Book contents
- Frontmatter
- Contents
- Preface 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 andcurvatures
- 9 Extensions and analogues of theBrunn–Minkowski theory
- 10 Affine constructions and inequalities
- Appendix Spherical harmonics
- References
- Notation index
- Author index
- Subject index
10 - Affine constructions and inequalities
Published online by Cambridge University Press: 05 December 2013
- Frontmatter
- Contents
- Preface 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 andcurvatures
- 9 Extensions and analogues of theBrunn–Minkowski theory
- 10 Affine constructions and inequalities
- Appendix Spherical harmonics
- References
- Notation index
- Author index
- Subject index
Summary
The mixed volume V (Kl,…,Kn), which is a central notion of the Brunn–Minkowski theory, remains unchanged if the same volume-preserving affine transformation of ℝn is applied to each of the convex bodies Kl,…,Kn. The general theory of mixed volumes thus belongs to the affine geometry of convex bodies.
This affine geometry of convex bodies has much more to offer. In fact, affine-invariant constructions, functionals and extremum problems for convex bodies are a rich source of questions and results of considerable geometric beauty. Moreover, surprising relations to some other fields and unexpected applications have surfaced. In some parts of this field, the Brunn–Minkowski theory may be of help, and its extensions considered in the previous chapter play a prominent role and have had considerable impact, while other parts require various tools and methods of their own, and some new approaches still need to be discovered.
This last chapter is meant as an outlook. We collect and present various aspects of the affine geometry of convex bodies, but give very few proofs. We hope that this survey will be helpful for interested readers to find their own way into this fascinating field and its original literature.
- Type
- Chapter
- Information
- Convex Bodies: The Brunn–Minkowski Theory , pp. 528 - 622Publisher: Cambridge University PressPrint publication year: 2013