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.