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Simple valuations on convex bodies

Published online by Cambridge University Press:  26 February 2010

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

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

MSC classification

Type
Research Article
Copyright
Copyright © University College London 1996

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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), 419430.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), 329339.CrossRefGoogle Scholar
4.McMullen, P.. Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3), 35 (1977), 113135.CrossRefGoogle Scholar
5.McMullen, P.. Continuous translation invariant valuations on the space of compact convex sets. Arch. Math., 34 (1980), 377384.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. 933988.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. 170247.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), 5378.CrossRefGoogle Scholar
10.Schneider, R.. Additive Transformationen konvexer Körper. Geom. Dedicata, 3 (1974), 221228.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