Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T13:25:22.258Z Has data issue: false hasContentIssue false
  • English
  • Français

Simple valuations on convex bodies

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Rolf Schneider
Rolf Schneider
Affiliation:
Mathematisches Institut, Albertstr. 23 b, D-79104 Freiburg i.Br., Germany.
Get access

Abstract

We determine all continuous translation invariant simple valuations on the space of convex bodies in d-dimensional Euclidean space.

MSC classification

Secondary: 52A39: Mixed volumes and related topics
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 32 - 39
Copyright
Copyright © University College London 1996

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.Goodey, P. and Weil, W.. The determination of convex bodies from the mean of random sections. Math, Proc. Camb. Phil. Soc., 112 (1992), 419–430.CrossRefGoogle Scholar
2.Hadwiger, H.. Vorlesungen über Inhalt, Oberflàche und Isoperimetrie (Springer, Berlin, 1957).CrossRefGoogle Scholar
3.Klain, D. A.. A short proof of Hadwiger's characterization theorem. Mathematika, 42 (1995), 329–339.CrossRefGoogle Scholar
4.McMullen, P.. Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3), 35 (1977), 113–135.CrossRefGoogle Scholar
5.McMullen, P.. Continuous translation invariant valuations on the space of compact convex sets. Arch. Math., 34 (1980), 377–384.CrossRefGoogle Scholar
6.McMullen, P.. Valuations and dissections. In Handbook of Convex Geometry, eds. Gruber, P. M. and Wills, J. M. (Elsevier, Amsterdam, 1993), pp. 933–988.CrossRefGoogle Scholar
7.McMullen, P. and Schneider, R.. Valuations on convex bodies. In Convexity and Its Applications, eds. Gruber, P. M. and Wills, J. M. (Birkhàuser, Basel, 1983), pp. 170–247.CrossRefGoogle Scholar
8.Müller, C.. Spherical Harmonics. Lecture Notes in Math. 17 (Springer, Berlin, 1966).CrossRefGoogle Scholar
9.Schneider, R.. Equivariant endomorphisms of the space of convex bodies. Trans. Amer. Math. Soc., 194 (1974), 53–78.CrossRefGoogle Scholar
10.Schneider, R.. Additive Transformationen konvexer Körper. Geom. Dedicata, 3 (1974), 221–228.CrossRefGoogle Scholar
11.Schneider, R.. Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia Math. Appl., 44 (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
12.Vilenkin, N. J.. Special Functions and the Theory of Group Representations. Transl. Math. Monographs, vol. 22 (Amer. Math. Soc, Providence, Rh. I., 1968).CrossRefGoogle Scholar