Convolutions and multiplier transformations of convex bodies
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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.References
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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
- MR Author ID: 764916
- Email: fschuster@osiris.tuwien.ac.at
- Received by editor(s): July 4, 2005
- Received by editor(s) in revised form: December 7, 2005
- Published electronically: May 11, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 5567-5591
- MSC (2000): Primary 52A20; Secondary 52A40, 43A90
- DOI: https://doi.org/10.1090/S0002-9947-07-04270-5
- MathSciNet review: 2327043