This is an announcement for the paper "Convolutions and multiplier transformations of convex bodies" by Franz E. Schuster.

Abstract: Rotation intertwining maps from the set of convex bodies in Rn 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.

Archive classification: math.MG math.DG math.FA

Mathematics Subject Classification: 52A20, 43A90, 52A40

Citation: Trans. Amer. Math. Soc. 359 (2007), 5567–5591

Submitted from: franz.schuster@tuwien.ac.at

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1207.7252

or

http://arXiv.org/abs/1207.7252