Lectures on Convex Geometry

Description

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. 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