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Diagrams and transforms applied to convex polytopes Lockeberg, Erik Ring

Abstract

The aim of this paper is to present a unified treatment of diagram techniques, particularly as applied to problems concerning convex polytopes. An attempt is made to summarize new results (listed below) of M. Perles, G.C. Shephard and P. McMullen. A diagram technique is one of several methods for associating to a finite subset of a finite-dimensional Euclidean space a finite subset of the same cardinality in another Euclidean space, in general of a different dimension. The second subset is a "diagram" or "transform" of the original set. The original set and its diagram are seen to be related in a symmetrical way to the kernel and image respectively of a certain linear map. A problem concerning a finite set corresponds to a problem concerning the diagram which may be easier to solve, particular if the diagram is of low dimension. Gale diagrams are used to enumerate d-polytopes with d+3 vertices, to construct an 8-polytope not rationally imbeddable, to investigate the symmetry group of polytopes and to obtain results concerning projectively unique polytopes. It is shown how positive diagrams may be used to investigate positive bases of Euclidean space. Zonal diagrams are used to investigate the neighbourliness of centrally symmetric polytopes. A diagram technique for dealing with polyhedral sets, including linear systems and metric properties of polyhedral sets. A geometric interpretation of the affine transform, central transform and zonal diagram in terms of regular polytope is given.

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