Convolutions and multiplier transformations of convex bodies

TL;DR

This paper characterizes rotation intertwining maps on convex bodies that are linear with respect to Minkowski and Blaschke addition, showing they are represented by spherical convolutions and classifying even homomorphisms.

Contribution

It provides a complete classification of Blaschke-Minkowski homomorphisms and reveals their structure via spherical convolution operators, extending understanding of convex body transformations.

Findings

All such maps are represented by spherical convolution operators.

Even Blaschke-Minkowski homomorphisms behave similarly to the projection body operator.

If such a homomorphism maps a convex body to a polytope, it is a scalar multiple of the projection body operator.

Abstract

Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.