ABSTRACT
The aim of this chapter is to present results about the volume of the convex hull of a pair of convex bodies. In Section 2.1 we introduce results of Rogers and Shephard, who in the 1950s in three subsequent papers defined various convex bodies associated to a convex body, and found the extremal values of their volumes on the family of convex bodies based on the convexity of some volume functional and Steiner symmetrization. After this we present a proof of a conjecture of Rogers and Shephard about the equality cases in their theorems, find a more general setting of their problems, and collect results about it. Section 2.2 describes a variant of the previous problems for normed spaces. In particular, using the notion of relative norm of a convex body, we collect results about the extremal values of four types of volume of the unit ball of a normed space. The problem of finding these values of one of these types leads to the famous Mahler Conjecture. We extend these notions to the translation bodies of convex bodies, defined in Section 2.1, and determine their extremal values in normed planes.
