Extended VC-dimension, and Radon and Tverberg type theorems for unions of convex sets

TL;DR

This paper introduces new theorems extending Radon and Tverberg results to unions of convex sets, utilizing an extended VC-dimension concept, and resolves a long-standing open problem from the 1970s.

Contribution

It presents a novel Radon type theorem for unions of convex sets and extends the VC-dimension notion to hypergraphs, leading to new Tverberg type results.

Findings

Proved a Radon type theorem for unions of convex sets

Extended VC-dimension concept for hypergraphs

Established a Tverberg type theorem for unions of convex sets

Abstract

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