Rotation invariant Minkowski classes of convex bodies
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.
Keywords
Cite
@article{arxiv.1207.7286,
title = {Rotation invariant Minkowski classes of convex bodies},
author = {Rolf Schneider and Franz E. Schuster},
journal= {arXiv preprint arXiv:1207.7286},
year = {2012}
}