The Number of Vertices of a Polyhedron

Abstract

where the aij, b1, c; are constants. In general, a set of linear inequalities defines a convex set in affine space; n-dimensional affine space A. is the set of points (cl, c2, * *, c.) where cl, C2, . . *, cn are real numbers. A set is convex in affine space if it contains the segment joining any two of its points. The convex set defined by (2) is the intersection of half spaces. It has vertices, edges, etc. Geometrically, a solution to the above problem is a point of the convex set defined by (2) which also maximizes (minimizes) (1). This point is generally a vertex of the boundary. For simplicity assume that (1) and (2) are normalized. Then this is the vertex (xi, x, * , x?) for which (1) assumes its maximum (or minimum) distance from the origin. Briefly, the is one of several iterative procedures used in solving this problem. The iterations of this process consist of translating the hyperplane corresponding to (1) in a parallel direction each time evaluating its distance from the origin at the intersected vertex of the convex set defined by the inequalities. The process is repeated in such a way that the translated hyperplane in each step yields improved results leading to the optimal distance. The simplex process also involves a useful criterion which eliminates some of the vertices of the convex set as possible trial points [3]. The number of iterations involved is decided for each problem separately. Unfortunately, so far, this number is only known to be dominated by Cn+m,n and as will be seen below this is an unsatisfactory estimate of the number of vertices of the convex sets of this problem. One would like an upper bound to the number of trial vertices knowing the number of inequalities and variables involved (the number of variables determines n, the dimension of the space). Hence the upper bound we wish to study here is of the form: (number of vertices)n< (a function of the number of inequalities) , n being the dimension of the space. Since each inequality defines at most a half-space with a hyperplane of dimension (n-1) as boundary, it suffices to study this problem in general in the form: (number of vertices)n< (a function of the number of (n-1) dimensional hyperplanes),. A rough upper bound which does not always take into consideration the polyhedral [1] property of the problem is available. If we denote by Fi(i = 0, 1, ... , n -1) the number of ith dimensional faces of a convex polyhedron in n-dimensional affine space then we have,