Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into some finite dimensional orthogonal representation.
Question: suppose now that $M$ is a manifold with corners. Does the conclusion still hold?
I can't imagine how such a statement could fail to generalize, but I'm hoping that there is some trick which would avoid having to rewrite the proof. (Or perhaps a reference?)