## ABSTRACT

In this chapter we present selected proofs of some theorems from Chapter 1 about the isoperimetric problem and the volume of polytopes. In Section 6.1 we prove Ball’s famous reverse isoperimetric inequality. In Section 6.2 we prove the monotonicity of the isoperimetric ratio under the Eikonal equation, and a dynamic variant of Lindelöf’s Condition. In Sections 6.3 and 6.4 we determine the largest volume of polytopes inscribed in the unit sphere of
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, respectively, with a small number of vertices. In Section 6.5 we characterize polytopes with n vertices whose symmetry group is isomorphic to the dihedral group D_{n}
. Section 6.6 presents the proof of the realization of every combinatorial class of 3-dimensional convex polyhedra by a midscribed polyhedron, that is, by a Koebe polyhedron. In Section 6.7 we prove that this realization is unique, up to Euclidean isometries, under the additional assumption that the barycenter of the tangency points of the polyhedron is the center of the midsphere. Finally, in Section 6.8, we prove results about the general problem of centering Koebe polyhedra.