## ABSTRACT

In this chapter we investigate the volumetric properties of (m, d)-scribed polytopes, that is, d-dimensional polytopes whose every m-face touches a Euclidean ball. Starting with a short historical overview of the isoperimetric problem in Section 1.1, in Section 1.2 we examine a problem proposed by L’huilier in the 18th century, which asks about finding the minimum volume polytopes circumscribed about a ball. By the famous Lindelöf Condition in Theorem 4, these polytopes are the ones having maximal isoperimetric ratio and a fixed number of facets. In this section, besides describing the results regarding L’huilier’s problem, we present a ‘dynamic’ version of Lindelöf’s theorem. This version offers a possible explanation of the strange, elongated shape of the interstellar asteroid ‘Oumuamua passing through the solar system. In Section 1.3 we investigate the ‘dual’ of L’huilier’s problem, and deal with maximum volume polytopes inscribed in a ball. A consequence of the Circle Packing Theorem is that every combinatorial class of 3-dimensional convex polytope can be represented by a polytope, called Koebe polyhedron, whose every edge is tangent to a unit sphere. In Section 1.4 we collect the properties of polyhedra and, in particular, present results about the existence of Koebe polyhedra in every combinatorial class with additional requirements.