Strong Restricted-Orientation Convexity

摘要

Strong O-convexity is a generalization of standard convexity, defined withrespect to a fixed set O of hyperplanar orientations. We explore the properties of strongly O-convex sets in two and more dimensions and develop a mathematical foundation of strong convexity. We characterize strongly 0-convex polytopes, flats, and halfspaces, establish the strong 0-convexity of the affine hull of a strongly O-convex set, and describe conditions under which two orientation sets yield the same collection of strongly 0-convex sets (orientation equivalence). We identify some of the major properties of standard convex sets that hold for strong O-convexity. In particular, we establish the following results: The intersection of a collection of strongly O-convex sets is strongly O-convex; For every point in the boundary of a strongly O-convex set, there is a supporting strongly 0-convex hyperplane through it; A closed set with a nonempty interior is strongly 0-convex if and only if it is the intersection of the strongly 0-convex halfspaces that contain it.