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 Carath\'{e}odory'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 .
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 .
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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
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. 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