We introduce O-halfspaces, which are an analog of halfspaces in the theory of O-convexity. We show that this notion generalizes standard halfspaces, explore properties of these generalized halfspaces, and demonstrate their relationship to O-convex sets. We also describe directed O-halfspaces, which are a subclass of O-halfspaces that has several special properties.
We first present some basic properties of O-halfspaces and compare them with the properties of standard halfspaces. We show that O-halfspaces may be disconnected, characterize an O-halfspace in terms of its connected components, and derive the upper bound on the number of components. We then study properties of the boundaries of O-halfspaces. Finally, we describe the complements of O-halfspaces and give a necessary and sufficient condition under which the complement of an O-halfspace is an O-halfspace.