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I have a problem that is geometrically obvious in the 2D case. It reads as

Suppose that $V$ is a $n$ dimensional Euclidean space, and $e_1,\cdots,e_n$ is a basis satisfying $(e_i,e_j)\leq 0$ for $i\neq j$.

Prove that

(1) there exists a non zero vector $v\in V$ such that $(e_i,v)\geq 0$ for $1\leq i\leq n$;

(2) Suppose $v=a_1e_1+\cdots+a_ne_n\in V$ is a vector satisfying the properties in (1), then $a_i\geq 0$, $i=1,\cdots,n$;

(3) Suppose $u=b_1e_1+\cdots+b_ne_n\in V$ is another vector satisfying the properties in (1), and define $$w=c_1e_1+\cdots+c_ne_n,\quad c_i=\min\{a_i,b_i\}.$$ Then $w$ satisfys the properties in (1) also.

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Ad 1. I think the vector $\sum e_i$ will do, and this can be proven with triangular inequality.

You may also need that the max inner product of $e_1$ with $e_2$ or $e_3$ will be smaller than with $e_1$ and $e_2+e_3$

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  • $\begingroup$ I find it is difficult to show $\sum e_i$ verifies... $\endgroup$
    – xldd
    Oct 17 '14 at 3:16

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