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Colorful Intersections and Tverberg Partitions

Michael Gene Dobbins, Andreas F. Holmsen, Dohyeon Lee

Abstract

The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime power. Suppose $F_1, F_2, \dots, F_m$ are families of convex sets in $\mathbb{R}^d$, each of size $n > (\frac{d}{m}+1)(k-1)$, such that for any choice $C_i\in F_i$ we have $\bigcap_{i=1}^mC_i\neq \emptyset$. Then, one of the families $F_i$ admits a Tverberg $k$-partition. That is, one of the $F_i$ can be partitioned into $k$ nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning $r$-dimensional transversals to families of convex sets in $\mathbb{R}^d$ that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.

Colorful Intersections and Tverberg Partitions

Abstract

The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers and a prime power. Suppose are families of convex sets in , each of size , such that for any choice we have . Then, one of the families admits a Tverberg -partition. That is, one of the can be partitioned into nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we also obtain a result concerning -dimensional transversals to families of convex sets in that satisfy the colorful Helly hypothesis, which extends the work of Karasev and Montejano.
Paper Structure (13 sections, 8 theorems, 60 equations, 1 figure)

This paper contains 13 sections, 8 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. An application to geometric transversals
  5. Proof of Theorem \ref{['thm:tverberg']}
  6. The configuration space
  7. Sarkaria's criterion
  8. The test map
  9. Using colorful intersections to avoid a subspace
  10. Finishing the proof
  11. Proof of Lemma \ref{['lem:pcomplex']}
  12. The complex of partial surjections
  13. Acyclic matchings

Key Result

Theorem 1.1

Given integers $d\geq m \geq 1$ and $k$ a prime power. Suppose $F_1, F_2, \dots, F_m$ are families of convex sets in $\mathbb{R}^d$, with $|F_i| = n > (\frac{d}{m}+1)(k-1)$, that satisfy the colorful intersection property. Then one of the $F_i$ admits a Tverberg $k$-partition.

Figures (1)

  • Figure 1: Above: The polyhedral cell of $C_{6,3}$ corresponding to the ordered partition $( \{1,2,5\}, \{3\}, \{4,6\} )$, which we express more succinctly by $(125 \: | \: 3 \: | \: 46)$. Below: The cell complex $C_{4,2}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • proof
  • Remark 1.6
  • Remark 1.7
  • Theorem : Volovikov volov-96
  • Lemma 2.1
  • ...and 6 more