Affine convex geometry – Part 2

The main objective of this chapter is to study some important (starlike or convex) bodies associated to a given convex body. These are the so-called associated bodies, and they play a central role in the affine geometry of convex bodies. For example, we state and prove the Petty projection inequality, which is stronger than the classical isoperimetric inequality. Recall that we regard the space \(\mathcal{C}\mathcal{C}^n\) (of non-empty compact convex subsets of \(\mathbb {R}^n\) ) as a metric space (with the Hausdorff distance) endowed with Minkowski addition and scalar multiplication operations. Subsets of \(\mathcal{C}\mathcal{C}^n\) which, in a certain sense, “preserve” these structures are known as Minkowski classes. We briefly investigate this concept in Section 10.1.