Totally Separable Packings of Convex Domains

Abstract

Two key combinatorial quantities that arise in the study of translative packings of convex bodies are the Hadwiger numbers and contact numbers. In this thesis, we will focus on computing these numbers for packings which are totally separable, particularly for two-dimensional bodies with smooth boundaries. The proofs require the use of various ideas from both combinatorics and analysis, such as minimum distance graphs, polyominoes, Birkhoff orthogonality, angular measures, and approximation via the Hausdorff metric.