Metrical Properties of Convex Bodies in Minkowski Spaces

The objective of this dissertation is the application of Minkowskian cross-section measures (i.e., section and projection measures in finite-dimensional linear normed spaces over the real field) to various topics of geometric convexity in Minkowski spaces, such as bodies of constant Minkowskian width, Minkowskian geometry of simplices, geometric inequalities and the corresponding optimization problems for convex bodies. First we examine one-dimensional Minkowskian cross-section measures deriving (in a unified manner) various properties of these measures. Some of these properties are extensions of the corresponding Euclidean properties, while others are purely Minkowskian. Further on, we discover some new results on the geometry of a simplex in Minkowski spaces, involving descriptions of the so-called tangent Minkowskian balls and of simplices with equal Minkowskian heights. We also give some (characteristic) properties of bodies of constant width in Minkowski planes and in higher dimensional Minkowski spaces. This part of investigation has relations to the well known \emph{Borsuk problem} from the combinatorial geometry and to the widely used monotonicity lemma from the theory of Minkowski spaces. Finally, we study bodies of given Minkowskian thickness ($=$ minimal width) having least possible volume. In the planar case a complete description of this class of bodies is given, while in case of arbitrary dimension sharp estimates for the coefficient in the corresponding geometric inequality are found.Die Dissertation befasst sich mit Problemen fuer spezielle konvexe Koerper in Minkowski-Raeumen (d.h. in endlich-dimensionalen Banach-Raeumen). Es wurden Klassen der Koerper mit verschiedenen metrischen Eigenschaften betrachtet (z.B., Koerper konstante Breite, reduzierte Koerper, Simplexe mit Inhaltsgleichen Facetten usw.) und einige kennzeichnende und andere Eigenschaften fuer diese Klassen herleitet