Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T05:21:39.459Z Has data issue: false hasContentIssue false
  • English
  • Français

Cell Complexes, Valuations, and the Euler Relation

Published online by Cambridge University Press:  20 November 2018

M. A. Perles  and
G. T. Sallee
M. A. Perles
Affiliation:
Hebrew University, Jerusalem, Israel
G. T. Sallee
Affiliation:
University of California, Davis, California
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.

1. Recently a number of functions have been shown to satisfy relations on polytopes similar to the classic Euler relation. Much of this work has been done by Shephard, and an excellent summary of results of this type may be found in [11]. For such functions, only continuity (with respect to the Hausdorff metric) is required to assure that it is a valuation, and the relationship between these two concepts was explored in [8]. It is our aim in this paper to extend the results obtained there to illustrate the relationship between valuations and the Euler relation on cell complexes.

To fix our notions, we will suppose that everything takes place in a given finite-dimensional Euclidean space X.

A polytope is the convex hull of a finite set of points and will be referred to as a d-polytope if it has dimension d. Polytopes have faces of all dimensions from 0 to d – 1 and each of these is in turn a polytope. A k-dimensional face will be termed simply a k-face.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 235 - 241
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Alexandroff, P. and Hopf, H., Topologie (Springer-Verlag, Berlin, 1935).Google Scholar
2. Griinbaum, B., Convex polytopes (Wile﹜7, New York, 1967).Google Scholar
3. Hadwiger, H., Vorlesungen ùber Inhalt, Oberflàche und Isoperimetrie (Springer-Verlag, Berlin, 1957).Google Scholar
4. Hadwiger, H., Vber additive Funktionale k-dimensionaler Eipolyeder, Publ. Math. Debrecen 3 (1953), 87–94.Google Scholar
5. Klee, V. L., The Euler characteristic in combinatorial geometry, Amer. Math. Monthly 70 (1963), 119–127.Google Scholar
6. Mani, P., On angle sums and Steiner points of polyhedra (to appear).Google Scholar
7. Sallee, G. T., A valuation property of Steiner points, Mathematika 13 (1966), 76–82.Google Scholar
8. Sallee, G. T., Polytopes, valuations, and the Euler relation, Can. J. Math. 20 (1968), 1412–1424.Google Scholar
9. Shephard, G. C., The Steiner point of a convex polytope, Can. J. Math. 18 (1966), 1294–1300.Google Scholar
10. Shephard, G. C., The mean width of a convex polytope, J. London Math. Soc. 43 (1968), 207-209.Google Scholar
11. Shephard, G. C. Euler-type relations for convex polytopes, Proc. London Math. Soc. (3) 18 (1968), 597–606.Google Scholar