Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

Abstract

For a given convex body K in \({\Bbb R}^3\) with C2 boundary, let Pcn be the circumscribed polytope of minimal volume with at most n edges, and let Pin be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of Pcn and Pin are asymptotically regular triangles and squares, respectively, in a suitable sense.