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

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

—each point in is the barycenter of a unique probability measure supported on , or equivalently

—the dual space of is an space (in the dual ordering).

A simplex is said to be **Bauer** if is closed in .

Oppositely, is said to be **Poulsen** if is dense in . (Poulsen in 1961 proved the existence of such simplex.)

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