We introduce a new notion of metric for the class of convex bodies in d -dimensional Euclidean space, measuring distance based on the intrinsic volume difference (or quermassintegral difference). This novel metric serves as a basis for establishing improved upper bounds for the asymptotic optimal approximation of the Euclidean unit ball by a polytope in general position. Our results demonstrate an improvement by a factor of the dimension when compared to classical results on the asymptotic approximation of the intrinsic volume difference with polytopes restricted to the domain of the ball. In our proof, we rely on a probabilistic construction to bound the asymptotic optimal polytope by random polytopes and utilize integral formulas by Blaschke--Petkantschin and Chern.