A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this talk will establish that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round. This work is joint with Ben Schmidt. Additionally, we will discuss the relationship between these half-geodesics (more generally 1/k-geodesics) and the Gromov-Hausdorff notion of convergence.