2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/152187I classify the Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature and whose geodesics are the great circles. Modulo diffeomorphism, there is a 2-parameter family of such Finsler structures, only one of which is homogeneous or symmetric, namely the Riemannian one. I discuss the history of the problem and its relation with Hilbert's Fourth Problem and the calculus of variations.AMS-TeX v2.1, amsppt.sty (v. 2.1d), 38 pagesDifferential Geometry53C60 (Primary) 53A20 (Secondary)text