  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.
