Vol. 37, No. 2, 1971

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Compact convex sets with the equal support property

John McDonald

Vol. 37 (1971), No. 2, 429–443

Simplexes may be characterized as follows: (C) X is a simplex if and only if each x X has a unique -maximal representing measure, where denotes the Choquel ordering on the set M(X) of positive regular Borel measures on X. In this paper, we study compact convex sets which satisfy a condition which is similar to that given in (C). Definition: X has the equal support property if, for each x X, any two -maximal representing measures for x have the same support. Some of our theorems are extensions to sets with the equal support property of results which hold for simplexes. Other results given here are analogous of theorems which hold for simplexes. We are especially interested in the relationships between the equal support property and a topology, called the structure topology, which was first defined for the set of extreme points of a simplex, but also makes sense for a wider class of compact convex sets.

Mathematical Subject Classification
Primary: 46A05
Received: 22 January 1970
Published: 1 May 1971
John McDonald