I have a system of linear inequalities $Ax>0$ where $A$ is a $m \times n$ matrix with $m \ge n$. Is there a simple way to check whether it has a solution?
What if one of the columns of $A$ is parametrized? In a more general case, I have one of the columns of $A(t)$ as $a(t) = tv+w$, where $t \in [0, 1]$. Is there a way to check whether all of the systems $$A(t)x>0, t \in [0,1]$$ have a solution?
Edit: Some additional info, $v$ is a multiple of $\vec{1}$ (vector of all ones).