Following the recent work of Develin and Sturmfels and others (see, e.g., [10, 16, 2, 11]), we investigate discrete geometric questions over the tropical semiring (R, min, +). Specifically, we obtain the following tropical analogues of classical theorems in convex geometry: a separation theorem for a pair of disjoint tropical polytopes by tropical halfspaces and tropical versions of Radon’s lemma, Helly’s theorem, the Centerpoint theorem, and Tverberg’s theorem, including algorithms to find tropical centerpoints and Tverberg points. We also prove tropical analogues of the colored Carathéodory and colored Tverberg theorems. Furthermore, we study the tropical analogues of k-sets and levels in halfspace arrangements and obtain tight bounds of Θ(n d−1) for the number of tropical halving sets in any fixed dimension d. 1

We prove several new results concerning k-sets of point sets on the 2-sphere (equivalently, for signed point sets in the plane) and k-sets in 3-space. Specific results include spherical generalizations of (i) Lovász’ lemma (regarding the number of spherical k-edges intersecting a given great circle) and of (ii) the crossing identity for k-edges due to Andrzejak et al. As a new ingredient compared to the planar case, the latter involves the winding number of k-facets around a given point in 3-space, as introduced by Lee and by Welzl, independently. As a corollary, we obtain a crossing identity for the number of pinched crossings (crossing pairs of triangles sharing one vertex) of k-facets in 3-space.

Many geometric problems are intrinsically linked to the issue of splitting or classifying points. We investigate two such families of problems in two separate branches of research. Guarding problems are motivated by the issue of guarding a region with security cameras or illuminating it with lights. Such problems have been studied for decades, but there are two significant guarding problems whose complexity is not completely understood. First, we investigate the problem of guarding simple polygons; this problem is known to be NP-complete but its approximability is not known. Second, we investigate the complexity of guarding monotone chains, also known as 1.5-dimensional terrains. Understanding the interaction of ‘visibility polygons ’ and how they separate point sets is crucial for the investigation of such problems. We resolve a significant open problem by proving strong NP-completeness for terrain guarding. We also present an approximation algorithm for guarding simple polygons with perimeter guards; this new algorithm improves the state of the art.