Let $x_1,x_2\ldots,x_n$ are vectors from $\mathbb{R}^d$. Also assume that $x_i^{\top}x_j \geq 0$ for all $i,j=1,2,\ldots,n$. I am wondering if there is an orthogonal matrix $W$ such that the entries of $Wx_i$'s are non-negative for $i=1,2,\ldots,n$.
I can prove the statement for $d=2$. Any ideas or counterexample for the cases when $d\geq 3$ will be very helpful.
I think you'll find that a circular, right-angled cone $C \subseteq \mathbb{R}^3$, say the one defined by $z \geq \sqrt{x^2+y^2}$, cannot be fit into any octant, even though $u^Tv \geq 0$ for all $u,v \in C$.