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Theory of Convex Bodies

by T. Bonnesen and W. Fenchel

Tranlated into English with the assistance of W. Fenchel from the 1974 revised German edition published by Springer-Verlag.

TABLE OF CONTENTS

1. Translator's Preface
   
2. Foreword
   
3. Preliminary Remarks on n-dimensional Geometry
   
4. Basic Concepts
   
   1. Convex Sets, Bodies and Cones
      
   2. Bounding Planes and Support Planes of a Closed Set
      
   3. The Convex Hull of a Closed Set
      
   4. Support Properties of Convex Bodies
      
5. Centroids and Convex Hulls
   
   1. Mass Distributions and their Centroids
      
   2. Representations of the Centroid of the Convex Hull
      
   3. Generation of the Convex Hull by Chords
      
   4. Centroids of Subbodies cut off by Hyperplanes and of Cross Sections
      
6. Classification of Boundary Points and Support Planes of a Convex Body
   
   1. Singular Boundary Points and Support Planes. Projections Cones and the Normal Cones. Vertices and Face Points
      
   2. Extreme Boundary Points and Support Planes
      
   3. Convex Polyhedra
      
   4. Cap Bodies and Tangential Bodies
      
7. Representation of Convex Bodies by Convex Functions
   
   1. Convex Functions and their Directional Derivatives
      
   2. The Distance Function of a Convex Body
      
   3. The Support Function of a Convex Body
      
   4. Representation of the Boundary Points of a Convex Body by Support Functions
      
   5. The Determination of a Convex Body by the Support Function
      
   6. Polar Bodies
      
8. Linear Combination of Convex Bodies. Linear and Concave Families
   
   1. Linear Combination of Support Bodies
      
   2. Linear Combinations of Convex Bodies
      
   3. Parallel Bodies of a Convex Body. Homothetic Bodies
      
   4. Behavior of Projections and Boundary Points under Linear Combinations.
      
   5. The Linear Combination of Degenerate Convex Bodies
      
   6. Linear and Concave Families of Convex Bodies
      
9. Approximation of Convex Bodies
   
   1. Convergent Sequences of Convex Bodies. The Blaschke Selection Theorem
      
   2. The Support Functions of Convergent Sequences of Bodies. The Function Space of Support Functions
      
   3. Approximation by Convex Polyhedra and Analytically Bounded Convex Bodies
      
10. Numbers and Figures Associated with Convex Bodies
    
    1. The Volume of a Convex Body
       
    2. The Volume of a Body in a Linear Family. Mixed Volumes
       
    3. Cress Sectional Measure. Projection Bodies
       
    4. The Surface Area of a Convex Body
       
    5. Cauchy's Surface Area Formula. Cross Sectional Measure Integrals
       
    6. Breadth, Diameter, Width.
       
    7. Centroids and Other Special Points of a Convex Body
       
    8. Circumscribed and Inscribed Balls. Minimal Annulus and other Related Figures
       
11. Integral Formulas for the Volume and the Mixed Volumes
    
    1. Formulas in Terms of the Coordinates of Points
       
    2. Representation of Mixed Volumes in Terms of Support Functions
       
    3. Curvature Functions and Curvature Integrals. Relative DIfferential Geometry
       
    4. Special Formulas. Geometric Probability Theory for Convex Bodies.
       
12. Symmetrization and Related Modifications of Convex Bodies
    
    1. Steiner Symmetrization and Annular Symmetrization
       
    2. Schwarz Rounding. Blaschke's Proof of the Brunn-Minkowski Theorem
       
    3. Central Symmetrization and Related Ideas
       
13. Inequalities, Extremal Problems and Covering Problems
    
    1. Generalities Concerning Extremal Problems
       
    2. Inequalities between two Quantities
       
    3. Inequalities involving more than two Quantities for a Region of the Plane
       
    4. Inequalities among several Quantities of Convex Bodies
       
    5. Coverings
       
14. The Brunn-Minkowski Theorem and the Minkowski Inequalities
    
    1. The Brunn-Minkowski Theorem
       
    2. Minkowski's Inequalities
       
    3. Refinements of the Brunn-Minkowski Theorem and of the Minkowski Inequalities
       
    4. More about the Case of the Plane
       
    5. More about Space. Hilbert's Proof of the Minkowski Inequalities
       
15. Special Cases, and Applications of the Brunn-Minkowski Theorem and of the Minkowski Inequalities
    
    1. The Volume of a Vector Body
       
    2. Estimates of Cross Sectional Measure Integrals in Terms of Width and Diameter
       
    3. The Surface Area of a the Bodies of a Linear Family
       
    4. Special Cases of the Minkowski Inequalities
       
    5. The Isoperimetric Problem
       
16. Determination of Convex Bodies by Curvature Functions
    
    1. Continuously Curved Bodies
       
    2. Uniqueness Theorems
       
    3. Existence Theorems
       
17. Convex Bodies with Center
    
    1. Characterizing Properties
       
    2. Convex Bodies with Center and Lattice Points
       
18. Bodies of Constant Breadth
    
    1. Characterizing and Other Properties
       
    2. Complete Sets
       
    3. Orbiforms
       
    4. Extremal Problems for Orbiforms
       
    5. Spheroforms
       
    6. Related Classes of Convex Bodies
       
19. Characteristic Properties of Manifolds of Second Degree
    
    1. Disk and Ball
       
    2. Ellipse and Ellipsoid
       
20. Differential Geometry of Convex Curves and Surfaces
    
    1. Curvature Properties of Convex Curves. The Four Vertex Theorem and Related Matters
       
    2. Surfaces of Positive Gaussian Curvature. Bending Questions.
       
21. Bibliography
    

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