Abstract
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings. 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.
We present a generalization of Kirchberger’s Theorem, 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. We also discuss further classical theorems from Convex Geometry in the context of simple topological drawings.
We introduce “generalized signotopes” as a generalization of simple topological drawings. As indicated by the name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines.
Raphael Steiner was funded by DFG-GRK 2434. Stefan Felsner and Manfred Scheucher were partially supported by DFG Grant FE 340/12-1. Manfred Scheucher was partially supported by the internal research funding “Post-Doc-Funding” from Technische Universität Berlin. We thank Alan Arroyo, Emo Welzl, Heiko Harborth, and Geza Tóth for helpful discussions. A special thanks goes to Patrick Schnider for his simplification of the construction in the proof of Proposition 3.
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Notes
- 1.
Arrangements of pseudolines obtained by such extensions are equivalent to pseudoconfigurations of points, and can be considered as oriented matroids of rank 3 (cf. Chapter 5.3 of [17]).
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Bergold, H., Felsner, S., Scheucher, M., Schröder, F., Steiner, R. (2020). Topological Drawings Meet Classical Theorems from Convex Geometry. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_22
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