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.
