Polar representations of compact groups and convex hulls of their orbits

TL;DR

This paper characterizes compact groups acting on real vector spaces based on when the convex hulls of their orbits form a semigroup under Minkowski addition, linking group properties to convex geometric structures.

Contribution

It provides a complete characterization of groups with this property, connecting polar and Coxeter group structures to convex hull semigroup behavior.

Findings

Finite groups with this property are exactly Coxeter groups.

Connected groups with this property are exactly polar groups.

General groups with this property are polar with Coxeter Weyl groups.

Abstract

The paper contains a characterization of compact groups $G\subseteq\GL(V)$, where $V$ is a finite dimensional real vector space, which have the following property \SP{}: the family of convex hulls of $G$-orbits is a semigroup with respect to the Minkowski addition. If $G$ is finite, then \SP{} holds if and only if $G$ is a Coxeter group; if $G$ is connected then \SP{} is true if and only if $G$ is polar. In general, $G$ satisfies \SP{} if and only if it is polar and its Weyl group is a Coxeter group.