC(T, X) can be expressed as the average of three extreme points if, and only if, dimT< dimX, where dimT is the covering dimension ofT. We also prove that, ifX is infinite-dimensional, the aforementioned representation of the points in the unit ball ofC(T, X) is always possible without restrictions on the topological spaceT. Finally, we deduce from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a representation as the mean of three retractions of the unit ball onto the unit sphere. The author wishes to express his gratitude to Dr. Juan Francisco Mena Jurado for many helpful suggestions during the preparation of this paper.