Let $G$ be a Lie group acting properly on a smooth manifold $M$, and equip $M$ with a Riemannian manifold that is adapted to the foliation by orbits. The celebrated theorem of Palais is that there exist slices for the action.

Is there any work that find estimates for the maximum radii of slices in the given metric, in terms of other geometric data?

I'm particularly interested in the case when we have an open set $U$ in $M$ that is invariant, and when we can say $U$ has a global slice. Is is possible to say that if the diameter of $U$ is smaller than some quantity, then it has a global slice?