APPROXIMATION OF CONVEX BODIES BY POLYTOPES ROLF SCHNEIDER AND JOHN ANDRE WIEACKER Every convex body can be approximated arbitrarily closely, in the Hausdorff metric, by polytopes. The present note proposes and investigates notions which are designed for describing the efficiency of such approximations. To be more precise, let X" denote the set of convex bodies (non-empty, compact, convex subsets) in ^-dimensional Euclidean space W (n ^ 2), and for K e $T' let ^ m {K) be the set of polytopes contained in K with at most m vertices. By d we denote the Hausdorff metric on jf", and for K e X n and m e N we write d m (K):=ini{d(K,P):Pe0> m (K)}. Now for K G X n and e > 0 we define m(K, s): = min {m e N : d m (K) < s} . By a result of Bronshteyn and Ivanov [1] there exists a constant c n depending only on n such that m(K,e)^c n (8/RY l - n) ' 2 (1) for all sufficiently small e > 0, where R denotes the radius of any ball containing K. (Strictly speaking, the approximating polytope constructed by Bronshteyn and Ivanov is not contained in K, but it is obvious how to obtain (1), with a different constant, from their result.) The precise asymptotic behaviour of m(K, e), as e -* 0, may be considered as a means of describing how efficiently a given convex body K can be approximated by polytopes. Guided by a certain analogy to the theory of generalized dimension (Hausdorff [5], Larman [7]), we measure this behaviour by utilizing the class H of Hausdorff functions. The elements of H are the increasing real-valued functions h defined on the nonnegative real numbers which are continuous on the right and satisfy h(0) = 0and/i(t) > Ofort > 0. Following Larman [7], we also introduce the set H* of blanketed Hausdorff functions. These are the Hausdorff functions h which satisfy the following condition. For any p > 0 there exists a positive real number k(p, h) such that h(pt)^k(p,h)h(t) (2) for all t ^ 0. Now for K e Jf" and he H we define a h (K):= Kminf m{K,e)h{e). e -> 0 + Received 1 April, 1980. [BULL. LONDON MATH. SOC, 13 (1981), 149-156]