## ABSTRACT

In this chapter we collect selected proofs of some theorems from Chapter 2. In Section 7.1 we investigate the properties of the volume of a linear parameter system, and the Steiner symmetrization of a convex body. We use these theorems to prove sharp estimates about the volume of translation, reflection, and associated (d + 1)-dimensional body of a d-dimensional convex body in Section 7.2. In Section 7.3 we find similar estimates for c_{i}
(K) for certain values of K, and characterize the plane convex bodies satisfying the translative constant volume property. In Section 7.4 we prove the Blaschke-Santaló inequality about the maximum of the Holmes-Thompson volume of the unit ball of a normed space. In Section 7.5 we examine the dual problem in the plane, and determine the minimum of the Holmes-Thompson volume of the unit disks of 2-dimensional norms. In Section 7.6 we prove sharp estimates for the Gromov’s mass and mass* of the unit disk of a normed plane, and also for the normed volumes of unit disks of Radon norms. Finally, in Section 7.7 we determine the minimum and the maximum values of the normed variants of the quantity c^{tr}
(K) both if K is an arbitrary plane convex body, and if it is assumed to be centrally symmetric.