拓扑绘图与凸几何经典定理的相遇
Topological Drawings meet Classical Theorems from Convex Geometry
AI总结：
本文研究凸几何经典定理在拓扑绘图中的应用，提出了纯组合性质的 Kirchberger 定理推广，引入广义 signotopes 结构，并展示了具有任意大 Helly 数的拓扑绘图及 Carathéodory 定理的新证明。
摘要：
在本文中，我们探讨了凸几何中的经典定理在拓扑绘图及更广泛背景下的应用。在完全图 $K_n$ 的简单拓扑绘图中，任意两条边最多共享一个点：要么是公共顶点，要么是它们交叉的点。简单拓扑绘图的三角形可以被视为凸集。这建立了与凸几何的联系。作为我们的主要成果，我们提出了一个具有纯组合性质的 Kirchberger 定理的推广。结果表明，这个经典定理同样适用于“广义 signotopes”——这是简单拓扑绘图的一种组合推广，我们在本文中对其进行了介绍和研究。正如其名称所示，它们是 signotopes 的推广，后者是在伪直线排列编码背景下研究的一种结构。我们还展示了一族具有任意大 Helly 数的简单拓扑绘图，给出了平面上 Carathéodory 定理拓扑推广的新证明，并进一步讨论了凸几何经典定理在简单拓扑绘图中的应用。
英文摘要：
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 Carathéodory's Theorem in the plane and discuss further classical theorems from Convex Geometry in the context of simple topological drawings.
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