After doing some searching around, I've come to realize that while there is a huge body of literature on hyperplane arrangements, not much is generally written about halfspace arrangements. Is there any reason for this?
Does one imply or relate to the other some how?
As an example, I might want to ask what the homology of some arbitrary set expression on an arragnement is: Given a set of affine halfspaces $A = \{A_1, A_2, A_3, \dots\}$ in $\mathbb{R}^n$ and some set expression $$E(A) = (A_1 \cap A_2 \cup A_3 \cap (A_2^c))^c \cap A_n$$ Then what is $H_p(E(A); \mathbb{Z})$?