I am interested in the following question:
Problem: Given a set $V$ of vectors in $\mathbb{R}^3$, what is a necessary and sufficient condition that there exists some $A \in O_3(\mathbb{R})$, such that for any $v \in V$, $Av$ lies in the octant $\{(x,y,z): x, y, z \geq 0\}$.
Clearly, one necessary condition is that for any $v, v' \in V$, we have $v \cdot v' \geq 0$. Now, if we are working in $\mathbb{R}$ or $\mathbb{R}^2$, this is actually a sufficient condition.
However, in 3D, the condition is no longer sufficient. For example, one can produce $5$ vectors $V = \{v_1, v_2, \cdots, v_5\}$ such that $v_i \cdot v_{i + 1} = 0$ and all other inner products are positive. In this case, such an $A$ does not exist.
I feel this question is of interest on its own. Furthermore, the problem can potentially settle a conjecture I am working on, so I would appreciate any thoughts!