Książki

Restricted-orientation 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 the properties of this generalized convexity in multidimensional Euclidean space, describes restricted-orientation analogs of lines, hyperplanes, flats, and halfspaces, and identify major properties of standard convex sets that also hold for restricted-orientation convexity. We then introduce the notion of strong restricted-orientation convexity, which is an alternative generalization of convexity, and show that its properties are also similar to those of standard convexity.

Restricted-Orientation Convexity

1 Introduction.- 1.1 Standard Convexity.- 1.2 Ortho-Convexity.- 1.3 Strong Ortho-Convexity.- 1.4 Convexity Spaces.- 1.5 Book Outline.- 2 Two Dimensions.- 2.1 O-Convex Sets.- 2.2 O-Halfplanes.- 2.3 Strongly O-Convex Sets.- 3 Computational Problems.- 3.1 Visibility and Convexity Testing.- 3.2 Strong O-Hull.- 3.3 Strong O-Kernel.- 3.4 Visibility from a Point.- 4 Higher Dimensions.- 4.1 Orientation Sets.- 4.2 O-Convexity and O-Connectedness.- 4.3 O-Connected Curves.- 4.4 Visibility.- 5 Generalized Halfspaces.- 5.1 O-Halfspaces.- 5.2 Directed O-Halfspaces.- 5.3 Boundary Convexity.- 5.4 Complementation.- 6 Strong Convexity.- 6.1 Strongly O-Convex Sets.- 6.2 Strongly O-Convex Flats.- 6.3 Strongly O-Convex Halfspaces.- 7 Closing Remarks.- 7.1 Main Results.- 7.2 Conjectures.- 7.3 Future Work.- References.