Analysis of the Jante’s Law Process and Proof of Conjecture: Coupling Y (⋅) and Z(⋅)

Table of Links

Abstract and Introduction

Preliminaries

Reduction to the case of uniform geometry

All original points are eventually removed, a. s.

Proof of Theorem 1

Coupling Y (⋅) and Z(⋅)

Acknowledgements and References

Appendix

6 Coupling Y (⋅) and Z(⋅)

In this section, we complete the proof of Theorem 2. By Lemma 10 it suffices to prove the absolute continuity of the limit point ξ for a B-valued Jante’s law process where B is bounded. So we shall assume throughout this section that B is bounded, and therefore has uniform geometry.

We will need an isoperimetric inequality for inner shells of convex bodies, for which we have been unable to find a reference. It concerns the following problem. Suppose you have a (possibly) hollow chocolate egg whose outer boundary is the boundary of a convex body. If all the chocolate is within distance r of the outer boundary of the egg, what is the maximum quantity of chocolate that can possibly be contained within a ball of radius R? See Figure 4.

8 Think of δ small and ∆ large.