........ published in NEWSLETTER # 53
[ NATO-PCO Home Page ] [ Table of Contents of NEWSLETTER # 53 ]
........ published in NEWSLETTER # 53
POLYTOPES: ABSTRACT, CONVEX AND COMPUTATIONAL
by Professor A.I. Weiss, York University, North York/Ontario (Canada)
Polygons and polyhedra were among the very earliest objects of systematic mathematical investigation, and there is nearly continuous history of their study. Their higher dimensional analogues are known as polytopes. The modern theory of polytopes has three main branches: abstract, convex and computational. In recent years the value of the interaction between the branches has become evident. This volume (NATO ASI SERIES C440) of eighteen research and survey articles, concerning the various aspects of polytopes aims to at least partially satisfy this need.
Abstract regular polytopes generalize the classical notion of a geometric regular polytope and regular tessellation to more complicated combinatorial structures. Most of the research in this area is quite recent and emerged as a natural extension of Coxeter's classical work on geometric regularity. This new exciting subject links geometry and combinatorics with group theory and topology and, apart from being interesting in its own right, has strong potential for further important applications in these and other areas of mathematics.
Although the investigation of convex polyhedra has been with us since antiquity it was in the middle of last century that the investigations focused on convexity. About the turn of the century, however, there was a decline in interest in the subject. Fortunately, a revival of interest has changed in theory beyond all recognition over the last thirty years. The two main aspects of this work are the metric and combinatorial study of convex polytopes. The research in both has grown at an astonishing rate. Some papers in this volume deal with the classical problems in convexity and describe some new results in this direction. Others deal with more recent developments in combinatorial theory of convex polytopes.
Convex polytopes are of increasing utility in areas of computational mathematics. Algorithms to compute convex hulls and to describe the vertices, facets etc. of convex polytopes are central themes in computational geometry. These computations arise in many applications, from linear programming to robotics and symbolic computations. These computational issues both draw on, and generate results in the theory of convex polytopes. Several articles in this volume describe these connections.
The volume contains lectures by the principal lecturers at the NATO Advanced Study Institute on polytopes, held in Toronto, Canada in August 1993. It also contains several submitted (and refereed) articles as well as a number of open problems posed at the symposium.
Reference books: C16, C411, C440, F82