Rotation Invariant Minkowski Classes of Convex Bodies

Abstract

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝ n which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝ n 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 T 1 , T 2 such that M + T 1 = T 2 , and T 1 , T 2 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.