# Topological Drawings meet Classical Theorems from Convex Geometry

arXiv: https://arxiv.org/abs/2005.12568 · HTML: https://mlsanity.com/paper/2005.12568/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2020.

## Built on by (most cited first, 1 of 1)

- [Finding hardness reductions automatically using SAT solvers](https://mlsanity.com/paper/2402.06397/) (2024)

## Abstract (arXiv)

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 Kn, 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'eodory's Theorem in the plane and discuss further classical theorems from Convex Geometry in the context of simple topological drawings.