I have two real vectors $v = (v_1,\ldots,v_n)$ and $u = (u_1,\ldots,u_n)$. I know that the dot product of $u$ and $v$ is larger than $\delta > 0$:
$\langle u,v \rangle \ge \delta$.
What would be an interesting condition on $u$, $v$ or both such that I have $u_i v_i \ge f(\delta)$ for each coordinate $i$ with some real function $f()$?
For example, one condition that I thought about is that for any $j \le n$ we have:
$\sum_{i \neq j} u_i v_i \le \delta/2$
and then we can get that $u_i v_i \ge \delta/2$ for every $i$ using triangle inequality.
You can assume that $||u|| = 1$ and that $||v||$ is bounded by some $M$ (L-2 norms here).
Any help appreciated.