Polar representations of compact groups and convex hulls of their orbits

The paper contains a characterization of compact groups G⊆GL(v)G⊆GL(v), where vv 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 equivalent to the property to be polar. In general, G satisfies SP if and only if it is polar and its Weyl group is a Coxeter group.