伪凸性的一种新离散理论
A new discrete theory of pseudoconvexity
AI总结：
本文延续几何超图研究，引入伪半平面超图的凸集概念，以纯组合方式证明了 Helly、Carathéodory 等经典凸性定理，为伪凸性提供了新理解。
摘要：
最近，可以通过伪半平面与有限点集的交点来定义的几何超图以纯组合的方式被定义。这导致将早期关于点和半平面的结果推广到伪半平面，包括多色着色和关于伪半平面的离散 Helly 型定理。在此我们继续这一研究路线，并引入此类伪半平面超图的凸集概念。在此背景下，我们证明了若干对应于经典凸性结果定理，即 Helly 定理、Carathéodory 定理、Kirchberger 定理、分离定理、Radon 定理和 Cup-Cap 定理。这些结果蕴含了关于使用伪半平面定义的平面中伪凸集的相应结果。结果表明，我们的大多数结果也可以使用定向拟阵和拓扑仿射平面（TAPs）来证明，但我们的方法与这两者不同。与定向拟阵相比，我们的理论基于顶点集的线性排序，这使得我们的定义和证明截然不同，或许更为初等。与作为连续对象的 TAPs 相比，我们的证明是纯组合的，风格同样迥异。总之，我们相信新方法能加深对这些基本凸性结果的理解。
英文摘要：
Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carathéodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.
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