This is an announcement for the paper "Rotation invariant Minkowski classes of convex bodies" by Rolf Schneider and Franz E. Schuster.

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.

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

Mathematics Subject Classification: 52A20, 33C55

Citation: Mathematika 54 (2007), 1–13

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/1207.7286

or

http://arXiv.org/abs/1207.7286