Recently, a geometric hypergraph, which can be defined by the intersection of a quasi-half-plane and a finite point set, was defined in a purely combinatorial way. This extended early results on points and half-planes to pseudo-half-planes, and included discrete Helly-type theorems on polychromatic colors and pseudo-half-planes.
Here we continue this line of work and introduce the notion of convex sets for such quasi-semi-planar hypergraphs. In this context, we will prove several results corresponding to the classical results on convexity: Helly’s theorem, Callas-Eodory’s theorem, Kirchberger’s theorem, separation theorem, Radon’s theorem, and Cup-Cap theorem. increase. These results imply the respective result for a pseudo-convex set in the plane defined using the pseudo-half-plane.
We find that most of our results can also be proved using directed matroids and topological affine planes (TAPs), but our approach differs from both of them. Compared to directed matroids, our theory is based on a linear ordering of vertex sets, which makes the definitions and proofs quite different and perhaps more rudimentary. Compared to TAP, which is a continuous object, our proof is purely combinatorial and has a completely different flavor. Overall, we believe our new approach can advance our understanding of these fundamental convexity consequences.