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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Convolutions and multiplier transformations of convex bodies

Author: Franz E. Schuster
Journal: Trans. Amer. Math. Soc. 359 (2007), 5567-5591
MSC (2000): Primary 52A20; Secondary 52A40, 43A90
Published electronically: May 11, 2007
MathSciNet review: 2327043
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Abstract: Rotation intertwining maps from the set of convex bodies in $ \mathbb{R}^n$ into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.

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Additional Information

Franz E. Schuster
Affiliation: Institut für Diskrete Mathematik and Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/1046, 1040 Wien, Austria

Keywords: Convex bodies, Minkowski addition, Blaschke addition, rotation intertwining map, spherical convolution, spherical harmonic, multiplier transformation, projection body, Petty conjecture
Received by editor(s): July 4, 2005
Received by editor(s) in revised form: December 7, 2005
Published electronically: May 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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