On the error of the Gaussian approximation for convolutions

Let $F_n$ be the distribution in $R^k$ of the sum of independent random vectors $\xi_1,\dots,\xi_n$, and $G_n$ be the normal distribution with the same means and соvariances as $F_n$.
If $\mathfrak L_{\Pi}$ is the Lévy–Prohorov distance between distributions in $R^k$ defined by the Euclidean norm in $R^k$, Theorem 1 yields the estimate
\begin{gather*} \mathfrak L_{\Pi}(F_n,G_n)\le ck^{1/4}\mu_1^{1/4}[|\ln\mu_1|^{1/2}+(\ln k)^{1/2}], \\ \mu_1=\mathbf E|\xi_1-\mathbf E \xi_1|^3+\dots+\mathbf E|\xi_n-\mathbf E \xi_n|^3, \end{gather*}
with $c$ being an absolute constant.
A similar bound holds when $\mathfrak L_{\Pi}$ is defined using a non-Hilbert but sufficiently smooth norm in $R^k$ (Theorem 2).
Finite-dimensional bounds are used in Section 2 to obtain coarse power convergence rate in the multidimensional invariance principle for a random walk.