Generalizing Halfspaces

Restricted-orientation convexity is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. We have studied the properties of restricted-orientation convex sets and demonstrated that this notion is a generalization of standard convexity. We now describe a restricted-orientation generalization of halfspaces and explore properties of these generalized halfspaces. In particular, we establish analogs of the following properties of standard halfspaces: -The intersection of a halfspace with every line is empty, a ray, or a line - Every halfspace is convex - A closed set with nonempty interior and convex boundary is a halfspace - The closure of the complement of a halfspace is a halfspace