  Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat
1-unit-side equilateral triangles glued along their boundaries is a local
extremum for the length of the shortest closed geodesic among the Riemannian
spheres with conical singularities of fixed area. We give an alternative proof
of this theorem, which does not make use of the uniformization theorem, and
extend the result to Finsler metrics.

Differential Geometry