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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 $t' = t(\{x^i\}) : t \le t'$

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Since e.g. $(x^1, y^2)$ could be very close to $0$, no such bound is possible.

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I agree. I think something should be said about being able to choose the $y^j$ such that the factors in the numerator don't cancel at least one of the small terms in the denominator, and one should probably assume the $x_i$ aren't all colinear, but yes, this can't have a finite upper bound in terms of the $x^j$. – Antonio Vargas May 25 '12 at 19:55

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