The Poulsen simplex is the unique nonsingleton metrizable Choquet simplex, up to a homeomorphism that preserves convex combinations, whose extreme points form a dense set. Poulsen (1961) constructed such a convex set, and Lindenstrauss et al. (1978) proved its uniqueness.
The density of the extreme points distinguishes this space from every nontrivial finite-dimensional simplex,
whose extreme points are just its finitely many
polytope vertices. Every metrizable Choquet
simplex is isomorphic, by a homeomorphism
preserving convex combinations, to a closed
set that is a face of the Poulsen simplex. Here a face is a convex
set 
such that, if a point of 
lies strictly between two points
of the simplex, both points belong to 
. Any homeomorphism
preserving convex combinations between two
proper faces that are closed sets extends to an automorphism of the whole simplex (Lindenstrauss et
al. 1978).
Sabok (2016, Question 9.4) asked whether the set closure of the convex hull of the distance profiles of the Urysohn sphere, using a countable set of coordinates that is also a dense set, is isomorphic to the Poulsen simplex by such a homeomorphism. Zhang and Yang (2026) reported that this convex set is not even a Choquet simplex. Independent specialist review of the complete argument and partial Lean development had not been reported as of Oct. 2, 2026.
-Algebras." Invent. Math. 204, 833-868,
2016.