Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat1-unit-side equilateral triangles glued along their boundaries is a localextremum for the length of the shortest closed geodesic among the Riemannianspheres with conical singularities of fixed area. We give an alternative proofof this theorem, which does not make use of the uniformization theorem, andextend the result to Finsler metrics.
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