A note on the fractional perimeter and interpolation

TL;DR

This paper explores the fractional perimeter as an interpolation between measure and perimeter, introducing a fractional Boxing inequality that links fractional perimeter with Hausdorff content and Sobolev space seminorms.

Contribution

It introduces a new fractional Boxing inequality connecting fractional perimeter, Hausdorff content, and Sobolev spaces, advancing understanding of fractional geometric measures.

Findings

Established a fractional Boxing inequality relating fractional perimeter and Hausdorff content.

Derived inequalities involving the Gagliardo seminorm of Sobolev spaces.

Provided a new perspective on fractional perimeter as an interpolation between measure and perimeter.

Abstract

We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces $W^{\alpha, 1}$ of order $0 < \alpha < 1$.