New Results in Tropical Discrete Geometry

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(nd−1) for the number of tropical halving sets in any fixed dimension d.