Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-15T18:22:13.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Polytopes with Centrally Symmetric Faces

Published online by Cambridge University Press:  20 November 2018

G. C. Shephard
G. C. Shephard*
Affiliation:
University of East Anglia, England
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If a convex polytope P is centrally symmetric, and has the property that all its faces (of every dimension) are centrally symmetric, then P is called a zonotope. Zonotopes have many interesting properties which have been investigated by Coxeter and other authors (see (4, §2.8 and §13.8) and (5) which contains a useful bibliography). In particular, it is known (5, §3) that a zonotope is completely characterized by the fact that all its two-dimensional faces are centrally symmetric.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 1206 - 1213
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Alexandrov, A. D., A theorem on convex polyhedra, Trudy Mat. Inst. Steklov, Sect. Math., 4 (1933), 87 (in Russian) (Kraus Reprint Ltd., Vaduz, 1963).Google Scholar
2. Bonnesen, T. and Fenchel, W., Theorie der konvexen Körper (Berlin, 1934; Reprint, New York, 1948).Google Scholar
3. Burckhardt, J. J., Über konvexe Körper mit Mittelpunkt, Vierteljschr. Natur. Ges. Zurich, 85 (1940), 149–154.Google Scholar
4. Coxeter, H. S. M., Regular polytopes (New York, 1948; 2nd éd., 1963).Google Scholar
5. Coxeter, H. S. M., The classification of zonohedra by means of projective diagrams, J. de Math., 41 (1962), 137–156.Google Scholar
6. Grünbaum, B., Convex polytopes (London, New York, and Sydney, in press).Google Scholar
7. Minkowski, H., Allgemeine Lehrsätze über die konvexen Polyë;der, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1897), 198–219 ; Gesammelte Abhandlungen, Vol. 2 (Leipzig and Berlin, 1911), pp. 103-121.Google Scholar
8. Perles, M. A. and Shephard, G. C., Angle sums of convex polytopes, Math. Scand. (to be published).Google Scholar
9. Rogers, C. A., Sections and projections of convex bodies, Portugal. Math., 24 (1965), 99–103.Google Scholar
10. Shephard, G. C., Angle deficiencies of convex polytopes, J. London Math. Soc. (to be published).Google Scholar