Estimates of the proximity of successive convolutions of the probability distributions on the convex sets and in the Prokhorov distance

Let $X_1,\dots, X_n,\dots$ be independent identically distributed $d$-dimensional random vectors with common distribution $F$. Then $S_n = X_1+\dots+X_n$ has distribution $F^n$ (degree is understood in the sense of convolutions). Let
$$\rho(F,G) = \sup_A |F\{A\} - G\{A\}|,$$
where the supremum is taken over all convex subsets of $\mathbb R^d$. Basic result is as follows. For any nontrivial distribution $F$ there is $c(F)$ such that
$$\rho(F^n, F^{n+1})\leq \frac{c(F)}{\sqrt n}$$
for any natural $n$. The distribution $F$ is considered trivial if it is concentrated on a hyperplane that does not contain the origin. Clearly, for such $F$
$$\rho(F^n, F^{n+1}) = 1.$$
A similar result is obtained for the Prokhorov distance between distributions normalized by the square root of $n$.