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Convex Bodies: The Brunn–Minkowski Theory, Second Edition

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Series Title
  4. Title
  5. Copyright
  6. Content
  7. Preface to the second edition
  8. Preface to the first edition
  9. General hints to the literature
  10. Conventions and notation
  11. 1 Basic convexity
    1. 1.1 Convex sets and combinations
    2. 1.2 The metric projection
    3. 1.3 Support and separation
    4. 1.4 Extremal representations
    5. 1.5 Convex functions
    6. 1.6 Duality
    7. 1.7 Functions representing convex sets
    8. 1.8 The Hausdorff metric
  12. 2 Boundary structure
    1. 2.1 Facial structure
    2. 2.2 Singularities
    3. 2.3 Segments in the boundary
    4. 2.4 Polytopes
    5. 2.5 Higher regularity and curvature
    6. 2.6 Generalized curvatures
    7. 2.7 Generic boundary structure
  13. 3 Minkowski addition
    1. 3.1 Minkowski addition and subtraction
    2. 3.2 Summands and decomposition
    3. 3.3 Additive maps
    4. 3.4 Approximation and addition
    5. 3.5 Minkowski classes and additive generation
  14. 4 Support measures and intrinsic volumes
    1. 4.1 Local parallel sets
    2. 4.2 Steiner formula and support measures
    3. 4.3 Extensions of support measures
    4. 4.4 Integral-geometric formulae
    5. 4.5 Local behaviour of curvature and area measures
  15. 5 Mixed volumes and related concepts
    1. 5.1 Mixed volumes and mixed area measures
    2. 5.2 Extensions of mixed volumes
    3. 5.3 Special formulae for mixed volumes
    4. 5.4 Moment vectors, curvature centroids, Minkowski tensors
    5. 5.5 Mixed discriminants
  16. 6 Valuations on convex bodies
    1. 6.1 Basic facts and examples
    2. 6.2 Extensions
    3. 6.3 Polynomiality
    4. 6.4 Translation invariant, continuous valuations
    5. 6.5 The modern theory of valuations
  17. 7 Inequalities for mixed volumes
    1. 7.1 The Brunn–Minkowski theorem
    2. 7.2 The Minkowski and isoperimetric inequalities
    3. 7.3 The Aleksandrov–Fenchel inequality
    4. 7.4 Consequences and improvements
    5. 7.5 Wulff shapes
    6. 7.6 Equality cases and stability
    7. 7.7 Linear inequalities
  18. 8 Determination by area measures and curvatures
    1. 8.1 Uniqueness results
    2. 8.2 Convex bodies with given surface area measures
    3. 8.3 The area measure of order one
    4. 8.4 The intermediate area measures
    5. 8.5 Stability and further uniqueness results
  19. 9 Extensions and analogues of the Brunn–Minkowski theory
    1. 9.1 The Lp Brunn–Minkowski theory
    2. 9.2 The Lp Minkowski problem and generalizations
    3. 9.3 The dual Brunn–Minkowski theory
    4. 9.4 Further combinations and functionals
    5. 9.5 Log-concave functions and generalizations
    6. 9.6 A glimpse of other ramifications
  20. 10 Affine constructions and inequalities
    1. 10.1 Covariogram and difference body
    2. 10.2 Qualitative characterizations of ellipsoids
    3. 10.3 Steiner symmetrization
    4. 10.4 Shadow systems
    5. 10.5 Curvature images and affine surface areas
    6. 10.6 Floating bodies and similar constructions
    7. 10.7 The volume product
    8. 10.8 Moment bodies and centroid bodies
    9. 10.9 Projection bodies
    10. 10.10 Intersection bodies
    11. 10.11 Volume comparison
    12. 10.12 Associated ellipsoids
    13. 10.13 Isotropic measures, special positions, reverse inequalities
    14. 10.14 Lp zonoids
    15. 10.15 From geometric to analytic inequalities
    16. 10.16 Characterization theorems
  21. Appendix Spherical harmonics
  22. References
  23. Notation index
  24. Author index
  25. Subject index