## Tag Archives: totally transitive

### 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 $\mathcal{P}$.