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Suppose I have $N$ vectors $v_1, \ldots, v_N$ in $\mathbb{R}^{d}$ such that for $i \neq 1, N$, $v_i$ is orthogonal to all $v_j$ with $j \neq i - 1, i + 1$ and $v_1$ is orthogonal to all $v_j$ with $j \neq 2$ and $v_N$ is orthogonal to all $v_j$ with $j \neq N - 1$. (Basically for each $v_i$, $v_i$ is orthogonal to all vectors except its neighbors.)

Is it true that there exists a constant $C>0$ independent of the $v_i$ and $N$ such that $$\left|\sum_{i = 1}^{N}v_i\right| \geq C\sum_{i = 1}^{N}|v_i|?$$

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  • $\begingroup$ Why not take $C = 0$? $\endgroup$ – Pawel Kowal Aug 24 '16 at 15:53
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This not correct unless $C=0$ which is a triviality and I think that the OP wanted a positive constant. Suppose that there is such constant $C>0$, take $N=3$, and define $v_1=(1,0,\ldots,0)$, $v_2=(-1,-1,0,\ldots,0)$, $v_3=(0,1,0,\ldots,0)$. Then your inequality yields a contradiction.

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