论文标题
拓扑图符合凸几何的古典定理
Topological Drawings meet Classical Theorems from Convex Geometry
论文作者
论文摘要
在本文中，我们在拓扑图及以后讨论了凸几何的古典定理。在完整图$ k_n $的简单拓扑图中，任何两个边缘最多共享：一个常见的顶点或它们越过的点。简单拓扑图的三角形可以看作是凸集。这给出了凸几何的链接。 作为我们的主要结果，我们提出了纯粹具有组合性质的Kirchberger定理的概括。事实证明，该古典定理也适用于“通用符号” - 简单拓扑图的组合概括，我们在本文的过程中介绍和调查。如该名称所示，它们是签名的概括，这是一种在编码伪oline的编码的背景下研究的结构。 我们还展示了一个简单拓扑图的家族，这些图纸是任意较大的Helly数字，以及在飞机上对Carathéodory定理进行拓扑概括的新证明，并在简单拓扑图的背景下讨论了来自凸几何的进一步的经典定理。
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.
