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On the Number of Vertices of a Convex Polytope

Published online by Cambridge University Press:  20 November 2018

Victor Klee
Victor Klee*
Affiliation:
University of Washington and Boeing Scientific Research Laboratories
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As is well known, the theory of linear inequalities is closely related to the study of convex polytopes. If the bounded subset P of euclidean d-space has a non-empty interior and is determined by i linear inequalities in d variables, then P is a d-dimensional convex polytope (here called a d-polytope) which may have as many as i faces of dimension d — 1, and the vertices of this polytope are exactly the basic solutions of the system of inequalities. Thus, to obtain an upper estimate of the size of the computation problem which must be faced in solving a system of linear inequalities, it suffices to find an upper bound for the number f0(P) of vertices of a d-polytope P which has a given number fd-1(P) of (d — l)-faces. A weak bound of this sort was found by Saaty (14), and several authors have posed the problem of finding a sharp estimate.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 701 - 720
Copyright
Copyright © Canadian Mathematical Society 1964

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