I have a polyhedron in $\mathbb{R}^n$ described by $n(n-1)$ inequalities of the form $Mx < \mathbb{1}$ where $M \in \mathbb{R}^{n(n-1)\times n}$ is a matrix such that $M \mathbb{1}=0 $ (rows sum up to 0) and $x \in \mathbb{R}^n$. Then I have the central hyperplane described by $x \mathbb{1}= x_1+x_2+ \dots+x_n=0$.
I would like to find some information about the intersection between these two sets. I can always say that the intersection is not empty because $x=0$ satisfies both conditions but is there any other useful information that I can extract? How can I exploit the fact that $M \mathbb{1}=0 $? Does this tell me that my polyhedron is unbounded?