Reifenberg’s isoperimetric inequality revisited

Abstract

We prove a generalization of Reifenberg’s isoperimetric inequality. The main result of this paper is used in Harrison and Pugh (General methods of elliptic minimization, Available on arxiv, 2016) 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.