Convexity

Title page
Preface
Chapter 1. GENERAL PBOPERTIES 0F CONVEX SETS
1 Preliminaries and notation
2 The definition of convexity and its relation to affine transformations
3 Intersections, closures and interiors of convex sets
4 Sections and projections of convex sets
6 The dimension of a convex,set. Barycentric coordinates
6 Intersections of convex sets with hyperplanes
7 Separation of convex seta and support hyperplanes
8 The convex cover
9 Duality in Euclidean space
10 Convex polytopes
11 Continuous mappings of convex sets. Regular convex sets
Chapter 2. HELLY'S THEOREM AND ITS APPLICATIONS
1 Radon's proof of Helly's theorem
2 Carathéodory's theorem
3 The relation of Helly's theorem to Carathéodory's theorem
4 Kirchberger's theorem
5 Horn's extensions of Helly's theorem
Chapter 3. GENERAL PBOPERTIES OF OONVEX FUNCTIONS
1 The definition of a convex function; boundedness and continuity
2 The directional derivatives of a convex function
3 Differential conditions for convexity
4 Planar convexity in terms of polar coordinates
5 The distance and support functions of a convex set
Chapter 4. APPROXIMATIONS TO CONVEX SETS. THE BLASCHKE SELECTION THEOREM
1 Classes of convex sets as metric spaces
2 The Blaschke selection theorem
3 Approximations by convex polytopes and regular convex sets
4 The volume of a convex set. Continuity
5 Sets and numbers aesociated with a convex set
Chapter 5. TRANSFORMATIONS AND COMBINATIONS OF CONVEX SBTS
1 Linear and concave arrays of convex sets
2 Mixed volumes
3 Surface area
4 Steiner symmetrization
5 The Brunn-Minkowski theorem. The Minkowski and Fenchel-A1exandroff inequalities
6 Central symmetrization
Chapter 6. SOME SPECIAL PROBLEMS
1 The isoperimetric inequality
2 The isoperimetric inequality in R²
3 Relations between the inradius, circumradius. Minimal width and diameter of a convex set
4: Plane convex sets
Chapter 7. SETS OF CONSTANT WIDTH
1 General properties
2 Plane sets of constant width
Notes
References
Index