  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.
