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, Carath\'eodory'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.

翻译：最近，通过伪半平面与有限点集的交集定义的几何超图被以纯组合的方式进行了定义。这导致将关于点和半平面的早期结果（包括多色染色和关于伪半平面的离散Helly型定理）扩展到了伪半平面。在此，我们延续这一研究方向，并引入了此类伪半平面超图中凸集的概念。在这一背景下，我们证明了与经典凸性结果相对应的若干定理，即Helly定理、Carathéodory定理、Kirchberger定理、分离定理、Radon定理以及Cup-Cap定理。这些结果蕴含了利用伪半平面在平面上定义的伪凸集的相关结果。结果表明，我们的多数结果也可利用定向拟阵和拓扑仿射平面进行证明，但我们的方法与之不同。与定向拟阵相比，我们的理论基于顶点集的线性序，这使得我们的定义和证明颇为不同，或许更为初等。与作为连续对象的拓扑仿射平面相比，我们的证明是纯组合的，且风格同样迥异。总之，我们相信这种新方法能够加深我们对这些基本凸性结果的理解。