Restricted-Orientation Convexity / Edition 1

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Overview

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. The authors explore the properties of this generalized convexity in multidimensional Euclidean space, and describ restricted-orientation analogs of lines, hyperplanes, flats, halfspaces, and identify major properties of standard convex sets that also hold for restricted-orientation convexity. They 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 that of standard convexity.

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Editorial Reviews

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"The well-organized, readable, interesting volume considers two generalizations of the concept of convexity in Rn, and their usual related concepts (hull, visibility, kernel, etc.). … The volume would be very good for a seminar studying the many results from the last two decades on these forms of generalized convexity. The book closes with suggestions and conjectures for the direction of future research." (John R. Reay, Mathematical Reviews, Issue 2007 j)

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Product Details

Meet the Author

Eugene Fink received his B.S. degree from Mount Allison University (Canada) in 1991, M.S. from the University of Waterloo (Canada) in 1992, and Ph.D. from Carnegie Mellon University (USA) in 1999. He has been an assistant professor in the Computer Science and Engineering Department at the University of South Florida (USA) since 1999. His research interests include computational geometry, artificial intelligence, machine learning, and e-commerce.

Derick Wood received his B.Sc. (1963) and Ph.D. (1968) from the University of Leeds (UK). He was a Postdoctoral Fellow at the Courant Institute, New York University (USA), from 1968 to 1970, and then joined McMaster University (Canada) in 1970. He was a professor at the University of Waterloo (Canada) from 1982 to 1992, at the University of Western Ontario (Canada) from 1992 to 1995, and at the Hong Kong University of Science and Technology since 1995. He has published widely in a number of research areas and written two textbooks, "Theory of Computation" (John Wiley, 1987) and "Data Structures, Algorithms, and Performance" (Addison-Wesley, 1993).

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Table of Contents

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.

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