Is it true that on every normed vector space $V$ of dimension $n$ there exists a basis of norm $1$ vectors $v_i$, such that $\|\sum_{i=1}^n\epsilon_iv_i\|\geq 1$ for all possible combinations of $\epsilon_i=\pm 1$? Clearly this is true for an orthonormal basis and it is also not hard to show that it is true if $n=2$.
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2$\begingroup$ Take the unit ball in your norm and take the circumscribed parallelepiped of the least volume. The points of tangency will lie at the centers of its faces and thus produce the basis you are looking for. $\endgroup$– fedjaCommented Jun 1, 2012 at 23:04
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2$\begingroup$ Fedja just gave you the construction that prove's Auerbach's lemma. $\endgroup$– Bill JohnsonCommented Jun 1, 2012 at 23:39
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$\begingroup$ Great, thank you! I was hoping that a result like this exists. $\endgroup$– DanielCommented Jun 2, 2012 at 13:41
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$\begingroup$ Is there a way to close questions like this, which are answered in comments? $\endgroup$– Yulia KuznetsovaCommented Jun 4, 2012 at 9:07
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