János Pach - Böcker

Discrete Geometry is the kind of area in which problems can be quickly expressed and understood by a wide audience, as usually only elementary geometry is involved. This long-awaited book is based on William O.J. Moser's problem collection that has been circulating in the community for many years. The authors state a much extended variety of problems. For each problem they provide the historical background and give comprehensive references. The book will be of great value to any graduate student and researcher in discrete geometry.Bernard Chazelle (Princeton University): "This will be a terrific book. I've had drafts of it over the years, and I can't wait to have the final version. It will be extremely popular in the discrete geometry community (where the authors are undisputed leaders). This project has my highest recommendation." Joel Spencer (NYU): "- Pach has that wonderful "Hungarian" sense of problem posing and solving. ' This is a very exciting project'"

While we were busy putting together the present collection of articles celebrating the twentieth birthday of our journal, Discrete & Computational Geometry, and, in a way, of the ?eld that has become known under the same name, two more years have elapsed. There is no doubt that DCG has crossed the line between childhood and adulthood. By the mid-1980s it became evident that the solution of many algorithmic qu- tions in the then newly emerging ?eld of computational geometry required classical methodsandresultsfromdiscreteandcombinatorialgeometry. Forinstance,visibility and ray shooting problems arising in computer graphics often reduce to Helly-type questions for line transversals; the complexity (hardness) of a variety of geometric algorithms depends on McMullen’s upper bound theorem on convex polytopes or on the maximum number of “halving lines” determined by 2n points in the plane, that is, the number of different ways a set of points can be cut by a straight line into two parts of the same size; proximity questions stemming from several application areas turn out to be intimately related to Erdos’ ? s classical questions on the distribution of distances determined by n points in the plane or in space. On the other hand, the algorithmic point of view has fertilized several ?elds of c- vexity and of discrete geometry which had lain fallow for some years, and has opened new research directions.

A complete, self-contained introduction to a powerful and resurging mathematical disciplineCombinatorial 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 theoryPacking and covering with congruent convex disksExtremal graph and hypergraph theoryDistribution of distances among finitely many pointsEpsilon-nets and Vapnik—Chervonenkis dimensionGeometric graph theoryGeometric discrepancy theoryAnd much more

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This 2005 volume, which contains 32 papers on a broad range of topics of interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems. This 2005 volume, which contains 32 papers on a broad range of topics of interest in the field, is an outgrowth of that synergism. It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension. There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.

This book 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.

In many applications of graph theory, graphs are regarded as geometric objects drawn in the plane or in some other surface. The traditional methods of "abstract" graph theory are often incapable of providing satisfactory answers to questions arising in such applications. In the past couple of decades, many powerful new combinatorial and topological techniques have been developed to tackle these problems. Today geometric graph theory is a burgeoning field with many striking results and appealing open questions. This contributed volume contains thirty original survey and research papers on important recent developments in geometric graph theory.  The contributions were thoroughly reviewed and written by excellent researchers in this field.

In many applications of graph theory, graphs are regarded as geometric objects drawn in the plane or in some other surface. The traditional methods of "abstract" graph theory are often incapable of providing satisfactory answers to questions arising in such applications. In the past couple of decades, many powerful new combinatorial and topological techniques have been developed to tackle these problems. Today geometric graph theory is a burgeoning field with many striking results and appealing open questions. This contributed volume contains thirty original survey and research papers on important recent developments in geometric graph theory.  The contributions were thoroughly reviewed and written by excellent researchers in this field.

In the Fall of 2023, the Erdős Center (Budapest) hosted a special semester on ”Discrete Geometry and Convexity”, which brought together some of the strongest experts in the field and many outstanding young researchers. The program featured intensive one-week mini-courses during a summer school, followed by conferences and workshops presenting cutting-edge research. Part I of the present volume includes the notes of three lecture series on: (1) approximation in discrete geometry, (2) on random polytopes, and (3) on a structure theory for graphs embedded in the plane. Part II starts with a classic: Matoušek’s until now unpublished elegant lecture notes concerning the algorithmic complexity of recognizing intersection graphs of segments and some other geometric objects. It is complemented by the first systematic and comprehensive survey of the corresponding complexity class: the existential theory of reals. This volume will be a valuable resource for graduate students, young researchers, and experts in related fields interested in discrete and computational geometry.

