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Rotation Invariant Minkowski Classes of Convex Bodies

Part of: General convexity

Published online by Cambridge University Press:  21 December 2009

Rolf Schneider  and
Franz E. Schuster
Rolf Schneider
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität, Eckerstraße 1, D-79104 Freiburg i. Br., Germany. E-mail: [email protected]
Franz E. Schuster
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8/104, A-1040 Wien, Austria. E-mail: [email protected]
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Abstract

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝn which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1, T2 such that M + T1 = T2, and T1, T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 33C55: Spherical harmonics
Type
Research Article
Information
Mathematika , Volume 54 , Issue 1-2 , December 2007 , pp. 1 - 13
Copyright
Copyright © University College London 2007

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