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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: rolf.schneider@math.uni-freiburg.de
Franz E. Schuster
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8/104, A-1040 Wien, Austria. E-mail: schuster@geometrie.tuwien.ac.at
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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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References

1Alesker, S., Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture. Geom. Funct. Anal. 11 (2001), 244–272.CrossRefGoogle Scholar
2Alesker, S., On GLn (ℝ)-invariant classes of convex bodies. Mathematika 50 (2003), 57–61.CrossRefGoogle Scholar
3Bell, J. and Gerhold, S., On the positivity of a linear recurrence sequence. Israel J. Math. 157 (2007), 333–345.CrossRefGoogle Scholar
4Gardner, R. J., Geometric Tomography. Cambridge University Press (Cambridge, 1995).Google Scholar
5Goodey, P. R. and Weil, W., Zonoids and generalisations. In Handbook of Convex Geometry (ed. Gruber, P. M. and Wills, J. M.), North-Holland (Amsterdam, 1993), 1297–1326.CrossRefGoogle Scholar
6Groemer, H., Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press (Cambridge, 1996).CrossRefGoogle Scholar
7Klain, D. A., A short proof of Hadwiger's characterization theorem. Mathematika 42 (1995), 329–339.CrossRefGoogle Scholar
8Klain, D. A., Even valuations on convex bodies. Trans. Amer. Math. Soc. 352 (1999), 71–93.CrossRefGoogle Scholar
9Lutwak, E., Centroid bodies and dual mixed volumes. Proc. London Math. Soc. (3) 60 (1990), 365–391.CrossRefGoogle Scholar
10Schneider, R., Zu einem Problem von Shephard über die Projektionen konvexer Körper. Math. Z. 101 (1967), 71–82.CrossRefGoogle Scholar
11Schneider, R., Equivariant endomorphisms of the space of convex bodies. Trans. Amer. Math. Soc. 194 (1974), 53–78.CrossRefGoogle Scholar
12Schneider, R., Convex Bodies: the Brunn-Minkowski Theory. Cambridge University Press (Cambridge, 1993).CrossRefGoogle Scholar
13Schneider, R., Simple valuations on convex bodies. Mathematika 43 (1996), 32–39.CrossRefGoogle Scholar
14Schneider, R. and Weil, W., Zonoids and related topics. In Convexity and its Applications (ed. Gruber, P. M. and Wills, J. M.), Birkhäuser (Basel, 1983), 296–317.CrossRefGoogle Scholar