A sharp quantitative version of Hales' isoperimetric honeycomb theorem

TL;DR

This paper provides a precise quantitative enhancement of Hales' honeycomb theorem, utilizing advanced inequalities and convergence results to better understand optimal tilings and bubble clusters in various geometric settings.

Contribution

It introduces a sharp quantitative version of Hales' theorem, combining polygonal isoperimetric inequalities with improved convergence theorems for planar bubble clusters.

Findings

Enhanced understanding of isoperimetric tilings on the torus

Quantitative bounds for planar bubble cluster convergence

Applications to almost flat Riemannian metrics

Abstract

We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.