Analytic properties and the asymptotic behavior of the area function of a Funk metric

Abstract

In Minkowski geometry the unit ball is a compact convex body $K$ containing the origin in its interior. The boundary of the body is formed by the unit vectors. We also have a so-called Minkowski functional to measure the length of vectors. By changing the origin in the interior of the body we have a smoothly varying family of Minkowski functionals. This is called the Funk metric. Under some regularity conditions the Minkowski functionals allow us to measure the volume (area) of the indicatrix bodies (hypersurfaces). Some homogenity properties provide the volume and the area to be proportional. The area as the function of the base point varying in the interior of $K$ is strictly convex [25]. This is called the area function of the Funk manifold. If the minimum is attained at the origin then $K$ is said to be balanced. The idea comes from the generalization of Brickell's theorem [6] for Finsler manifolds with balanced indicatrices [25]. As a continuation of [25] we are going to investigate analytic properties and the asymptotic behavior of the area function of a Funk manifold. We prove that the area function is locally analytic and the area can be arbitrary large near to the boundary of $K$. Therefore the minimum always attained at a uniquely determined interior point of $K$. If we apply the result to the indicatrices of a Finsler manifold point by point then the uniquely defined minima of the area functions constitute a vector field. We prove that it is differentiable. Therefore each indicatrix body can be translated in such a way that the translated body is balanced and we always have an associated Finsler manifold with balanced indicatrices. Finsler manifolds having balanced indicatrices represent a class of Finsler spaces such that the so-called Brickell's conjecture holds [6], see also [25].