Synopses & Reviews
About the Author - Karoly Bezdek received his Dr.rer.nat.(1980) and Habilitation (1997) degrees in mathematics from the Eötvös Loránd University, in Budapest and his Candidate of Mathematical Sciences (1985) and Doctor of Mathematical Sciences (1994) degrees from the Hungarian Academy of Sciences. He is the author of more than 100 research papers and currently he is professor and Canada Research Chair of mathematics at the University of Calgary. About the book: This multipurpose book can serve as a textbook for a semester long graduate level course giving a brief introduction to Discrete Geometry. It also can serve as a research monograph that leads the reader to the frontiers of the most recent research developments in the classical core part of discrete geometry. Finally, the forty-some selected research problems offer a great chance to use the book as a short problem book aimed at advanced undergraduate and graduate students as well as researchers. The text is centered around four major and by now classical problems in discrete geometry. The first is the problem of densest sphere packings, which has more than 100 years of mathematically rich history. The second major problem is typically quoted under the approximately 50 years old illumination conjecture of V. Boltyanski and H. Hadwiger. The third topic is on covering by planks and cylinders with emphases on the affine invariant version of Tarski's plank problem, which was raised by T. Bang more than 50 years ago. The fourth topic is centered around the Kneser-Poulsen Conjecture, which also is approximately 50 years old. All four topics witnessed very recent breakthrough results, explaining their major role in this book.
Review
From the reviews:"The present volume actually surveys packing and covering problems in Euclidean space and close cousins. ... Bezdek ... surveys the state of the art, best results, and outstanding conjectures for a host of problems. ... Summing Up: Recommended. Academic audiences, upper-division undergraduates through researchers/faculty." (D. V. Feldman, Choice, Vol. 48 (5), January, 2011)
Synopsis
The goal of this book is to provide focused material for a semester long graduate level course, which can be regarded as a brief introduction to Discrete Geometry. There are a large number of exercises and the book can also act as a short problem book aimed at advanced undergraduate and graduate students as well as researchers. This text is centered around three major problems of Discrete Geometry. The first is the problem of Densest Sphere Packings, which has more than 100 years of mathematically rich history. The second major problem is typically quoted under the roughly 50 years old Illumination Conjecture of V. Boltyanski and H. Hadwiger, and the third topic is centered around another nearly 50 years old conjecture called the Kneser-Poulsen Conjecture. All three topics witnessed very recent breakthrough results, explaining their major role in this book.
Synopsis
This monograph leads the reader to the frontiers of the very latest research developments in what is regarded as the central zone of discrete geometry. It is constructed around four classic problems in the subject, including the Kneser-Poulsen Conjecture.
Synopsis
This multipurpose book can serve as a textbook for a semester long graduate level course giving a brief introduction to Discrete Geometry. It also can serve as a research monograph that leads the reader to the frontiers of the most recent research developments in the classical core part of discrete geometry. Finally, the forty-some selected research problems offer a great chance to use the book as a short problem book aimed at advanced undergraduate and graduate students as well as researchers. The text is centered around four major and by now classical problems in discrete geometry. The first is the problem of densest sphere packings, which has more than 100 years of mathematically rich history. The second major problem is typically quoted under the approximately 50 years old illumination conjecture of V. Boltyanski and H. Hadwiger. The third topic is on covering by planks and cylinders with emphases on the affine invariant version of Tarski's plank problem, which was raised by T. Bang more than 50 years ago. The fourth topic is centered around the Kneser-Poulsen Conjecture, which also is approximately 50 years old. All four topics witnessed very recent breakthrough results, explaining their major role in this book.
Table of Contents
Preface.- Part I.- Sphere Packings.- Finite Packings by Translates of Convex Bodies.- Coverings by Homothetic Bodies - Illumination and Related Topics.- Coverings by Planks and Cylinders.- On the Volume of Finie Arrangements of Spheres.- Ball-Polyhedra as Interesctions of Congruent Balls.- Part II.- Selected Proofs on Sphere Packagings.- Selected Proofs on Finite Packagings of Translates of Convex Bodies.- Selected Proofs on Illumination and Related Topics.- Selected Proofs on Coverings by Planks and Cylinders.- Selected Proofs on the Kesner-Poulsen Conjecture.- Selected Proofs on Ball-Polyhedra.- References.