Quantities of the form | F(X) - G(X) | are estimated, where F and G are the convolutions of certain k-dimensional probability distributions, while X is a convex polyhedron in R^k. Estimates of the form | F(X) - G(X) | ≤ c(k)εβ(F, G, X) are proved, differing from the known ones by the presence of the factor β(F, G, X) in the right-hand side, which may turn out to be small if the polyhedron X is small in a definite sense.