Fundamentals of Restricted-Orientation Convexity

Abstract

A restricted-orientation convex set, also called an O-convex set, is a set of points whose intersection with lines from some fixed set is empty or connected. The notion of O-convexity generalizes standard convexity and orthogonal convexity. We explore some of the basic properties of O-convex sets in two and higher dimensions. We also study O-connected sets, which are restricted O-convex sets with several special properties. We introduce and investigate restricted-orientation analogs of lines, flats, and hyperplanes, and characterize O-convex and O-connected sets in terms of their intersections with hyperplanes. We then explore properties of O-connected curves; in particular, we show when replacing a segment of an O-connected curve with a new curvilinear segment yields an O-connected curve and when the catenation of several curvilinear segments forms an O-connected segment. We use these results to characterize an O-connected set in terms of O-connected segments, joining pairs of its points, that are wholly contained in the set. We also identify some of the major properties of standard convex sets that hold for O-convexity. In particular, we establish the following results: The intersection of a collection of O-convex sets is an O-convex set; every O-connected curvilinear segment is a segment of some O-connected curve; for every two points of an O-convex set, there is an O-convex segment joining them that is wholly contained in the set.