Strictly convex functions on complete Finsler manifolds (Q345696)

Suppose that the Finsler manifold \((M,F)\) is geodesically complete with respect to forward geodesics. A function \(\phi:M\longrightarrow\mathbb{R}\) is called a convex function if for any geodesic segment \(\gamma:[a,b]\longrightarrow M\), the function \(\phi\circ\gamma:[a,b]\longrightarrow\mathbb{R}\) is convex, which means that for any \(0\leq\lambda\leq1\) and for all \(a\leq s\leq t\leq b\) we have: \[ \phi\circ\gamma((1-\lambda)s+\lambda t)\leq (1-\lambda)\phi\circ\gamma(s)+\lambda \phi\circ\gamma(t). \] If the above inequality is strict, for any \(\gamma\) and any \(0<\lambda<1\), then \(\phi\) is called strictly convex. A convex function \(\phi:M\longrightarrow\mathbb{R}\) is called exhaustion if for any \(a\in\phi(M)\), the set \(\phi^{-1}(-\infty,a]\) is compact. In this paper, it is shown that if \(\phi:M\longrightarrow\mathbb{R}\) is a strictly convex exhaustion function then every compact subgroup of the group of isometries \(I(M,F)\) has a common fixed point and the exponential map is proper at every point.