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

Abstract

In this paper we consider metric fillings of convex bodies. We show that convex bodies $C\subset {\mathbb{R}}^n$ 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 a further application of this result, we answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.

25 pages, 1 figure, v3: we have added Corollary 1.3 concerning the LV-rigidity of the sphere. In version 2 we had already added Theorem 1.1, a filling minimality result for convex bodies