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A new discrete theory of pseudoconvexity

Balázs Keszegh

TL;DR

This work develops a purely combinatorial framework for discrete convexity via pseudohalfplane hypergraphs, introducing ABA-free structure, topsets, bottomsets, and extremal vertices to define pseudoconvex sets. It proves discrete analogues of the classical convexity results—Helly, Carathéodory, Kirchberger, Separation, Radon, and Cup-Cap—for pseudoconvex sets, using extension lemmas and extremal-vertex analysis. The results connect to geometric models such as TAPs and rank-3 acyclic oriented matroids, while remaining intrinsic to hypergraph combinatorics, offering potential algorithmic applications. The paper also establishes the tightness of several results and outlines pathways to higher dimensions and broader convexity frameworks. Overall, it provides a robust, elementary approach to discrete convexity that mirrors classical theorems in a broader combinatorial setting.

Abstract

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.

A new discrete theory of pseudoconvexity

TL;DR

This work develops a purely combinatorial framework for discrete convexity via pseudohalfplane hypergraphs, introducing ABA-free structure, topsets, bottomsets, and extremal vertices to define pseudoconvex sets. It proves discrete analogues of the classical convexity results—Helly, Carathéodory, Kirchberger, Separation, Radon, and Cup-Cap—for pseudoconvex sets, using extension lemmas and extremal-vertex analysis. The results connect to geometric models such as TAPs and rank-3 acyclic oriented matroids, while remaining intrinsic to hypergraph combinatorics, offering potential algorithmic applications. The paper also establishes the tightness of several results and outlines pathways to higher dimensions and broader convexity frameworks. Overall, it provides a robust, elementary approach to discrete convexity that mirrors classical theorems in a broader combinatorial setting.

Abstract

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.
Paper Structure (20 sections, 29 theorems, 5 figures)

This paper contains 20 sections, 29 theorems, 5 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Results
  4. Helly theorems for pseudohalfplanes
  5. Helly theorems for pseudoconvex sets
  6. Carathéodory's, Kirchberger's and the Separation theorem
  7. Radon's theorem
  8. Cup-Cap Theorem
  9. Geometric variants
  10. Properties of pseudohalfplane hypergraphs
  11. Extensions of pseudohalfplane hypergraphs
  12. Properties of the extremal vertices
  13. Proofs
  14. Proofs of Helly theorems about pseudoconvex sets
  15. Proofs of generalizations of further classical results
  16. ...and 5 more sections

Key Result

Lemma 4

abafree If $\mathcal{F}$ is ABA-free, then every $A\in \mathcal{F}$ contains an unskippable vertex.

Figures (5)

  • Figure 1: Realization with upwards halfplanes a pseudohalfplane hypergraph in which $v\in Conv(S')$ yet $v$ is not in the convex hull of any proper subset of the vertices of $S'$.
  • Figure 2: Lemma \ref{['lem:extension']}, Middle + Side case.
  • Figure 3: Lemma \ref{['lem:extension']}, Middle + Middle case.
  • Figure 4: Lemma \ref{['lem:extension']}, Side + Same Side case.
  • Figure 5: Lemma \ref{['lem:extension']}, Side + Opposite Side case.

Theorems & Definitions (77)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 10: Primal Discrete Helly theorem for pseudohalfplanes, $3\rightarrow +1$
  • ...and 67 more