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.