Convex and Discrete Geometry

Convex and Discrete Geometry is an area of mathematics situated between analysis, geometry and discrete mathematics with numerous relations to other areas. The book gives an overview of major results, methods and ideas of convex and discrete geometry and its applications. Besides being a graduate-level introduction to the field, it is a practical source of information and orientation for convex geometers. It should also be of use to people working in other areas of mathematics and in the applied fields.

• Convex body
• combinatorial geometry
• convex function
• convex polytope
• discrete geometry
• geometric lattice

From the reviews:

"The aim of the present book is to give an overview of basic methods and results of convex analysis and discrete geometry and their applications. … A large bibliography of 1052 titles, each of them being referred to in the text, tries to cover all the facets of the subject, from its origins to the present day state. … the clear and pleasant style of presentation, make the book accessible to a large audience, including graduate students, teachers and researchers in various areas of mathematics." (S. Cobzas, Studia Universitatis Babes-Bolyai, Vol. LII (4), 2007)

"Convex geometry is a classical area of mathematics that dates back to antiquity. … the author’s main goal is to expose applications of convex geometry to other areas of mathematics and science, in particular number theory, mathematical physics, crystallography, tomography, optimization, and computational geometry. … The book can be used for graduate courses in convex, discrete geometry and geometry of numbers. For experts and people working on applications, the book is a very valuable source of open problems, new directions and references." (Aleksandr Koldobsky, Mathematical Reviews, Issue 2008 f)

"This excellent book presents a complete and coherent overview of principal ideas, results and methods of convex and discrete geometry. … The book is highly recommended to everyone who is interested in convex geometry: it may be viewed as a graduate-level introduction as well as a source of both classical and actual results in the subject." (Vasyl Gorkaviy, Zentralblatt MATH, Vol. 1139 (17), 2008)

“In this monograph, the author gives a broad and colourful picture of classic and contemporary convex geometry and discrete geometry. … presents a wide variety of highly interesting and often surprising results. … can be used as textbook for different courses in geometry. … reader will find more and more results which he or she has not seen beforein a monograph and perhaps not even in the original literature. … a book that addresses itself to a wider public and is accessible to students.” (R. Schneider, International Mathematical News, Issue 12, December, 2009)

“This almost 600 page monograph introduces the reader to the present state of the theory of convex geometry in its many facts. … All proofs are carefully chosen and presented with all details. … The book will be appreciated by specialists in the field … . The book can be used as a base for some special courses at universities and for further preparation of PhD students. It is a book that should be available in any mathematical oriented library.” (EMS Newsletter, December, 2009)

“A fascinating overview of major results, methods and ideas of convex and discrete geometry and some of their applications. … present this enormous amount of topics and results in a clear and lucid way by sprinkling the text with various comments, historic and heuristic remarks … . All in all an impressive book, which by no means may serve only as work of reference but can be used very well as a rich source for lecturers and as graduate-level introduction to the field.” (G. Kowol, Monatshefte für Mathematik, Vol. 156 (3), March, 2009)

• Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Vienna, Austria

  Peter M. Gruber

1959-66 Study of mathematics and physics, Univ Vienna, Univ Kansas

1996 PhD, Univ Vienna

1966-71 Assistant, Techn.Univ.Vienna

1968 Award of the ÖMG

1969 (Junior) Kardinal Innitzer Award

1970- Docent, Techn. Univ. Vienna

1971-76 Full Professor of Mathematics, Univ. Linz

1976- Full Professor of Mathematical Analysis, Techn. Univ. Vienna

1978-82 President, Austrian Math. Soc.

1981-87 Head, Division of Mathematics, Techn. Univ. Vienna

1985 Hon.Member, Accademia Nazionale di Scienze, Letter e Arti, Modena

1988 Corr. Member, Austrian Academy of Sciences

1991 Full Member, Austrian Academy of Sciences

2000 Hon. Doctorate, Univ. Turin

2001 Hon. Doctorate, Univ. Siegen

2001 Memorial Medal, Fac. Math and Physics, Charles Univ. Prague

2002 Korr. Member, Bayer. Akad. Wiss.

2003 Foreign Member, Russia Acad. Sciences

More than 100 articles and books in the geometry of numbers, convex and discrete geometry, and analysis. Extended visits to Budapest, Bologna, Toronto, Hobart (Tasmania), Chandigarh, Turin, Messina, Moscow-St.Petersburg, Warsaw, Sofia, Guanajuato, Peking, Tel Aviv-Jerusalem, Vancouver, Heraklion, Alicante.

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