论文标题
新的离散理论的伪有源理论
A new discrete theory of pseudoconvexity
论文作者
论文摘要
最近，可以通过纯粹的组合方式定义了可以通过伪平面与有限点集的相交定义的几何超图。这导致了早期关于点和半平面的结果扩展到伪集，包括多色色彩以及关于伪alflanflanes的离散性Helly-type定理。 在这里，我们继续进行这一研究，并介绍了此类Pseudohalflane超图的凸组概念。在这种情况下，我们证明了几个结果，这些结果与有关凸的经典结果相对应，即Helly的定理，Carathéodory的定理，Kirchberger的定理，分离定理，Radon的定理和Cup-Cap定理。这些结果暗示了使用伪alfplanes定义的平面中的假子共振值集的相应结果。 事实证明，我们的大多数结果也可以使用定向的矩形和拓扑仿射平面（TAP）证明，但我们的方法与两者都不同。与定向的矩形相比，我们的理论基于顶点集的线性顺序，这使我们的定义和证明完全不同，甚至更基本。与是连续物体的水龙头相比，我们的证明纯粹是组合的，风味又完全不同。总而言之，我们相信我们的新方法可以进一步了解这些基本凸性结果。
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.
