Extended VC-dimension and Radon Type Theorems for Unions of Convex Sets

We define and study an extension of the notion of the VC-dimension of a hypergraph and apply it to establish a Tverberg type theorem for unions of convex sets. We also prove a new Radon type theorem for unions of convex sets, settling an open problem posed by Kalai in the 1970s.

Joint work with Shakhar Smorodinsky.