  For a hyperplane arrangement in a real vector space, the coefficients of its
Poincar\'{e} polynomial have many interpretations. An interesting one is
provided by the Varchenko-Gel'fand ring, which is the ring of functions from
the chambers of the arrangement to the integers with pointwise addition and
multiplication. Varchenko and Gel'fand gave a simple presentation for this
ring, along with a filtration and associated graded ring whose Hilbert series
is the Poincar\'{e} polynomial. We generalize these results to cones defined by
intersections of halfspaces of some of the hyperplanes and prove a novel result
for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is
supersolvable the associated graded ring of the arrangement is Koszul.
