Let $G$ be an $n$-dimensional compact Lie group. Let $G \times M \rightarrow M$ be a group action on a smooth manifold $M$, let $f \in C^{\infty} (M; \mathbb{R})$.
Construct a function $f_{avg}: M \rightarrow \mathbb{R}$ with the following properties:
1) $f_{avg}$ is constant on the orbits of $G$.
2) $\forall p \in M$, we have $\inf_{q \in G \cdot p} f(q) \leq f_{avg} (p) \leq \sup_{q \in G \cdot p} f(q)$.
I have no idea how to do this. For all $g \in G$ and all $p \in M$, I need $f_{avg} (g \cdot p ) = f_{avg} (p)$. Obviously, I have to define $f_{avg}$ in terms of the given $f$. In a previous problem, I showed that there exists a left-invariant volume form $\omega$ on $G$, satisfying $\int_{G} \omega = 1$. But I am not sure if I have to use that here.
I don't know if it's the right answer, but my idea would be to take the mean of $f$ along the orbits. By that I mean :
As $G$ is a compact Lie group, it has a Haar measure (with volume 1), say $\mu$. I would take $f_{avg}(p) = \int_G f(g\cdot p) \mathrm{d}\mu(g)$.
By construction, it is invariant under $G$, and as it's the mean value, it would be between the infimum and the supremum of the values of $f$ on the orbit.