Depth, Crossings and Conflicts in Discrete Geometry

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Depth, Crossings and Conflicts in Discrete Geometry is written by Marek Sulovsk 'y and published by Logos Verlag Berlin GmbH. It's available with International Standard Book Number or ISBN identification 383253119X (ISBN 10) and 9783832531195 (ISBN 13).

Discrete geometry has been among the fastest growing fields of mathematics in the last decades. One of the most fascinating objects studied in discrete geometry are k-sets. Not only are they extremely difficult to understand but they also play an important role in estimating the running time of several geometric algorithms. This thesis presents developments in three areas related to k-sets. First, it examines the circle containment problem of Urrutia and Neumann-Lara and reveals its relationships to geometric partitioning problems and centre regions. Next, it investigates k-sets in low dimensions and generalises the k-edge crossing identity of Andrzejak et al. to the sphere. Last, it studies conflict-free colourings of geometric hypergraphs and extends many results on this topic to more restrictive list colouring variants.