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Preface

Restricted-orientation convexity, also called *O*-convexity, is the
study of geometric objects whose intersections with lines from some fixed
set are connected. This notion generalizes standard convexity and several
types of nontraditional convexity. We explore this generalized convexity
in multidimensional Euclidean space and identify the properties of standard
convex sets that also hold for restricted-orientation convexity.
The purpose of the book is to present the current results on restricted-orientation
convexity to the research community and discuss related open problems.
The book requires only basic knowledge in geometry; the reader should be
familiar with the notion of higher-dimensional Euclidean space and with
basic objects in this space, such as lines, balls, and hyperplanes. We
use geometric techniques in most proofs, which are accessible to all mathematics
and computer-science researchers and graduate students.

*O*-convexity: We begin with basic properties of *O*-convex
sets, and then introduce *O*-connected sets, which are a subclass
of *O*-convex sets. We study restricted-orientation analogs of lines,
flats and hyperplanes, and characterize *O*-convex and *O*-connected
sets in terms of their intersections with hyperplanes. We also explore
properties of *O*-connected curves; in particular, we determine when
the replacement of a segment of an *O*-connected curve gives a new
*O*-connected
curve, and when the catenation of several curvilinear segments gives an
*O*-connected segment. We use these results to characterize an *O*-convex
set in terms of *O*-convex segments joining its points, and an *O*-connected
set in terms of *O*-connected segments.

*O*-halfspaces: We introduce *O*-halfspaces, which
are a generalization of standard halfspaces, defined as geometric objects
whose intersection with every line from some fixed set is empty, a ray
or a line. We give basic properties of *O*-halfspaces and compare
them with standard halfspaces; in particular, we show that *O*-halfspaces
may be disconnected and characterize them through their connected components.
We also characterize *O*-halfspaces in terms of *O*-convexity
of their boundaries, and give a condition under which the complement of
an *O*-halfspace is an *O*-halfspace.

**Strong ***O*-convexity: We also introduce the notion of strong
*O*-convexity,
which is an alternative generalization of convexity. We describe properties
of strongly *O*-convex flats and halfspaces, and establish the strong
*O*-convexity of the affine hull of a strongly
*O*-convex set.
We then show that, for every point in the boundary of a strongly *O*-convex
set, there is a supporting strongly *O*-convex hyperplane through
it. Finally, we characterize strongly *O*-convex sets in terms of
the intersections of strongly *O*-convex halfspaces.