Reifenberg's Isoperimetric Inequality Revisited

TL;DR

This paper generalizes Reifenberg's isoperimetric inequality and applies it to prove the existence of minimizers in an axiomatic Plateau problem involving anisotropic surface energies.

Contribution

It introduces a generalized isoperimetric inequality and demonstrates its use in establishing minimizer existence for an axiomatic surface problem with anisotropic weights.

Findings

Established a generalized Reifenberg inequality

Proved existence of minimizers for anisotropic Plateau problem

Extended classical isoperimetric results to a broader setting

Abstract

We prove a generalization of Reifenberg's isoperimetric inequality. The main result of this paper is used to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces which satisfies a set of axioms, namely being closed under certain deformations and Hausdorff limits. This problem is known as the axiomatic Plateau problem.