## Abstract

A family S of convex sets in the plane defines a hypergraph H = (S, Ɛ) with S as a vertex set and Ɛ as the set of hyperedges as follows. Every subfamily S^{'} ⊂ S defines a hyperedge in E if and only if there exists a halfspace h that fully contains S^{'}, and no other set of S is fully contained in h. In this case, we say that h realizes S^{'}. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in ℝ^{d}, for d ≥ 3. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.

Original language | American English |
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Article number | 2 |

Journal | DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2021 |

## Keywords

- Convex sets
- Epsilon nets
- Halfplanes
- VC-dimension

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science
- Discrete Mathematics and Combinatorics