Topological Drawings meet Classical Theorems from Convex Geometry
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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.
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