On the proximity of distributions of successive sums on convex sets and in the Prokhorov metric

Let $X_1$, $X_2$, ... be independent identically distributed random vectors in a $d$-dimensional Euclidean space with distribution $F$. Then $S_n=X_1+...+X_n$ has distribution $F^n$ (degrees of measures are understood in the sense of convolution). Let $R(F,G)=\sup|F(A)-G(A)|$, where the supremum is taken over all convex subsets of $d$-dimensional Euclidean space. Then for any nontrivial distributions $F$ there is $c(F)$ depending only on $F$ and such that $R(F^n,F^{n+1})$ does not exceed $c(F)$ divided by the square root of $n$, for any natural $n$. A distribution $F$ is considered trivial if it is concentrated on an affine hyperplane that does not contain the origin. It is clear that for such $F$ we have $R(F^n,F^{n+1})=1$.
A similar result is also obtained for the Prokhorov distance between the distributions of vectors $S_n$ and $S_{n+1}$ normalized by the square root of $n$. Moreover, the statement remains true for arbitrary distributions, including trivial ones.