## Generalizing halfspaces

### Eugene Fink and Derick Wood

In Proceedings of the Eighth Canadian Conference
on Computational Geometry, pages 211-216, 1996.

### Abstract

Restricted-orientation convexity is the study of geometric
objects whose intersection with lines from some fixed set is empty or
connected. We have studied the properties of restricted-orientation
convex sets and demonstrated that this notion is a generalization of
standard convexity. We now describe a restricted-orientation
generalization of halfspaces and explore properties of these generalized
halfspaces. In particular, we establish analogs of the following
properties of standard halfspaces:

- The intersection of a halfspace with every line is empty,
a ray, or a line
- Every halfspace is convex
- A closed set with nonempty interior and convex boundary
is a halfspace
- The closure of the complement of a halfspace is a halfspace