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