I have a set of points in N-dimensional space. I want to find out whether they could, in principle, be rotated to lie solely in the positive orthant of space. Is there a property of these points that would guarantee this rotation exists?
If this question proves too difficult, perhaps another related one will be easier. I have the linear dynamical system and starting point that generated these points. Is there a property of linear dynamical systems which, if satisfied, ensure that under some rotation the trajectory will lie solely in the positive orthant?
To give an idea of the kind of thing I'm looking for (if it exists!): in 2D if the dot products between the set of points are all positive you can guarantee that there's some rotation that could rotate the set of points into the positive quadrant, and if a dot product is negative you know there is no such rotation. This is a nice easy test I can apply. However, this approach doesn't generalise (in 3D points can have all positive dot products, but cannot be rotated into the positive octant), can you think of one that will generalise?
Later edit - a few things I've been considering:
People study positive linear systems: linear dynamical systems, $\dot{\boldsymbol{x}}(t) = \boldsymbol{Ax}(t)$, in which the trajectory, $\boldsymbol{x}(t)$, never leaves the positive orthant. It turns out if the matrix $\boldsymbol{A}$ is Metzler, meaning all off-diagonal elements are non-negative, then a trajectory starting in the positive orthant will stay there. This is related to the Perron-Frobenius Theorem which I don't know so much about.
So it seems another way of framing my question would be: you have your linear dynamical system in an arbitrary basis. Can you tell whether the dynamics matrix $\boldsymbol{A}$ is Metzler under some rotation? The Perron-Frobenius theorem talks about rotation invariant things like eigenvalue spectra which gives me hope that this could be a constructive route, but I haven't been able to fit things together yet... Maybe the clever people on this site will be able to!
(Which is similar to this old question: Converse of Perron Frobenius Theorem: Necessary and Sufficient Conditions for positivity (or non negativity))
Not a solution but some simple comments rather.
First off, one can normalize vectors $x_k$ i.e. multiply them by a positive scalar so that we can assume that $||x_k||$ = 1. Then we look for a unitary operator, say $U$ such that $U(x_k)$ has non-negative coordinates for any $k \in \{1,2,....m\}$. If such $U$ can be found then for any subset $S \subset \{1,2,...,m\}$ we would have $$||\sum_{k \in S} U(x_k)|| >= (\sum_{k \in S} ||U(x_k)||)^{\frac{1}{2}} $$ I mean, $||.||$ here is assumed to be Euclidean i.e. $l_2$ norm and for such norms it is not difficult to see that the above inequality holds (again $U(x_k)$ has nonnegative coordinates). Since $U$ is unitary we can now formulate necessary condition for the set of (normalized) points to be rotated into positive orthant: $$||\sum_{k \in S} x_k|| >= \sqrt |S| $$ for any subset $S \subset \{1,2,...,m\}$
It is easy to see that this conditions is also sufficient in $R^{1}$ and $R^{2}$.
For n=1 it means that all vectors point in the same direction, and for n=2 we just take two vectors with the biggest angle between them and the condition above insures that this angle is not bigger than $\frac{\pi}{2}$.