Let $x$ be real vector with $\|x\|_1=x_1+\ldots +x_{2n}$.
How to bound from above $(x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})$ by $l_2$ norm of the vector $x$.
Of course, using $\|x\|\leq\sqrt {2n}\|x\|_2$ I can bound $ (x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})\leq\|x\|^2_1\leq 2n\|x\|_2^2 $
But I would like to get an upper bount not greater then $1/2\|x\|_2^2$. Is it possible to get such a bound?