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Helly's Theorem--A Very Early Introduction

Eric L. Grinberg

Abstract

We propose an interpretation of, and approach to, Helly's theorem that can be included quite early in the undergraduate curriculum. At the same time, the approach connects with contemporary models of data privacy and with sampling methods used in epidemiology. The presentation is intended to be accessible to teachers and their students.

Helly's Theorem--A Very Early Introduction

Abstract

We propose an interpretation of, and approach to, Helly's theorem that can be included quite early in the undergraduate curriculum. At the same time, the approach connects with contemporary models of data privacy and with sampling methods used in epidemiology. The presentation is intended to be accessible to teachers and their students.

Paper Structure

This paper contains 7 sections, 3 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Overdetermined systems: sampling for consistency.
  3. Conventions and notations.
  4. A tetrahedral example.
  5. Four equations suffice.
  6. Follow-up exercises.
  7. Recalling Venn Diagrams

Key Result

Proposition 1

Any $3$-equation subsystem of the tetrahedral linear system tetrahedron-system is consistent, but the system as a whole is inconsistent. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure :
  • Figure :
  • Figure :
  • Figure : Figure H

Theorems & Definitions (6)

  • Proposition
  • Theorem 1: Helly for Planes in $\mathbb R^3$
  • proof
  • Theorem 2: Minimalist Helly
  • proof
  • Remark