Simultaneous random polytope approximation characterizes the Euclidean ball

In this work, we present a new geometric characterization of the Euclidean ball based on approximation by random polytopes. We investigate when a single boundary sampling density is asymptotically optimal for two random polytope approximation problems, and show that for a convex body in $\R^d$ with $C^2$ boundary and everywhere positive principal curvatures, one fixed density minimizes the leading expected errors for any two distinct intrinsic volume differences of inscribed random polytopes if and only if the body is a Euclidean ball. We show that an analogous characterization holds for the volume and mean width differences of circumscribed random polytopes, and that uniform sampling gives the converse statement in both models. We also determine the unique density minimizing the average of two normalized leading errors, and obtain an explicit positive gap for every nonball. The proofs combine classical random polytope asymptotics, Hölder's inequality, and a curvature rigidity argument based on the Hsiung--Minkowski identities and Newton--Maclaurin inequalities.