At first, let us see the following matheoverflow question,
The question is whether (combinatorial) regular dodecahedron can be realized as a Delzant polytope or not.
A combinatorial type of a polytope means a face-lattice structure.
When I learned Delzant polytopes, in fact, my first question was that what kind of combinatorial type allow Delzant realization.
Obviously the first requirement is that it should be a rational polytope. Fortunately, all 3-dimensional (combinatorial) polytopes always rational realization. Moreover, 3-dimensional polytopes are identified with 3-vertex connected planar graph by Steiniz theorem, So let me restrict the question to only 3-dimensional cases only.
On superficial consideration, it seems quite doable to try check that a 3-dimensional polyhedron allow Delzant realization or not.
I think it is very natural question, but I couldn't find any literature. Is there any study concering combinatorial type of polytopes and Delzant condition? Is it trivially true? or is it meaningless question?