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Fundamentals of restricted-orientation convexity

Information Sciences, 92,
pages 175-196, 1996.
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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 a subclass
of 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 demonstrate that 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, and, for every
two points of an O-convex set, there is an O-convex segment joining them
that is wholly contained in the set.