Let's say I have a system of inequalities: $Ax \leq g$ for some $A \in \mathbb{R}^{4\times4}$, $x \in \mathbb{R}^4$, $g \in \mathbb{R}^4$, and $A$ is full rank. Here, the $\leq$ denotes element-wise inequality.
Specifically, I know that $x$ lies in a two-dimensional subspace of $\mathbb{R}^4$ (determined by the null space of some matrix $N$). What I'm interested in is the dimension of the solution to the above system of inequalities. More succinctly, I'm interested in the dimension of the set
$ \left\{ x \in \mathbb{R}^4 \ \vert \ x \in {\rm Null}(N) , \ Ax \leq g\right\} $
Understanding more about this set would be nice too, but the dimension would suffice. I'm really not sure how to approach this problem... at all. This problem arose while I was trying to analyze the set of solutions to a linear program, if you're curious.