Depth, Crossings and Conflicts in Discrete Geometry

Abstract

This thesis studies the following three combinatorial problems in discrete geometry: (i) centre points and depth of sets of points with respect to half-spaces or balls, (ii) crossings, their higher dimensional analogues and their significance in the context of k-sets, and (iii) conflict-free colourings of geometric hypergraphs. First, we consider the circle containment problem introduced by Urrutia and Neumann-Lara in the late 1980s. The original question in the plane is the following: What is the largest number f(n) such that in any set P of n ≥ 2 points in general position in the plane, there is a pair {a, b} ⊆ P of disc depth at least f(n), i.e. such that any disc D containing {a, b} has |D ∩ P | ≥ f(n)? We study a similar notion of depth with respect to various ranges (discs, balls, half-spaces) and improve the lower bounds on the higher dimensional variant of the problem – showing that every point set in R contains a subset of ⌊ 2 ⌋ points of ball depth roughly n d2 . This improves the previous lower bounds whose leading term was inversely exponentially dependent on the dimension d. As an intermediate step towards this improvement, we use random sampling and the bounds for the number of (≤ k)-sets to show that every point set in R has a small subset of high half-space depth. This has additional corollaries for several related geometric partitioning problems as well as for the structure of centre regions. Next, we look into crossings, their generalisations, and crossing identities of k-edges and k-facets. As the main result in this direction, we extend an identity of Andrzejak et al. onto the sphere. The original identity relates the number of k-edges in a set of points in the plane to the number of crossings between them and gives an alternative way of proving an O(nk) upper bound on the number of k-edges in the plane. We show a similar identity on the sphere. Unlike in the plane, we need to consider an additional quantity measuring the “non-planarity” of the point set: its g-vector. There are two immediate consequences of the new identity: (i) It allows us to interpolate between the k-edge upper bound in the plane and the trivial (but tight) upper bound on the sphere, depending on the values gk. (ii) Summing up our spherical