Algebra Seminar talk
Let $K^n$ denote the set of convex bodies (compact and convex sets) in $\mathbb R^n$. This set is endowed with two important structures. There is an algebraic structure, Minkowski addition +, and a topological one, Hausdorff distance. The cone $(K^n , +)$ is complicated - most convex bodies are extremal elements. Investigating endomorphisms of this cone, with possibly additional properties, is one approach to gain a better understanding of its structure.
In 1974 Schneider introduced the notion of Minkowski endomorphisms. A Minkowski endomorphism is a continuous, and SO(n)-equivariant map $\Phi : K^n \to K^n$ that is Minkowski additive, i.e. satisfying
$\Phi(K + L) =\Phi K + \Phi L$ for all $ K, L \in K^ n$.
While Schneider was able to fully characterize Minkowski endomorphisms in the plane, in higher dimensions much less is known.
In this talk, we will discuss a representation result for Minkowski endomorphisms that goes back to Kiderlen. Then we are going to establish (prove if time permits) that Minkowski endomorphisms are necessarily uniformly continuous in the Hausdorff metric. As a consequence we obtain that any Minkowski endomorphism can be described by a (signed) Borel measure on the sphere. Finally, we will talk about necessary and sufficient conditions for such a measure to generate a Minkowski endomorphism.