On the dependence of the Berry-Esseen bound on dimension

Let X be a random vector with values in R-d. Assume that X has mean zero and identity covariance. Write beta = E\X\(3). Let S-n be a normalized sum of n independent copies of X. For Delta(n) = sup(Ais an element ofC) \P{Sn is an element of A} - v(A)\, where C is the class of convex subsets of R-d, and v is the standard d-dimensional normal distribution, we prove a Berry-Esseen bound Delta(n) less than or equal to 400d(1/4) beta/rootn. Whether one can remove or replace the factor d(1/4) by a better one (eventually by 1), remains an open question. (C) 2002 Published by Elsevier Science B.V.