The error of polytopal approximation with respect to the symmetric difference metric and theL p metric

LetM be a convex body in ℝ ^d withC ₊ ³ boundary. Polytopal approximation ofM with respect to the symmetric difference metric (or theL _p metric) is considered, if the approximating polytope has at mostn facets (or at mostn vertices). The asymptotic behavior of the distance of the best approximating polytope is well-known; it is of order . This paper provides an estimate of order for the error term. Supported by OTKA, Hungary. The paper was written during a visit at University College London.