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Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact?

Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i x_i=0$ if and only if $c_i\le\sum_{j\neq i}c_j, \forall i$.

The necessity is easy to prove. (My proof of the necessity: If $\sum_{i=1}^n c_i x_i=0$, then $c_i x_i = -\sum_{j\neq i}c_j x_j$. So $\| c_i x_i \|=c_i\le\sum_{j\neq i}\| c_j x_j\|=\sum_{j\neq i}c_j)$.

But how to prove the sufficiency? That is: if $c_i\le\sum_{j\neq i}c_j, \forall i$, can we always find some $x_i$ such that $\sum_{i=1}^n c_i x_i=0$? It seems a very basic result, but not easy to prove for me.

Thank you for your help. Shiyu

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    The statement could be written a lot clearer. Given the $\{c_i\}$, you are asking when some configuration of $\{x_i\}$ exists. Also, you require d>1. The critical case is $d=2$.2011-03-28

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Hint:

Assume $c_1 \le c_2 \le \dots \le c_n$.

Can you try to find a $k$ such that $c_1 + \dots + c_{k-1}$, $c_k$ and $c_{k+1} + \dots + c_n$ form the sides of a triangle?

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    @Shiyu: I don't think so. $1,2,3,4$ we can get two triangles: $1+2,3,4$ and $2,1+3,4$2011-03-26