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Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $π$ and equals $π$ if and only if the metric is Riemannian and Zoll.