Approximation of smooth convex bodies by circumscribed polytopes with respect to the surface area

摘要

Let K be a convex body with C 2 boundary in the Euclidean d-space. Following the work of L. Fejes Tóth, R. Vitale, R. Schneider, P.M. Gruber, S. Glasauer and M. Ludwig, best approximation of K by polytopes of restricted number of vertices or facets is well-understood if the approximation is with respect to the volume or the mean width. In this paper we consider the circumscribed polytope P _(n) of n facets with minimal surface area, and present an asymptotic formula in terms of n for the difference of surface areas of P _(n) and K.