Consider the intersection of half space $H_{1}=\{x\in\mathbb{R}^{n}|a_{1}^{T}x>mb_{+}\}\cap\{x\in\mathbb{R}^{n}|a_{1}^{T}x<mb_{-}\}$, where $0\neq a_{1}\in\mathbb{R}^{n}$, $m>1$ and $b_{-}>b_{+}$ so $H_{1}$ is non-empty. Consider $H_{2},..., H_{L}$ defined in similar way with $L>n$, where $H_{i}=\{x\in\mathbb{R}^{n}|a_{i}^{T}x>mb_{+}\}\cap\{x\in\mathbb{R}^{n}|a_{i}^{T}x<mb_{-}\}$. Assume $a_{i}$ are non-negative vectors and $a_{i}\neq a_{j}$ if $i\neq j$. Can we always find large enough $m$ such that $\cap_{i=1}^{L}H_{i}$ is non-empty?
Here is what I think. Intersection of each pair of $H_{i}, H_{j}$ is non-empty, increasing $m$ will translate the intersection and increase the volume of the intersection. Since all $a_{i}$ are non-negative, the translation has no effect on the intersection, but the increasing the volume will lead to all intersections intersect. I am not sure if I am correct though.