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 $n=2$.
