1
$\begingroup$

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.

  • 0
    btw, your example is not true. If for any $j \leq n$ you have $\sum_{i\neq j} u_i v_i \leq \delta / 2$, then summing over $j$ you get $$\sum_j \sum_{i\neq j} u_iv_i = (n-1) \langle u,v\rangle \leq n\delta / 2$$ or equivalently $\langle u,v\rangle \leq n\delta / (2n - 2) < \delta$ if $n > 2$, which contradicts your assumption that the inner product is bounded below by $\delta$.2011-02-14

1 Answers 1