A characterization of round spheres in terms of half-geodesics

A half-geodesic is a closed geodesic that realizes the distance between every pair of its points. The great circles on the round sphere are examples of half-geodesics. First I’ll show that the Blaschke condition (injectivity radius equals diameter) is equivalent to every geodesic being a half-geodesic. Then I’ll provide a characterization of the round metric on the sphere via half-geodesics. These results use techniques from Morse theory and recent results by Radeschi-Wilking and Lin-Schmidt.