This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski 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.
Author(s): Daniel Hug, Wolfgang Weil
Series: Graduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 287
Tags: Convexity, Geometry
Preface
Contents
List of Symbols
Preliminaries and Notation
1 Convex Sets
1.1 Algebraic Properties
Exercises and Supplements for Sect. 1.1
1.2 Combinatorial Properties
Exercises and Supplements for Sect. 1.2
1.3 Topological Properties
Exercises and Supplements for Sect. 1.3
1.4 Support and Separation
Exercises and Supplements for Sect. 1.4
1.5 Extremal Representations
Exercises and Supplements for Sect. 1.5
2 Convex Functions
2.1 Properties and Operations
Exercises and Supplements for Sect. 2.1
2.2 Regularity
Exercises and Supplements for Sect. 2.2
2.3 The Support Function
Exercises and Supplements for Sect. 2.3
3 Brunn–Minkowski Theory
3.1 The Space of Convex Bodies
Exercises and Supplements for Sect. 3.1
3.2 Volume and Surface Area
Exercises and Supplements for Sect. 3.2
3.3 Mixed Volumes
Exercises and Supplements for Sect. 3.3
3.4 The Brunn–Minkowski Theorem
Exercises and Supplements for Sect. 3.4
3.5 The Alexandrov–Fenchel Inequality
Strongly Isomorphic Polytopes
Mixed Volumes of Strongly Isomorphic Polytopes
Exercises and Supplements for Sect. 3.5
3.6 Steiner Symmetrization
Exercises and Supplements for Sect. 3.6
4 From Area Measures to Valuations
4.1 Mixed Area Measures
Exercises and Supplements for Sect.4.1
4.2 An Existence and Uniqueness Result
Exercises and Supplements for Sect.4.2
4.3 A Local Steiner Formula
Exercises and Supplements for Sect.4.3
4.4 Projection Bodies and Zonoids
Exercises and Supplements for Sect.4.4
4.5 Valuations
Exercises and Supplements for Sect.4.5
5 Integral-Geometric Formulas
5.1 Invariant Measures
Exercises and Supplements for Sect.5.1
5.2 Projection Formulas
Exercises and Supplements for Sect.5.2
5.3 Section Formulas
Exercises and Supplements for Sect.5.3
5.4 Kinematic Formulas
Exercises and Supplements for Sect.5.4
6 Solutions of Selected Exercises
6.1 Solutions of Exercises for Chap. 1
Exercise 1.1.3
Exercise 1.1.7
Exercise 1.1.8
Exercise 1.1.13
Exercise 1.2.7
Exercise 1.2.11
Exercise 1.3.3
Exercise 1.4.3
Exercise 1.4.8
Exercise 1.4.11
Exercise 1.4.12
Exercise 1.4.13
Exercise 1.5.3
6.2 Solutions of Exercises for Chap. 2
Exercise 2.1.2
Exercise 2.2.4
Exercise 2.2.5
Exercise 2.2.6
Exercise 2.2.7
Exercise 2.2.13
Exercise 2.3.1
Exercise 2.3.5
6.3 Solutions of Exercises for Chap. 3
Exercise 3.1.8
Exercise 3.1.15
Exercise 3.2.4
Exercise 3.3.1
Exercise 3.3.9
Exercise 3.4.2
Exercise 3.4.11
Exercise 3.4.13
6.4 Solutions of Exercises for Chap. 4
Exercise 4.2.3
Exercise 4.4.3
Exercise 4.4.6
Exercise 4.4.8
Exercise 4.5.4
6.5 Solutions of Exercises for Chap. 5
Exercise 5.1.6
References
Index
