A new discrete theory of pseudoconvexity

Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were in purely combinatorial way. This led to extensions earlier results about points and halfplanes pseudohalfplanes, including polychromatic colorings discrete Helly-type theorems pseudohalfplanes. Here we continue this line research introduce the notion convex sets such pseudohalfplane hypergraphs. In context prove several corresponding classical convexity, namely Helly's Theorem, Carath\'eodory's Kirchberger's Separation Radon's Theorem Cup-Cap Theorem. These imply respective pseudoconvex plane using It turns out most our also proved oriented matroids topological affine planes (TAPs) but approach is different from both them. Compared matroids, theory based on linear ordering vertex which makes definitions proofs quite perhaps more elementary. TAPs, are continuous objects, again flavor. Altogether, believe new further understanding these fundamental convexity results.