Tag Archives: density of order a

Some short notes

5. Let E be a locally convex topological linear space, K\subset E be a compact convex subset of E and \partial_e K be the set of the extreme points of K. Let A(K) be the set of all affine continuous functions on K. Endowed with the supremum norm, A(K) is a Banach space.

Then K is said to be a (Choquet) simplex if

—each point in K is the barycenter of a unique probability measure supported on \partial_e K, or equivalently
—the dual space of A(K) is an L^1 space (in the dual ordering).

A simplex K is said to be Bauer if \partial_e K is closed in K.
Oppositely, K is said to be Poulsen if \partial_e K is dense in K. (Poulsen in 1961 proved the existence of such simplex.)

Lindenstrauss, Olsen and Sternfeld showed in 1978 here that given two Poulsen simplices P and Q, there is an affine homeomorphism h:P\to Q. In other words, there exists a unique Poulsen simplex (up to affine homeomorphisms), say \mathcal{P}.

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