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Combinatorial Properties of Associated Zonotopes

Published online by Cambridge University Press:  20 November 2018

G. C. Shephard
G. C. Shephard*
Affiliation:
University of East Anglia, Norwich, NOR 88C England
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Let S1 . . . ,Sr be r line segments, each of non-zero length, in n-dimensional euclidean space Rn. If a polytope Z is defined as the vector (Minkowski) sum

(1) Z = S1 + . . . + Sr,

then the segments Si will be called the components of Z. Since we do not wish to exclude the possibility that some of the components may be parallel, the polytope Z may be written in the form (1) in many different ways. For this reason it is convenient to define a zonotope to be the polytope Z together with some specified set of components {S1 , . . . , Sr}. Figures 1, 2 and 3 show some zonotopes of 1, 2 and 3 dimensions with 4, 5 and 6 components.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 2 , April 1974 , pp. 302 - 321
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Canham, R. J., Arrangements of hyperplanes in projective and euclidean spaces, Ph.D. Thesis, University of East Anglia, 1972.Google Scholar
2. Coxeter, H. S. M., Regular polytopes (London-New York, 1948; Second edition, 1963; Third edition, Dover, 1973).Google Scholar
3. Coxeter, H. S. M., The classification of zonohedra by means of projective diagrams, J. Math. Pures Appl. 41 (1962), 137–156; reprinted in Twelve geometric essays (Southern Illinois University Press, Illinois-London-Amsterdam, 1968).Google Scholar
4. Grünbaum, B., Convex polytopes (John Wiley & Sons, London-New York-Sydney, 1967).Google Scholar
5. Hansen, S., A generalization of a theorem of Sylvester on the lines determined by a finite point set, Math. Scand. 16 (1965), 175–180.Google Scholar
6. McMullen, P., On zonotopes, Trans. Amer. Math. Soc. 159 (1971), 91–110.Google Scholar
7. McMullen, P. and Shephard, G. C., Convex polytopes and the upper bound conjecture , London Math. Soc. Lecture Notes, Volume 3 (Cambridge University Press, Cambridge-London, 1971).Google Scholar
8. Shephard, G. C., Polytopes with centrally symmetric faces Can. J. Math. 19 (1967), 1206– 1213.Google Scholar
9. Shephard, G. C., Diagrams for positive bases, J. London Math. Soc. 4 (1971), 165–175.Google Scholar
10. Winder, R. O., Partitions of n-space by hyperplanes, SIAM J. Appl. Math. 14 (1966), 811–818.Google Scholar