A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric g on the two-sphere, let be the length of the shortest closed geodesic and 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 l(min)(g)(2) <= pi Area(S-2, g) <= l(max)(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. (AU)