I have a (convex) polyhedron $P \in \mathbb{R}^n$, defined by a set of linear inequalities. $P$ can be degenerate i.e. some of the inequalities can actually imply some equality. I have two points $x,y \in \mathbb{R}^n$ such that $x,y \in P$. How to check whether $x$ and $y$ belong to same facet ( or face) or not ? Can we do this computationally ?
I can assume $x,y$ to be extreme points of $P$, if that helps.
I have an approach but I don't know whether it will always work. Let the inequalities be defined as $a_i x \leq b_i$ for $i=1,2,\ldots,k$ where $a_i \in \mathbb{R}^n$ and $b_i \in \mathbb{R}$. We can check for all inequalites by putting x and y in them. i.e. if $a_i x=a_i y=b_i$ then the belong to same facet. Is there any issue with this approach.