Biclosed sets in real hyperplane arrangements

The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Bjorner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets of the arrangement. Two families of examples of arrangements with a lattice of chambers are simplicial and supersolvable arrangements. For these arrangements, we prove that the chambers correspond to biclosed sets, a weakening of the biconvex property.