Topological Drawings meet Classical Theorems from Convex Geometry
Helena Bergold, Stefan Felsner, Manfred Scheucher, Felix Schröder, Raphael Steiner · arXiv (Cornell University) · 2020
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. As our main result, we present a generalization of Kirchberger's Theorem that is of purely combinatorial nature. It turned out that this classical theorem also applies to "generalized signotopes" - a combinatorial generalization of simple topological drawings, which we introduce and investigate in the course of this article. As indicated by the name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines. We also present a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory's Theorem in the plane and discuss further classical theorems from Convex Geometry in the context of simple topological drawings.