Three-Dimensional Restricted-Orientation Convexity1

A restricted-orientation convex set is a set of points whose intersection with lines from some fixed set is empty or connected. This notion generalizes both standard convexity and orthogonal convexity. We explore basic properties of restricted-orientation convex sets in three dimensions. In particular, we establish analogs of the following properties of standard convex sets: The intersection of a convex set with every line is empty or connected The intersection of a collection of convex sets is a convex set For every two points of a convex set, the straight segment joining them is contained in the set Convex sets are contractable