The volume and lengths on a three sphere
arXiv:math/0003102
Abstract
We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.
The revision improves the main result and simplifies the proof