Overview

"Triangulations appear in many different parts of mathematics and computer science since they are the natural way to decompose a region of space into smaller, easy-to-handle pieces. From volume computations and meshing to algebra and topology, there are many natural situations in which one has a ?xed set of points that can be used as vertices for the triangulation. Typically one wants to ?nd an optimal triangulation of those points or to explore the set of their all triangulations. The given points may represent the sites for a Delaunay triangulation computation, d thetest pointsfora surfacereconstruction, ora set ofmonomials, representedaslattice pointsinZ, inanalgebra- geometric meaning. A central theme of this book is to use the rich geometric structure of the space of triangulations of a given set of points to solve computational problems (e.g., counting the number of triangulations or ?nding optimal triangulations with respect to various criteria), and for setting up connections to novel applications in algebra, computer science, combinatorics, and optimization. Thus at the heart of the book is a comprehensive treatment of the theory of regular subdivisions, secondary polytopes, ?ips, chambers, and their interactions. Again, we ?rmly believe that understandingthe fundamentsof geometry and combinatoricspays up for algorithmsand applications."""

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9783662502419

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Reviews

From the reviews: Focusing on the structure of the set of all possible triangulations ... the current study sits at the threshold of geometry and combinatorics ... . offering terra firma to students still struggling with abstraction, the central theorem ... only dates to 1989, so the present elaboration carries readers to the frontiers of research. ... It is unusual to find such a leisurely, generous exposition of a new subject, as replete with illustrations as contemporary calculus textbooks. ... Summing Up: Recommended. Upper-division undergraduates through professionals. (D. V. Feldman, Choice, Vol. 49 (1), September, 2011) This book masterfully presents the theory of triangulations of (the convex hull of) a point set alongside many appealing applications in algebra, computer science, combinatorics, and optimization. ... The writing is thorough and engaging, assisted by clear (and numerous) illustrations, and many exercises for the reader. Graduate students and researchers in any area in which triangulations of points set configurations play a role will find this book a comprehensive and most useful reference. (Matthias Beck, Zentralblatt MATH, Vol. 1207, 2011)

Author Information

"J.A. De Loera is a professor of mathematics at the University of California, Davis. His work approaches difficult computational problems in discrete mathematics and optimization using tools from algebra and convex geometry. His research has been recognized by an Alexander von Humboldt Fellowship and several national and international grants. He is an associate editor of the journal ""Discrete Optimization"". Jörg Rambau is the chair professor of Wirtschaftsmathematik (Business Mathematics) at the Universität of Bayreuth since 2004. Before that he was associate head of the optimization department at the Zuse Institute Berlin (ZIB). His research encompasses problems in applied optimization, algorithmic discrete mathematics and combinatorial geometry. He is the creator of the state of the art program for triangulation computations TOPCOM. He is associate editor of the ""Jahresberichte der Deutschen Mathematiker-Vereinigung"". Francisco Santos, a professor at the Universidad de Cantabria Spain, received the Young Researcher award from the Universidad Complutense de Madrid in 2003 and was an invited speaker in the Combinatorics Section of the International Congress of Mathematicians in 2006. He is well-known for his explicit constructions of polytopes with disconnected spaces of triangulations, some of which are featured in this book. He is an editor of Springer Verlag's journal ""Discrete and Computational Geometry""."

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