Parameterabhängige dünne Überdeckungen konvexer Körper

The work deals with coverings of translates of centrally symmetric convex bodies in the d -dimensional euclidean space E d . A pointset C n ⊂ E d with n elements is called covering configuration (arrangement) with respect to K if conv C n ⊂ C n + K . With respect to the parametric density ϑ ( K, C n , ρ ) = n ⋅ V ( K )/ V (conv C n + ρK ) optimal (thin) covering configurations are considered, i.e. configurations with minimal density. In the euclidean plane such optimal configurations are fulldimensional for small parameter ρ , while large parameter lead to nearly one dimensional arrangements. For strictly convex bodies the last statement can be extended to arbitrary dimensions d ≥ 2. Those nearly one dimensional optimal configurations are called bones and will be treated detailed. Finally the last chapter deals with covering arrangements which belong to lattice coverings with respect to K . In this case for d = 2 and strictly convex K the optimal configurations can be characterized for all parameter ρ . They are either one dimensional, socalled doublesausages or fulldimensional, depending on the parameter. In the end for the euclidean d -ball the lattices are characterized, for which the optimal arrangement is one dimensional, for large parameter.