  We propose to take a look at a new approach to the study of integral
polyhedra. The main idea is to give an integral representation, or matrix model
representation, for the key combinatorial characteristics of integral
polytopes. Based on the well-known geometric interpretations of matrix model
digram techniques, we construct a new model that enumerates triangulations,
subdivisions, and numbers of integral points of integral polygons. This
approach allows us to look at their combinatorics from a new perspective,
motivated by knowledge about matrix models and their integrability. We show how
analogs of Virasoro constraints appear in the resulting model. Moreover, we
make a natural generalization of this matrix model to the case of polytopes of
an arbitrary dimension, considering already a tensor model. We also obtain an
analogue of Virasoro constraints for it and discuss their role in the
solvability of these models. The deep connection between the geometry of convex
polyhedra and toric geometry is the main reference point in the construction of
these models. We present considerations on specific ways of applying this
approach to the description of Batyrev'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 varieties.
