Lectures on Convex Geometry

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops BrunnMinkowski theory, with an exposition of mixed volumes, the BrunnMinkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book.

Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry.

Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

The text provides a self-contained and efficient one-semester introduction to the main concepts and results in convex geometry. The selected topics highlight the interactions between geometry and analysis, treating several topics for the first time in an introductory textbook. Suggestions for further reading and a large number of solved exercises complement the main text

Autorentext
Prof. Dr. Daniel Hug (1965) obtained his Ph.D. in Mathematics (1994) and Habilitation (2000) at Univ. Freiburg. He was an assistant Professor at TU Vienna (2000), trained and acted as a High School Teacher (20052007), was Professor in Duisburg-Essen (2007), Associate Professor in Karlsruhe (20072011), and has been a Professor in Karlsruhe since 2011.

Prof. Dr. Wolfgang Weil (19452018) obtained his Ph.D. in Mathematics at Univ. Frankfurt/Main in 1971 and his Habilitation in Freiburg (1976). He was an Assistant Professor in Berlin and Freiburg, Akad. Rat in Freiburg (19781980), and was a Professor in Karlsruhe from 1980. He was a Guest Professor in Norman, Oklahoma, USA (1985 and 1990).

Klappentext

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops BrunnMinkowski theory, with an exposition of mixed volumes, the BrunnMinkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book.

Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry.

Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

Inhalt
Preface.- Preliminaries and Notation.- 1. Convex Sets.- 2. Convex Functions.- 3. Brunn-Minkowski Theory.- 4. From Area Measures to Valuations.- 5. Integral Geometric Formulas.-6. Solutions of Selected Exercises.- References.- Index.