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

Noga Alon, Shakhar Smorodinsky

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.

Extended VC-dimension, and Radon and Tverberg type theorems 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.

Paper Structure

This paper contains 11 sections, 16 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Structure
  4. Preliminaries: VC-dimension and $r$-VC-dimension
  5. Proof of Theorem \ref{['thm:main']}
  6. A Generalized Tverberg Theorem
  7. Improved bounds in special cases
  8. The Asymmetric Case $f(d,s,1)$
  9. Upper Bounds
  10. Lower Bounds
  11. Concluding remarks and open problems

Key Result

Theorem 1.1

(Tverberg's TheoremTverberg1966) Let $r \geq 2$ be a fixed integer and $d \geq 1$. Then for any set $P$ of $(r-1)(d+1)+1$ points in $\mathbb R^d$ there exists a partition of $P$ into $r$ pairwise disjoint sets $P = \bigcup_{i=1}^r P_i$ such that $\bigcap_{i=1}^r \text{CH}(P_i) \neq \emptyset$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • Definition 2.1: VC-dimension
  • Remark 2.2
  • ...and 27 more