Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

Abstract

In this paper we consider metric fillings of boundaries of convex bodies. We show that convex bodies are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As further applications of this result, we prove a variant of Lipschitz-volume rigidity for round spheres and answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.