Local extremality of the Calabi–Croke sphere for the length of the shortest closed geodesic.

Recently, Balacheff [‘A local optimal diastolic inequality on the two-sphere’, J. Topol. Anal. 2 (2010) 109–121] 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 carries over to the Lipschitz distance topology. Furthermore, we extend the result to Finsler metrics. [ABSTRACT FROM AUTHOR]