Divisible points of compact convex sets (Q1087781)

A point in a compact convex set K in a real Hausdorff locally convex space is termed divisible if it occurs as the resultant (or barycentre) of each of a pair of mutually singular boundary probability measures. Roughly speaking, such points are the centres of mass of two disjointly supported weightings of the extreme points of K. The role which D(K) (the set of all divisible points of K) plays in determining the boundary probability measures associated with an arbitrary point is revealed. Also, the equivalence of D(K) with a set recently introduced by Reay for the particular case of convex polytopes is demonstrated. For polytopes, D(K) is a continuous image of a real projective space. A conjecture concerning D(K) is settled affirmatively for finite dimensional cubes.