Volume approximation of convex bodies by inscribed polytopes

Abstract

A convex body C in d-dimensional Euclidean space E ~ is a compact convex subset of lE a with non-empty interior. We say that C is of differentiability class ~k if its boundary considered as a manifold is of class ~k. The symmetric difference metric 6 s, also called Nikodym metric, on the space of convex bodies is defined for any two convex bodies as the volume of their symmetric difference. Given a convex body C, let ~ = ~ ( C ) (n = d + l, d + 2,...) denote the family of convex polytopes P having at most n vertices and being inscribed into C, that is, all vertices are on the boundary bdC of C. It is well known that for each n there is a polytope P, ~ ~ such that ~s(C, P.) = ~s(C, ~,~)( = inf {~s(C, P): P ~ ~.~}).