We propose to consider a new approach to the study of integral polyhedra. The main idea is to give an integral or matrix model representation of the important combinatorial properties of the integral polytope. Based on the well-known geometric interpretation of the matrix-model diagram method, we construct a new model that enumerates the number of integration points in the triangulation, subdivision, and integration polygons. This approach allows us to look at their combinatorics from a new perspective, motivated by our knowledge of matrix models and their integrability. We show how the analogue of the Virasoro constraint appears in the resulting model. Moreover, already considering the tensor model, we naturally generalize this matrix model to the case of arbitrary-dimensional polytopes. Also for that he takes analogues of the Virasoro constraints and describes their role in the solvability of these models. The deep relationship between convex polyhedron geometry and toric geometry is the main reference point in the construction of these models. We consider a concrete application of this approach to the description of Batilev’s mirror pairs. All this allows us to formulate many interesting directions in the study of the connection between matrix/tensor models and the geometry of toric manifolds.
