Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, September 19, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Nicolas Grelier

On the VC-dimension of convex sets and half-spaces

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 that 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 any Euclidean space of dimension at least 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.

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