A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric $g$ on the two-sphere, let ${\mathrm{\ell }}_{\min }(g)$ be the length of the shortest closed geodesic and ${\mathrm{\ell }}_{\max }(g)$ 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 $${\mathrm{\ell }}_{\mathrm{m}\mathrm{i}\mathrm{n}}(g{)}^2\le \pi \mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}(S^2,g)\le {\mathrm{\ell }}_{\max }(g{)}^2,$$ 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.