Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T05:24:53.911Z Has data issue: false hasContentIssue false
  • English
  • Français

Decomposition of convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Wolfgang Weil
Wolfgang Weil
Affiliation:
Albert-Ludwigs-Universität, Freiburg
Get access

Extract

1. A convex body K in n-dimensional Euclidean space En (nonvoid, compact, and convex subset of En) is uniquely determined, up to translations, by its mixed volumes with other convex bodies. Therefore one might expect that relations between two convex bodies K and L correspond to relations between the mixed volumes of K, L, and other bodies. In this paper we give conditions, formulated in terms of mixed volumes, which are necessary and sufficient for a certain property of decomposability, namely, the property that L is a summand of K, and we solve some related problems.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 21 , Issue 1 , June 1974 , pp. 19 - 25
Copyright
Copyright © University College London 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Aleksandrov, A. D.. “Zur Theorie der gemischten Volumina von konvexen Körpern, I”, Mat. Sb., N. S., 2 (1937), 947–972. (Russian with German summary.)Google Scholar
2.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper (Berlin, 1934).Google Scholar
3.Erard, P. J.. “Über die zweite Fundamentalform von Flächen im Raum”, Dissertation (Zürich, 1968).Google Scholar
4.Fenchel, W. and Jessen, B.. “Mengenfunktionen und konvexe Körper”, Danske Vid. Selsk. Mat.-Fys. Medd., 16 (1938), 3.Google Scholar
5.Shephard, G. C.. “Decomposable Convex Polyhedra”, Mathematika, 10 (1963), 89–95.CrossRefGoogle Scholar
6.Weil, W.. “Ein Approximationssatz für konvexe Körper”, Manuscripta math., 8 (1973), 335–362.CrossRefGoogle Scholar
7.Weil, W.. “Über den Vektorraum der Differenzen von Stützfunktionen konvexer Körper”, Math. Nachr., 59 (1974), 353–369.CrossRefGoogle Scholar