# A Boxing Inequality for the Fractional Perimeter

## TL;DR

This paper establishes a new inequality linking Hausdorff content and fractional perimeter, leading to trace and Sobolev inequalities in fractional Sobolev spaces, with results extending classical inequalities as  approaches 0 or 1.

## Contribution

The paper introduces a novel Boxing inequality for fractional perimeter, connecting geometric measure theory with fractional Sobolev space embeddings and inequalities.

## Key findings

- Proves the Boxing inequality for fractional perimeter.
- Derives trace inequalities and Sobolev embeddings from the inequality.
- Extends classical inequalities to the fractional setting with asymptotic analysis.

## Abstract

We prove the Boxing inequality: $$\mathcal{H}^{d-\alpha}_\infty(U) \leq C\alpha(1-\alpha)\int_U \int_{\mathbb{R}^{d} \setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{|y-z|^{\alpha+d}},$$ for every $\alpha \in (0,1)$ and every bounded open subset $U \subset \mathbb{R}^d$, where $\mathcal{H}^{d-\alpha}_\infty(U)$ is the Hausdorff content of $U$ of dimension $d -\alpha$ and the constant $C > 0$ depends only on $d$. We then show how this estimate implies a trace inequality in the fractional Sobolev space $W^{\alpha, 1}(\mathbb{R}^d)$ that includes Sobolev's $L^{\frac{d}{d - \alpha}}$ embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as $\alpha$ tends to $0$ and $1$, recovering in particular the classical inequalities of first order. Their counterparts in the full range $\alpha \in (0, d)$ are also investigated.