Volumetric Discrete Geometry

Volume of geometric objects plays an important role in applied and theoretical mathematics. This is particularly true in the relatively new branch of discrete geometry, where volume is often used to find new topics for research. Volumetric Discrete Geometry demonstrates the recent aspects of volume, introduces problems related to it, and presents methods to apply it to other geometric problems. Part I of the text consists of survey chapters of selected topics on volume and is suitable for advanced undergraduate students. Part II has chapters of selected proofs of theorems stated in Part I and is oriented for graduate level students wishing to learn about the latest research on the topic. Chapters can be studied independently from each other. Provides a list of 30 open problems to promote research Features more than 60 research exercises Ideally suited for researchers and students of combinatorics, geometry and discrete mathematics

Geometry is a classical core part of mathematics which, with its birth, marked the beginning of the mathematical sciences. Thus, not surprisingly, geometry has played a key role in many important developments of mathematics in the past, as well as in present times. While focusing on modern mathematics, one has to emphasize the increasing role of discrete mathematics, or equivalently, the broad movement to establish discrete analogues of major components of mathematics. In this way, the works of a number of outstanding mathema- cians including H. S. M. Coxeter (Canada), C. A. Rogers (United Kingdom), and L. Fejes-T oth (Hungary) led to the new and fast developing eld called discrete geometry. One can brie y describe this branch of geometry as the study of discrete arrangements of geometric objects in Euclidean, as well as in non-Euclidean spaces. This, as a classical core part, also includes the theory of polytopes and tilings in addition to the theory of packing and covering. D- crete geometry is driven by problems often featuring a very clear visual and applied character. The solutions use a variety of methods of modern mat- matics, including convex and combinatorial geometry, coding theory, calculus of variations, di erential geometry, group theory, and topology, as well as geometric analysis and number theory.

Volume of geometric objects plays an important role in applied and theoretical mathematics. This is particularly true in the relatively new branch of discrete geometry, where volume is often used to find new topics for research. Volumetric Discrete Geometry demonstrates the recent aspects of volume, introduces problems related to it, and presents methods to apply it to other geometric problems. Part I of the text consists of survey chapters of selected topics on volume and is suitable for advanced undergraduate students. Part II has chapters of selected proofs of theorems stated in Part I and is oriented for graduate level students wishing to learn about the latest research on the topic. Chapters can be studied independently from each other. Provides a list of 30 open problems to promote research Features more than 60 research exercises Ideally suited for researchers and students of combinatorics, geometry and discrete mathematics

Celebrating the work of Professor W. Kuperberg, this reference explores packing and covering theory, tilings, combinatorial and computational geometry, and convexity, featuring an extensive collection of problems compiled at the Discrete Geometry Special Session of the American Mathematical Society in New Orleans, Louisiana. Discrete Geometry analyzes packings and coverings with congruent convex bodies , arrangements on the sphere, line transversals, Euclidean and spherical tilings, geometric graphs, polygons and polyhedra, and fixing systems for convex figures. This text also offers research and contributions from more than 50 esteemed international authorities, making it a valuable addition to any mathematical library.

The book is centered around two major conjectures of discrete geometry: the Hadwiger-Levi conjecture (1955) and the Kneser-Poulsen conjecture (1955). Although both conjectures have been solved only in dimension two and are open in higher dimensions, they have already influenced a great deal of research in discrete geometry and surely will continue to do so. The book gives a detailed account of all major results already achieved with complete proofs and emphasizing the role of volumetric methods/inequalities. The book s main purpose is to present the relevant frontline research in discrete geometry while generating wider interest in two fundamental conjectures of discrete geometry. "

"The book is centered around two major conjectures of discrete geometry: the Hadwiger-Levi conjecture (1955) and the Kneser-Poulsen conjecture (1955). Although both conjectures have been solved only in dimension two and are open in higher dimensions, they have already influenced a great deal of research in discrete geometry and surely will continue to do so. The book gives a detailed account of all major results already achieved with complete proofs and emphasizing the role of volumetric methods/inequalities. The book’s main purpose is to present the relevant frontline research in discrete geometry while generating wider interest in two fundamental conjectures of discrete geometry."--Provided by publisher.

This book constitutes the refereed proceedings of the 6th International Workshop on Discrete Geometry for Computer Imagery, DGCI'96, held in Lyon, France, in November 1996. Computer imaging essentially depends on discrete models for coding, processing, recognition, representation, etc. The volume presents 24 revised full papers selected from 41 submissions together with 3 invited contributions and a tutorial paper, which bridges the gap between theory and practice. The issues addressed are topology, geometry, shape representation, 3D surfaces and volumes, models for discrete space, image transformation and generation.

This book contains recent contributions to the fields of rigidity and symmetry with two primary focuses: to present the mathematically rigorous treatment of rigidity of structures and to explore the interaction of geometry, algebra and combinatorics. Contributions present recent trends and advances in discrete geometry, particularly in the theory of polytopes. The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry, group theory, classical geometry, hyperbolic geometry and topology. Overall, the book shows how researchers from diverse backgrounds explore connections among the various discrete structures with symmetry as the unifying theme. The volume will be a valuable source as an introduction to the ideas of both combinatorial and geometric rigidity theory and its applications, incorporating the surprising impact of symmetry. It will appeal to students at both the advanced undergraduate and graduate levels, as well as post docs, structural engineers and chemists.