  A Finsler space is said to be geodesically reversible if each oriented
geodesic can be reparametrized as a geodesic with the reverse orientation. A
reversible Finsler space is geodesically reversible, but the converse need not
be true.
  In this note, building on recent work of LeBrun and Mason, it is shown that a
geodesically reversible Finsler metric of constant flag curvature on the
2-sphere is necessarily projectively flat.
  As a corollary, using a previous result of the author, it is shown that a
reversible Finsler metric of constant flag curvature on the 2-sphere is
necessarily a Riemannian metric of constant Gauss curvature, thus settling a
long-standing problem in Finsler geometry.
