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Topological Drawings Meet Classical Theorems from Convex Geometry

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Abstract

In this article we discuss classic 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 classic 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 their 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 classic theorems from Convex Geometry in the context of simple topological drawings.

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Notes

  1. The authors of [8] use the term face-convex instead of cell-convex.

  2. Arrangements supporting a drawing of \(K_n\) are also known as pseudoconfigurations of points and can be considered as oriented matroids of rank 3 (cf. [17, Chapt. 5.3]). For a formal definition of arrangements of pseudolines, we refer the interested reader to [19] or [17, Chapt. 6].

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Helena Bergold

  2. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Stefan Felsner, Manfred Scheucher, Felix Schröder & Raphael Steiner

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  1. Helena Bergold
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  2. Stefan Felsner
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Correspondence to Helena Bergold.

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We thank Alan Arroyo, Emo Welzl, Heiko Harborth, and Geza Tóth for inspiring discussions and the reviewers for helpful comments. A special thanks goes to Patrick Schnider for his simplification of the construction in the proof of Proposition 3. R. Steiner and H. Bergold were funded by DFG-GRK 2434. S. Felsner and M. Scheucher were supported by the DFG Grant FE 340/12-1. M. Scheucher was supported by the DFG Grant SCHE 2214/1-1 and by the internal research funding “Post-Doc-Funding” from Technische Universität Berlin.

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Bergold, H., Felsner, S., Scheucher, M. et al. Topological Drawings Meet Classical Theorems from Convex Geometry. Discrete Comput Geom 70, 1121–1143 (2023). https://doi.org/10.1007/s00454-022-00408-6

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