$\text{BMO}$-type norms and anisotropic surface measures

Abstract

The purpose of this note is to present an anisotropic variant of the BMO-type norm introduced in [4] and to show its relation with a surface measure, which is indeed a multiple of the perimeter in the isotropic case. This is done in the spirit of the new characterization of the perimeter of a measurable set in ${\mathbb{R}}^n$ recently studied by Ambrosio, Bourgain, Brezis and Figalli in [2].