Diastolic inequalities for the distance function min-max width of spheres

In this article, we investigate inequalities involving the min-max width associated to the distance function and the area of two-dimensional Riemannian spheres. This geometric invariant naturally arises as min-max methods are applied for the distance function defined on the space of pairs of points. Our methods involve careful constructions of sweepouts that can be performed under the hypothesis that the Riemannian sphere does not contain stable closed geodesics. Improved bounds are obtained under further geometric assumptions (positive curvature, pinching condition, and antipodal symmetry). In the cases of a pinching condition or antipodal symmetry, our inequalities are optimal.