Discrete geometry in normed spaces [Elektronische Ressource] / by Margarita Spirova

DISCRETE GEOMETRY INNORMED SPACESbyDr. Margarita SpirovaHABILITATION THESISMathematische Fakult¨atTechnische Universit¨at ChemnitziiPrefaceSummaryThis work refers to ball-intersections bodies as well as covering, packing, and kissingproblems related to balls and spheres in normed spaces. A quick introduction to thesetopics and an overview of our results is given in Section 1.1 of Chapter 1. The neededbackground knowledge is collected in Section 1.2, also in Chapter 1. In Chapter 2 wedefine ball-intersection bodies and investigate special classes of them: ball-hulls, ball-intersections, equilateral ball-polyhedra, complete bodies and bodies of constant width.Thus, relations between the ball-hull and the ball-intersection of a set are given. Weextend a minimal property of a special class of equilateral ball-polyhedra, known asTheorem of Chakerian, to all normed planes. In order to investigate bodies of constantwidth, we develop a concept of affine orthogonality, which is new even for the Euclideansubcase. In Chapter 2 we solve kissing, covering, and packing problems. For a givenfamilyofcirclesandlineswefindatleastone, butforsomefamilieseven allcircleskissingall the members of this family. For that reason we prove that a strictly convex, smoothnormed plane is a topological M¨obius plane. We give an exact geometric descriptionof the maximal radius of all homothets of the unit disc that can be covered by 3 or 4translates of it.