Bodies of Constant Width

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, T 2 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.