Inequalities for the distribution of the length of random vector sums

Starting from a combinatorial proof of the inequality
$$ \mathbf P(|\xi+\eta|\ge x)\ge\frac{1}{2}\mathbf P^2(|\xi|\ge x). $$
where $\xi$ and $\eta$ are independent random vectors in a $d$-dimensional Euclidean space, continuous analogues of the combinatorial model are constructed, which enable to deduce inequalities similar to the above.