On concentration of distributions of sums of independent random vectors on bounded sets

Bounds are obtained for the concentration function
$$ Q_n (A) =\sup_{x\in\mathbf{R}^k}{\mathbf P}(S_n \in A + x) $$
of sums $S_n=X_1+\cdots+X_n $ of independent random vectors $X_1,\ldots,X_n$ with values in the $k$-dimensional Euclidean space $\mathbf{R}^k$ on bounded Borel sets $A$ in $\mathbf{R}^k$.