Generalized Halfspaces in Restricted-Orientation Convexityz

Restricted-orientation convexity (O-convexity) is the study of geometric objects whose intersections with lines from some xed set O of orientations are empty or connected. The notion of O-convexity generalizes standard convexity, as well as several other types of non-traditional convexity. We introduce and study O-halfspaces, which are analogs of standard halfspaces in the theory of O-convexity, and directed O-halfspaces, which are restricted O-halfspaces. We explore some of the basic properties of them and outline their relationships to O-convex sets and O-connected sets, which are restricted O-convex sets, in two and more dimensions. For O-halfspaces, we prove that: Every O-halfspace is O-convex; if O has the point-intersection property, then the number of connected components of an O-halfspace in d dimensions is at most 2 d?1 , and this bound is attainable; and the closed complement of an O-halfspace is an O-halfspace if and only if the boundary of the O-halfspace is O-convex. In addition, for directed O-halfspaces, we prove that: Every O-halfspace is O-connected, and the closed complement of a directed O-halfspace is a directed O-halfspace. Abstract Restricted-orientation convexity (O-convexity) is the study of geometric objects whose intersections with lines from some xed set O of orientations are empty or connected. The notion of O-convexity generalizes standard convexity, as well as several other types of nontraditional convexity. We introduce and study O-halfspaces, which are analogs of standard halfspaces in the theory of O-convexity, and directed O-halfspaces, which are restricted O-halfspaces. We explore some of the basic properties of them and outline their relationships to O-convex sets and O-connected sets, which are restricted O-convex sets, in two and more dimensions.