Minimal area of Finsler disks with minimizing geodesics

Abstract

We show that the Holmes–Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $\frac{6}{\pi }{r}^2$. Furthermore, we construct examples showing that the inequality is sharp and observe that equality is attained by a non-rotationally-symmetric metric. This contrasts with Berger’s conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we include a proof of the integral geometry formulas of Blaschke and Santaló for Finsler manifolds with almost no trapped geodesics.