Convexity from the Geometric Point of View
Published on 15. July 2024
Book
Hardback
XIX, 1184 pages
978-3-031-50506-5 (ISBN)
Description
This text gives a comprehensive introduction to the "common core" of convex geometry. Basic concepts and tools which are present in all branches of that field are presented with a highly didactic approach. Mainly directed to graduate and advanced undergraduates, the book is self-contained in such a way that it can be read by anyone who has standard undergraduate knowledge of analysis and of linear algebra. Additionally, it can be used as a single reference for a complete introduction to convex geometry, and the content coverage is sufficiently broad that the reader may gain a glimpse of the entire breadth of the field and various subfields. The book is suitable as a primary text for courses in convex geometry and also in discrete geometry (including polytopes). It is also appropriate for survey type courses in Banach space theory, convex analysis, differential geometry, and applications of measure theory. Solutions to all exercises are available to instructors who adopt the text for coursework.
Most chapters use the same structure with the first part presenting theory and the next containing a healthy range of exercises. Some of the exercises may even be considered as short introductions to ideas which are not covered in the theory portion. Each chapter has a notes section offering a rich narrative to accompany the theory, illuminating the development of ideas, and providing overviews to the literature concerning the covered topics. In most cases, these notes bring the reader to the research front. The text includes many figures that illustrate concepts and some parts of the proofs, enabling the reader to have a better understanding of the geometric meaning of the ideas. An appendix containing basic (and geometric) measure theory collects useful information for convex geometers.
Most chapters use the same structure with the first part presenting theory and the next containing a healthy range of exercises. Some of the exercises may even be considered as short introductions to ideas which are not covered in the theory portion. Each chapter has a notes section offering a rich narrative to accompany the theory, illuminating the development of ideas, and providing overviews to the literature concerning the covered topics. In most cases, these notes bring the reader to the research front. The text includes many figures that illustrate concepts and some parts of the proofs, enabling the reader to have a better understanding of the geometric meaning of the ideas. An appendix containing basic (and geometric) measure theory collects useful information for convex geometers.
Reviews / Votes
"This impressive piece of work will be valuable for initiation and deeper study, as well as a detailed guide to historical and active research areas for anybody interested in the geometry and measure-theory of finite dimensional convex sets, and will probably remain a major reference for a long time." (Frank Plastria, zbMATH 1571.52001, 2026)
"This book on classical convex geometry is written in a clear and attractive form. It presents proofs of important results in a very detailed manner. An impressive collection of exercises and notes is given at the end of each chapter." (Marek Lassak, Mathematical Reviews, February, 2026)
More details
Series
Edition
2024 ed.
Language
English
Place of publication
Cham
Switzerland
Publishing group
Springer International Publishing
Target group
Primary & secondary/elementary & high school
Illustrations
13 farbige Abbildungen, 73 s/w Abbildungen
XIX, 1184 p. 86 illus., 13 illus. in color.
Dimensions
Height: 25.4 cm
Width: 17.8 cm
ISBN-13
978-3-031-50506-5 (9783031505065)
DOI
10.1007/978-3-031-50507-2
Schweitzer Classification
Other editions
Additional editions
Vitor Balestro | Horst Martini | Ralph Teixeira
Convexity from the Geometric Point of View
Book
07/2025
Birkhäuser
€106.99
Available immediately
Vitor Balestro | Horst Martini | Ralph Teixeira
Convexity from the Geometric Point of View
E-Book
07/2024
Birkhäuser
€106.99
Available for download
Persons
Vitor Balestro
is a Professor of Mathematics at Federal Fluminense University in Niterói, Brazil. He received his PhD in Mathematics from the same university in 2016. In Mathematics, his research interests include convex geometry, geometry of finite dimensional normed spaces, convex analysis and functional analysis. Outside of Mathematics, his main interests are computer programming, computer science, music, and literature.
Horst Martini Before his retirement in 2020, Horst Martini was Full Professor of Mathematics at the Chemnitz University of Technology in Germany (Chair of Geometry). After receiving his Ph.D. in Dresden, in 1988 he obtained his habilitation from the University of Jena, and in 1993 he received his position in Chemnitz. His research interests include convex geometry, discrete geometry, functional analysis, and classical subfields of geometry, as well as applications in optimization and related fields. He also studies certain fields in history of mathematics. For about fifteen years he was editor in chief f the Springer journal Contributions to Algebra and Geometry, and in 2015 he received an honorary professorship from the Harbin University f Science and Technology (China). His passions include travelling to interesting places of geographical and historical significance and studying different genres of good music.
Ralph Teixeira is a Professor of Mathematics at Federal Fluminense University in Niterói, Brazil. He received his PhD in Mathematics from Harvard University in 1998, and has since worked in many areas, including Mathematics applied to Computer Vision, Differential and Discrete Geometry, and lately Convex Geometry. Second to his wife, his main love is computer gaming, but he would never admit this in public.
Horst Martini Before his retirement in 2020, Horst Martini was Full Professor of Mathematics at the Chemnitz University of Technology in Germany (Chair of Geometry). After receiving his Ph.D. in Dresden, in 1988 he obtained his habilitation from the University of Jena, and in 1993 he received his position in Chemnitz. His research interests include convex geometry, discrete geometry, functional analysis, and classical subfields of geometry, as well as applications in optimization and related fields. He also studies certain fields in history of mathematics. For about fifteen years he was editor in chief f the Springer journal Contributions to Algebra and Geometry, and in 2015 he received an honorary professorship from the Harbin University f Science and Technology (China). His passions include travelling to interesting places of geographical and historical significance and studying different genres of good music.
Ralph Teixeira is a Professor of Mathematics at Federal Fluminense University in Niterói, Brazil. He received his PhD in Mathematics from Harvard University in 1998, and has since worked in many areas, including Mathematics applied to Computer Vision, Differential and Discrete Geometry, and lately Convex Geometry. Second to his wife, his main love is computer gaming, but he would never admit this in public.
Content
Preface.- 1. Convex functions.- 2. Convex sets.- 3. A first look into polytopes.- 4. Volume and area.- 5. Classical inequalities.- 6. Mixed volumes- 7. Mixed surface area measures.- 8. The Alexandrov-Frechel inequality.- 9. Affine convex geometry Part 1.- 10. Affine convex geometry Part 2.- 11. Further selected topics.-12. Historical steps of development of convexity as a field.- A. Measure theory for convex geometers.- References.- Index.