What is The Set of Interior Points of the Polytope

This chapter describes how the structure of a polytope of dimension n consisting of points of the boundary complex including a set of faces from zero to n - 1 and a set of interior points that are not belonging to the boundary complex is considered. The value is equal to the number of elements of the boundary complex, which the given element belongs, having dimension one greater than the given element of the boundary complex is denoted coefficient incidence of the given element. It is proven that the coefficient incidence of an element of dimension i of the boundary complex of an n - cube and n - simplex is equal to the difference of the dimension of the cube or simplex n and the dimension of this element i. The incidence coefficient of elements of n – cross - polytopes is substantially higher than this difference.