Reimann surfaces with shortest geodesic of maximal length

I describe Riemann surfaces of constant curvature –1 with the property that the length of its shortest simple closed geodesic is maximal with respect to an open neighborhood in the corresponding Teichmüller space. I give examples of such surfaces. In particular, examples are presented which are modelled upon (Euclidean) polyhedra. This problem is a non-Euclidean analogue of the well known best lattice sphere packing problem.Supported by the Schweizerischer Nationalfonds zur Förderung wissenschaftlicher Forschung