On multidimensional analogues of the S. N. Bernstein inequalities

In the paper the method proposed in [1] of proving of multidimensional variants of the S. N. Bernstein inequalities is generalized. Instead of the estimate
$$ \mathbf P(|Y_n|\ge r)\le C\exp\biggl\{-\frac{r^2}{8e^2L^2}\biggr\} \quad C=1+\frac{e^{5/12}}{\pi\sqrt2}\cdot\frac\Lambda{L^2}, $$
where $Y_n$'s are normalized sums of independent identically distributed vectors with the restrictions (0.1), a more precise one is established:
$$ \mathbf P(|Y_n|\ge r)\le C\exp\biggl\{-\frac{r^2}{8e^2L^2}\ \frac{\ln\frac{2L^2}\Lambda}{\ln2}\biggr\},\quad C=1+\frac{e^{13/24}}{2\pi}\sum_{s=2}^\infty s^{s/2}(e\ln2)^{-s}\sqrt s $$

A case is considered when it is possible to release from the dependence of the estimate on the dimensionality. It is indicated when the results obtained may be extended to the case of unequally distributed terms. Examples of application of various forms of the inequalities are given.