# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.00108