Convex Bodies: The Brunn-Minkowski Theory

At the heart of this monograph is the Brunn-Minkowski theory, which can be used to great effect in studying such ideas as volume and surface area and their generalizations. In particular, the notions of mixed volume and mixed area measure arise naturally and the fundamental inequalities that are satisfied by mixed volumes are considered here in detail. The author presents a comprehensive introduction to convex bodies, including full proofs for some deeper theorems. The book provides hints and pointers to connections with other fields and an exhaustive reference list. This second edition has been considerably expanded to reflect the rapid developments of the past two decades. It includes new chapters on valuations on convex bodies, on extensions like the Lp Brunn-Minkowski theory, and on affine constructions and inequalities. There are also many supplements and updates to the original chapters, and a substantial expansion of chapter notes and references. Review of the first edition: 'Neither one of [the old classics] may be considered a substitute for the excellent detailed monograph written by Rolf Schneider. I recommend this book to everyone who appreciates the beauty of convexity theory or who uses the strength of geometric inequalities, and to any expert who needs a reliable reference book for his/her research.' V. Milman, Bulletin of the American Mathematical Society Review of the first edition: 'Professor Schneider's book is the first comprehensive account of the Brunn-Minkowski theory and will immediately become the standard reference for the Aleksandrov-Fenchel inequalities and the current knowledge concerning the cases of equality and estimates of their stability. The book is aimed at a broad audience from graduate students to working professionals. The presentation is very clear and I enjoyed reading it.' Bulletin of the London Mathematical SocietyPreface to the second edition; Preface to the first edition; General hints to the literature; Conventions and notation; 1. Basic convexity; 2. Boundary structure; 3. Minkowski addition; 4. Support measures and intrinsic volumes; 5. Mixed volumes and related concepts; 6. Valuations on convex bodies; 7. Inequalities for mixed volumes; 8. Determination by area measures and curvatures; 9. Extensions and analogues of the Brunn-Minkowski theory; 10. Affine constructions and inequalities; Appendix. Spherical harmonics; References; Notation index; Author index; Subject index.