A remark on convex functions and isometry groups of Lorentzian manifolds

Let $M$ be a Lorentzian manifold, $G$ be a connected and compact subgroup of the isometries of $M$, and $f$ be a differentiable function on $M$. We show that if $f$ is strictly convex without minimum along causal geodesics, then all $G$-orbits are fixed points of $G$ or they are spacelike submanifolds of $M$. Also, if $f$ is $G$-bounded and strictly convex along causal geodesics and it has a minimum point, then $G$ has a fixed point or it has a spacelike orbit.