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I have proved a simple Lemma that I need for a larger result, and I was wondering whether it is actually another more famous result in disguise.

The lemma says that for any set of vectors in $\mathbb{R}^n$, it is possible to choose a set of coefficients in ${-1,+1}$ such that the weighted sum of the vectors is close to the origin. The interesting thing (to me, non-mathematician) is that how close depends on the dimensionality of the space and not on the number of vectors.

Lemma Consider a set of $m\geq1$ vectors $\{v_{i}\}_{i=1}^{m}$ in $\mathbb{R}^{n}$, $n\geq1$, such that $||v_{i}||_{\infty}\leq c$. Then there exists a vector of coefficients $\{x_{i}\}_{i=1}^{m}$ , $x_{i}\in\{-1,+1\}$, such that $\left\Vert \sum_{i=1}^{m}x_{i}v_{i}\right\Vert \leq nc$ `.

(Note that the bound depends on the dimension $n$, but not the number of vectors $m$.)

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up vote 10 down vote accepted

I don't have access right now to search tools, but the result is is due to Steinitz and might be described in

Kadets, Mikhail I.; Kadets, Vladimir M. (1997). "Chapter 2.1 Steinitz's theorem on the sum range of a series, Chapter 7 The Steinitz theorem and B-convexity". Series in Banach spaces: Conditional and unconditional convergence. Operator Theory: Advances and Applications. 94.

Anyway, the right book by V. Kadets contains the result and best known estimates for the constant in terms of the dimension.

EDIT: I asked Gideon Schechtman for a good reference, and he provided


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Thanks! It seems that the book contains exactly this result (as much as I can see from the Google Books preview; I will order it for our library). – Andrea Apr 8 '12 at 18:29

It sounds like a Littlewood-Offord-type result to me, but I'll have to leave it to the experts to weigh in on this.

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Thanks for mentioning Littlewood-Offord problems (, it might be useful to me in the future. – Andrea Apr 8 '12 at 18:34

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