Faces of invariant convex sets in representations of nontrivial copolarity

Let $(V,G)$ be an orthogonal representation of a compact Lie group $G$ with nontrivial copolarity, and $\mathrm{\Sigma }$ a fat section of $(V,G)$. If $E$ is a $G$-invariant compact convex set in $V$, then $P=E\cap \mathrm{\Sigma }$ is a convex set in $\mathrm{\Sigma }$. We prove that up to conjugacy the face structure of $E$ is completely determined by that of $P$ and that a face of $E$ is exposed if and only if the corresponding face of $P$ is exposed. Our result generalizes the result proved by Leonardo Biliotti, Alessandro Ghigi and Peter Heinzner in the case where $(V,G)$ is a polar representation.