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In this paper we consider metric fillings of convex bodies. We show thatconvex bodies $C\subset \mathbb{R}^n$ are the unique minimal fillings of theirboundary metrics among all integral current spaces. To this end, we also provethat convex bodies enjoy the Lipschitz-volume rigidity property within thecategory of integral current spaces, which is well known in the smoothcategory. As a further application of this result, we answer a question ofPerales concerning the intrinsic flat convergence of minimizing sequences forthe Plateau problem.

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