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A Choquet simplex is a closed, convex and metrizable subset of a locally convex Hausdorff topological vector space in which every point is a barycenter of a unique probability measure supported on the set of extreme points. The Poulsen simplex is a unique nontrivial compact Choquet simplex with a dense set of extreme points. This was proved by Lindenstrauss, Olsen and Sternfeld (Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, vi, 91–114); see also http://www.ams.org/mathscinet-getitem?mr=500918. The Poulsen simplex has many remarkable properties. Is there a similar object in the category of not necessarily compact (but bounded) Choquet simplices?

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This is not a complete answer but rather a long comment. The classical (compact) Poulsen simplex is a very homogeneous object. This heuristic observation can be indeed made precise via the theory of Fraïssé limits. Clinton Conley and Asger Törnquist proved that the Poulsen simplex is indeed a Fraïssé limit for the class of compact, finite-dimensional simplices. (This is not published yet as far as I know.)

I am not sure what do you mean a non-compact simplex but the general rule of thumb to define a non-compact version of the Poulsen simplex should be as follows:

  • consider the class of non-compact simplices that you consider small / finitary (simplices in $\mathbb{R}^n$ with removed boundary?),

  • check whether this class has the amalgamation property,

  • if so, think of the Fraïssé limit of this class as a variant of the Poulsen simplex appropriate for your class of finitary simplices.

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