On the VC-dimension of convex sets and half-spaces
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A family S of convex sets in the plane defines a hypergraph H = (S,E) as follows. Every subfamily S'⊂ 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 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 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.
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