Besicovitch-type inequality for closed geodesics on 2-dimensional spheres

Abstract: We prove the existence of a constant $C > 0$ such that for any Riemannian metric $g$ on a 2-dimensional sphere $S^2$, there exist two distinct closed geodesics with lengths $L_{1}$ and $L_{2}$ satisfying $L_{1} L_{2} \leq C \cdot \operatorname{Area}(S^2, g)$.