Topological Drawings meet Classical Theorems from Convex Geometry

TL;DR

This paper explores the connections between topological graph drawings and convex geometry, generalizing classical theorems like Kirchberger's and Carathéodory's to topological and combinatorial settings.

Contribution

It introduces generalized signotopes, extends classical convex geometry theorems to topological drawings, and provides new insights into their combinatorial and topological properties.

Findings

Generalization of Kirchberger's Theorem to signotopes

Construction of topological drawings with arbitrarily large Helly number

New proof of topological Carathéodory's Theorem

Abstract

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 Carath\'{e}odory's Theorem in the plane and discuss further classical theorems from Convex Geometry in the context of simple topological drawings.