A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric g on the two-sphere, let ℓmin(g) be the length of the shortest closed geodesic and ℓmax(g) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalities ℓmin(g)2≤π Area(S2,g)≤ℓmax(g)2, hold, and each of these two inequalities is an equality if and only if the metric g is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.