Reifenberg flatness for almost-minimizers of the perimeter under minimal assumptions

TL;DR

This paper proves that almost-minimizers of the perimeter are Reifenberg flat under minimal assumptions, showing that small excess at one scale ensures small excess at all smaller scales.

Contribution

It establishes Reifenberg flatness for almost-minimizers of the perimeter using a weak notion of minimality, extending previous results.

Findings

Almost-minimizers are Reifenberg flat.

Small excess at one scale implies small excess at all smaller scales.

Results hold under very weak minimality assumptions.

Abstract

The aim of this note is to prove that almost-minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all smaller scales.