I have a set of vectors $x^1, \ldots , x^m \in R^n$
$x^i_j \ne 0$. I know them!
I generate random vectors $y^1 , \ldots , y^m \in R^n$ , but $||y|| = 1$
It is possible that for some $i,j$, $y^i_j = 0$
$(x,y)$ - scalar multiplication
Now, finally I want to estimate this expression: $t = \sqrt[k]{\frac{(x^1,y^1)\cdots(x^k,y^k)}{(x^1,y^2)(x^2,y^3)\cdots(x^k,y^1)}},$ for any $k \le m $ and any ${y^1 , \ldots , y^m}$
I want find some $t' = t(\{x^i\}) : t \le t'$