# On the VC-dimension of half-spaces with respect to convex sets

## TL;DR

This paper investigates the VC-dimension of hypergraphs formed by convex sets in the plane, revealing bounds for disjoint and intersecting sets, and exploring implications for geometric set cover and range-query problems.

## Contribution

It establishes tight bounds on the VC-dimension for disjoint convex sets and segments, and shows unbounded VC-dimension for intersecting convex sets in higher dimensions.

## Key findings

- VC-dimension for disjoint convex sets in the plane is exactly 3.
- VC-dimension for convex sets in the plane can be unbounded.
- Maximum VC-dimension for five segments in the plane is 5.

## Abstract

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H 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 > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always 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.