On families of Finsler metrics

In this paper, we answer some natural questions concerning symmetrisation and more general combinations of Finsler metrics, with a view to applications to Funk and Hilbert geometries and metrics on Teichmüller spaces. The metrics on Teichmüller space that we consider are the Thurston metric and the earthquake metric, both introduced by Thurston. The first metric has been thoroughly studied over the last couple of decades, and the second over the last few years. The Funk metric and its symmetrisation, the Hilbert metric, are classical metrics that have been investigated in the contexts of hyperbolic geometry and geometric function theory and, more recently, in the settings of computational geometry, information geometry, and machine learning. For a general nonsymmetric Finsler metric on a smooth manifold, we introduce two different families of metrics that contain, as special cases, the arithmetic and max symmetrisations, respectively, of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, including their geodesics and completeness, the conditions under which such metrics are Finsler, and the shapes of their unit balls when they are Finsler. We address such questions, in particular, in the setting of Funk and Hilbert geometries and in that of the Teichmüller spaces of several kinds of surfaces equipped with Thurston-like asymmetric metrics.