Arranging Vectors in $\mathbb{R}^n$ to Minimize Dot Product

Question

Given two nonzero vectors $u,v\in \mathbb{R}^n$, a necessary and sufficient condition to impose on $u,v$ in order to guarantee that for all $x\in\mathbb{R}^n$, either $x\cdot u\leq0\text{ or }x\cdot v\leq0$ can be formulated as follows: $$\frac{u\cdot v}{|u||v|}=-1\iff\nexists x\in\mathbb{R}^n\text{ s.t. } x\cdot u>0\text{ and }x\cdot v>0$$ How could one generalize this to three vectors in $\mathbb{R}^n$? Or to $m$ vectors in $\mathbb{R}^n$? In other words, what necessary and sufficient conditions must be imposed on the nonzero vectors $u_1,\dots,u_m\in \mathbb{R}^n$ such that $\nexists x\in\mathbb{R}^n$ which satisfies $\forall i\leq m\text{, } x\cdot u_i>0$?

Answer 1

This condition is satisfied if and only if there exists $0\neq (\lambda_1,\dots,\lambda_m)\in\mathbb{R}_{\geq 0}^m$ such that $\sum_i \lambda_i u_i=0$.

Proof : Consider the $m\times n$ matrix $A$ where the $i$-th row is $u_i$. Then $A$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ which maps $x$ to $v$ with $v_i=\langle u_i,x\rangle$. The image of this map is a linear subspace $H$ of $\mathbb{R}^m$. If there exists a vector $0\neq \lambda\in\mathbb{R}_{\geq 0}^m\cap H^{\perp}$, then we have $H\cap\mathbb{R}_{>0}^m=\varnothing$. This proves sufficiency. Also observe that if $H\cap\mathbb{R}_{>0}^m=\varnothing$, then there also exists a hyperplane $H\subseteq V$ such that $V\cap \mathbb{R}_{>0}^m=\varnothing$. For such a hyperplane, the existence of a nonzero normal vector in $\mathbb{R}_{\geq 0}^m$ is clear and the result follows.