About the error term for best approximation with respect to the Hausdorff related metrics

Let M be a convex body with C-+(3) boundary in R-d, d greater than or equal to 3, and consider a polytope P-n (or P-(n)) with at most n vertices (at m ost n facets) minimizing the Hausdorff distance from M. It has long been kn own that as n tends to infinity, there exist asymptotic formulae of order n (-2/(d-1)) for the Hausdorff distances delta (H)(P-n, M) and delta (H)(P-(n ), M). In this paper a bound of order n(-5/(2(d-1))) is given for the error of the asymptotic formulae. This bound is clearly not the best possible, a nd Gruber [9] conjectured that if the boundary of M is sufficiently smooth, then there exist asymptotic expansions for delta (H)(P-n, M) and delta (H) (P-(n), M). With the help of quasiconformal mappings, we show for the three -dimensional unit ball that the error is at least f(n) . n-(2) where f(n) t ends to infinity. Therefore in this case, no asymptotic expansion exists in terms of n(-2/(d-1)) = n(-1).