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

Polar representations of compact groups and convex hulls of their orbits