This volume grew out of the Fall 2023 special semester Discrete Geometry and Convexity at the Erdős Center of the Alfréd Rényi Institute in Budapest, where leading experts and outstanding young researchers gathered for an intensive program of mini-courses, workshops, and conferences. New Probes into Discrete and Convex Geometry brings together 17 invited survey articles that map the current landscape of the field and highlight some of its most dynamic frontiers. The chapters range from convexity and packing problems to combinatorial and piecewise-linear topology, from Helly-Tverberg theory and its topological methods to questions motivated by computer science, data analysis, and geometric optimization. Along the way, readers will find both polished expositions of major tools and guided tours of active research areas, rich with open problems and new perspectives. Written by world-class researchers, these surveys offer an accessible entry point for graduate students and newcomers, while providing specialists with a concise reference to recent advances.

An impressive collection of original research papers in discrete and computational geometry, contributed by many leading researchers in these fields, as a tribute to Jacob E. Goodman and Richard Pollack, two of the 'founding fathers' of the area, on the occasion of their 2/3 x 100 birthdays. The topics covered by the 41 papers provide professionals and graduate students with a comprehensive presentation of the state of the art in most aspects of discrete and computational geometry, including geometric algorithms, study of arrangements, geometric graph theory, quantitative and algorithmic real algebraic geometry, with important connections to algebraic geometry, convexity, polyhedral combinatorics, the theory of packing, covering, and tiling. The book serves as an invaluable source of reference in this discipline.

The 12th International Symposium on Graph Drawing (GD 2004)was held d- ing September 29-October 2, 2004, at City College, CUNY, in the heart of Harlem, New York City. GD 2004 attracted 94 participants from 19 countries. In response to the call for papers, the program committee received 86 re- larsubmissionsdescribingoriginalresearchand/orsystemdemonstrations.Each submissionwasreviewedbyatleastthreeprogramcommitteemembersandc- ments were returned to the authors. Following extensive e-mail discussions, the program committee accepted 39 long papers (11 pages each in the proceedings) and 12 short papers (6 pages each). In addition, 4 posters were displayed and discussed in the conference exhibition room (2 pages each in the proceedings). Theprogramcommittee ofGD 2004invitedtwo distinguishedlecturers.P- fessorPaulSeymourfromPrincetonUniversitypresenteda newcharacterization ofclaw-freegraphs(jointworkwithMariaChudnovsky).ProfessorErikDemaine from MIT reported on his joint work with Fedor Fomin, MohammadTaghi - jiaghayi and Dimitrios Thilikos, concerning fast (often subexponential) ?x- parameter algorithms and polynomial approximation schemes for broad classes of NP-hard problems in topological graph theory. A survey of the subject by Professors Demaine and Hajiaghayi is included in this volume. As usual, the annual graph drawing contest was held during the conference. This time the contest had two distinct tracks: the graph drawing challenge and the freestyle contest. A report is included in the proceedings.

The present volume is a collection of a dozen survey articles, dedicated to the memory of the famous Hungarian geometer, László Fejes Tóth, on the 99th anniversary of his birth. Each article reviews recent progress in an important field in intuitive, discrete, and convex geometry. The mathematical work and perspectives of all editors and most contributors of this volume were deeply influenced by László Fejes Tóth.

An impressive collection of original research papers in discrete and computational geometry, contributed by many leading researchers in these fields, as a tribute to Jacob E. Goodman and Richard Pollack, two of the 'founding fathers' of the area, on the occasion of their 2/3 x 100 birthdays. The topics covered by the 41 papers provide professionals and graduate students with a comprehensive presentation of the state of the art in most aspects of discrete and computational geometry, including geometric algorithms, study of arrangements, geometric graph theory, quantitative and algorithmic real algebraic geometry, with important connections to algebraic geometry, convexity, polyhedral combinatorics, the theory of packing, covering, and tiling. The book serves as an invaluable source of reference in this discipline.

Discrete and computational geometry are two fields which in recent years have benefitted from the interaction between mathematics and computer science. The results are applicable in areas such as motion planning, robotics, scene analysis, and computer aided design. The book consists of twelve chapters summarizing the most recent results and methods in discrete and computational geometry. All authors are well-known experts in these fields. They give concise and self-contained surveys of the most efficient combinatorical, probabilistic and topological methods that can be used to design effective geometric algorithms for the applications mentioned above. Most of the methods and results discussed in the book have not appeared in any previously published monograph. In particular, this book contains the first systematic treatment of epsilon-nets, geometric tranversal theory, partitions of Euclidean spaces and a general method for the analysis of randomized geometric algorithms. Apart from mathematicians working in discrete and computational geometry this book will also be of great use to computer scientists and engineers, who would like to learn about the most recent results.

The present volume is a collection of a dozen survey articles, dedicated to the memory of the famous Hungarian geometer, László Fejes Tóth, on the 99th anniversary of his birth. Each article reviews recent progress in an important field in intuitive, discrete, and convex geometry. The mathematical work and perspectives of all editors and most contributors of this volume were deeply influenced by László Fejes Tóth.