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

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.