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

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.