I have a linear program: a system of linear inequalities of the form
$$Ax \le b, \qquad x \ge 0.$$
where $x \in \mathbb{R}^n$, $b \in \mathbb{R}^m$, and $A$ is a $m\times n$ matrix.
I am looking for an upper-bound on the size of the smallest solution to this system of inequalities. In other words, I'd like to find a constant $c$ where I can make some statement like
"If this system has any solution, then it has a solution $x$ such that $||x|| \le c$."
where $c$ depends somehow on $A,B$, and where you can pick any norm of your choice. Are there any known techniques or results of this form?
In my specific application, I happen to know that every entry in $A$ is $-1$, $0$, or $+1$. Intuitively, it seems like the bad case that can force $c$ to be big is where I have inequalities that correspond to hyperplanes that have a small angle between them; however, if the entries of $A$ are all in $\{-1,0,+1\}$, it seems like that bad case can't happen, which is why it seems like there might be some hope for some kind of nice result along these lines. (For instance, if taking $c=\sum_i b_i$ and using the $L_{\infty}$ norm let me make such a statement, that would be the kind of result I would find useful.) Is anything along these lines known?