# Algebra Seminar talk

2017-03-17

Felix **Dorrek***Minkowski endomorphisms*

Abstract:

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.