Generalized halfspaces in restricted-orientation convexity

Eugene Fink and Derick Wood

Journal of Geometry, 62, pages 99-120, 1998.


Restricted-orientation convexity, also called O-convexity, is the study of geometric objects whose intersection with lines from some fixed set is empty or connected. The notion of O-convexity generalizes standard convexity as well as several other types of nontraditional convexity.

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.