Rotation invariant Minkowski classes of convex bodies

TL;DR

This paper demonstrates that non-centrally symmetric convex bodies can be approximated by universal bodies within rotation invariant Minkowski classes, strengthening previous results with a simpler proof.

Contribution

It shows that every non-centrally symmetric convex body has a close linear image that is universal, enhancing and simplifying earlier findings.

Findings

Non-centrally symmetric bodies can be approximated by universal bodies.

Universal bodies form a dense class in the space of convex bodies.

The results extend to centrally symmetric convex bodies with modifications.

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.