At first, let us see the following matheoverflow question,

About a Delzant polytope. (In particular dodecahedron)

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?


This is the only theorem i know.

[Theorem] Let P be a 3-dimensional Delzant polytope. Then P has at least one triangular or quadrangular face.

see [C. Delaunay], On Hyperbolicity of Toric Real Threefolds, IMRN International Mathematics Research Notices 2005, No. 51.

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