János Pach

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Although discrete geometry has a rich history extending more than 150 years, it abounds in open problems that even a high-school student can understand and appreciate. Some of these problems are notoriously difficult and are intimately related to deep questions in other fields of mathematics. But many problems, even old ones, can be solved by a clever undergraduate or a high-school student equipped with an ingenious idea and the kinds of skills used in a mathematical olympiad. Research Problems in Discrete Geometry is the result of a 25-year-old project initiated by the late Leo Moser. It is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research. Important features include: - More than 500 open problems, some old, others new and never before published - Each chapter divided into self-contained sections, each section ending with an extensive bibliography - A great selection of research problems for graduate students looking for a dissertation topic - A comprehensive survey of discrete geometry, highlighting the frontiers and future of research - More than 120 figures - A preface to an earlier version written by the late Paul Erdos Peter Brass is Associate Professor of Computer Science at the City College of New York. William O. J. Moser is Professor Emeritus at McGill University. Janos Pach is Distinguished Professor at The City College of New York, Research Professor at the Courant Institute, NYU, and Senior Research Fellow at the Rényi Institute,Budapest.

Although discrete geometry has a rich history extending more than 150 years, it abounds in open problems that even a high-school student can understand and appreciate. Some of these problems are notoriously difficult and are intimately related to deep questions in other fields of mathematics. But many problems, even old ones, can be solved by a clever undergraduate or a high-school student equipped with an ingenious idea and the kinds of skills used in a mathematical olympiad. Research Problems in Discrete Geometry is the result of a 25-year-old project initiated by the late Leo Moser. It is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems. This book is intended as a source book for both professional mathematicians and graduate students who love beautiful mathematical questions, are willing to spend sleepless nights thinking about them, and who would like to get involved in mathematical research. Important features include: - More than 500 open problems, some old, others new and never before published - Each chapter divided into self-contained sections, each section ending with an extensive bibliography - A great selection of research problems for graduate students looking for a dissertation topic - A comprehensive survey of discrete geometry, highlighting the frontiers and future of research - More than 120 figures - A preface to an earlier version written by the late Paul Erdos Peter Brass is Associate Professor of Computer Science at the City College of New York. William O. J. Moser is Professor Emeritus at McGill University. Janos Pach is Distinguished Professor at The City College of New York, Research Professor at the Courant Institute, NYU, and Senior Research Fellow at the Rényi Institute,Budapest.

A complete, self-contained introduction to a powerful and resurging mathematical discipline Combinatorial Geometry presents and explains with complete proofs some of the most important results and methods of this relatively young mathematical discipline, started by Minkowski, Fejes Tóth, Rogers, and Erd???s. Nearly half the results presented in this book were discovered over the past twenty years, and most have never before appeared in any monograph. Combinatorial Geometry will be of particular interest to mathematicians, computer scientists, physicists, and materials scientists interested in computational geometry, robotics, scene analysis, and computer-aided design. It is also a superb textbook, complete with end-of-chapter problems and hints to their solutions that help students clarify their understanding and test their mastery of the material. Topics covered include:*Geometric number theory*Packing and covering with congruent convex disks*Extremal graph and hypergraph theory*Distribution of distances among finitely many points*Epsilon-nets and Vapnik—Chervonenkis dimension*Geometric graph theory*Geometric discrepancy theory*And much more