  I 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.
