On the speed of convergence in the multi-dimensional central limit theorem

Estimates, which are uniform with respect to the class of convex sets, of the rate of convergence in the multi-dimensional central limit theorem for a sequence of identically distributed independent random variables with finite third moments are obtained. In the particular case corresponding to distribution functions a better estimate than that given by the general theorem is obtained. An example is constructed to show that the estimate obtained in this case, in some sence, cannot be improved