I am assuming that the common lineality space of the hyperplane arrangement has already been factored out. So we are looking at an ``effective" arrangement in lower dimensions where all the polyhedra formed are convex hulls of $0$-dimensional faces.
Now the $i^{th}$ hyperplane in this lower dimension can be thought of as the zero-set of the equation $\vec{a}_i.\vec{x}=b_i$ for some $\vec{a}_i$ and $b_i$. Now for this hyperplane the $2$ spaces on its two sides are defined by $\vec{a}_i.\vec{x} \leq b_i$ and $\vec{a}_i.\vec{x} \geq b_i$. If there are $p$ vertex points in this polyhedral complex so induced and there are $k$ planes inducing it then one can create a ``slack matrix" of dimension, $p \times 2k$ whose rows correspond to the points and the columns correspond to each of the two half-space inequalities corresponding to each plane. One can reverse inequalities and write all half-spaces as some $\vec{\alpha}.\vec{x} \leq \beta$. Then corresponding to the row of the vertex point $v$ and the column corresponding to $(\vec{\alpha},\beta )$ we have the entry $\beta - \vec{\alpha}.\vec{v}$.
- Is this matrix standard? Is its rank or any such properties known?
Its clear that the slack matrix of each of the polyhedron induced is sitting as a submatrix of this matrix defined above.
