Boundary Complexes and Interior Points of the Polytopes

Introduction

A finite family C of convex polytopes will be called a complex under the condition

• 1.

  Each face of an element of the family C is an element of the family C;

• 2.

  The intersection of any two elements of C is a face of both.

Let P is an n – polytope, i.e. a convex polytope of dimension n. We denote by B (P) the boundary complex P, that is, a complex consisting of all faces of P with dimension n - 1 or less (Grunbaum, 1967). The points not belonging to B (P) will be called interior points of P. The set of interior points of the polytope P will be denoted by V. Thus, we have equality . Since all the faces of a polytope from vertices (which in chemical compounds are images of atoms, molecules or functional groups) to facets enter the boundary complex, it is of interest to establish relationships between faces of different dimensions in the boundary complex B (P).

For each element of the boundary complex B (P) can to introduce of the notation its dimension

There is denoted the incidence value of the element of the boundary complex to the elements of the boundary complex having dimension This 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 .

The Structure Of The Boundary Complex Of A N – Cube

If polytope P is point (vertex), so evidence that n = 0, If polytope P is segment with vertices so the boundary complex B (P) is points . In this case are Can to introduce the origin of coordinates O in the middle of the segment []. Thus, if put of length of the segment equal one, so the boundary elements (points) have coordinates The set of interior points in this case is If n = 2 so polytope P is square (Figure 1). There elements of the boundary complex are vertices , and edges , . The dimensions of the vertices equal zero The dimensions of the edges equal one The incidence value of the elements of the boundary complex equal corresponding If to introduce the origin of coordinates O in centrum of square and put of length of edges equal one so vertices of square have coordinates (x, y):