 # Boundary Complexes and Interior Points of the Polytopes

DOI: 10.4018/978-1-5225-6968-8.ch002

## Abstract

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.
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## 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).

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 .

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## Key Terms in this Chapter

The Boundary Complex of Polytope With Dimension n: The set of faces of a polytope with dimension from zero to n - 1.

The Set of Interior Points of the Polytope: The set of points belonging to a polytope, but not belonging to its boundary complex.

Cluster: A chemical compound that is intermediate between a molecule and a volumetric body.

The Coefficient Incidence of an Element of the Boundary Complex: The number of elements of the boundary complex to which the given element of the boundary complex belongs, having a dimension one greater than this element.

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