  Recently, Brezis, Van Schaftingen and the second author established a new
formula for the $\dot{W}^{1,p}$ norm of a function in
$C^{\infty}_c(\mathbb{R}^N)$. The formula was obtained by replacing the
$L^p(\mathbb{R}^{2N})$ norm in the Gagliardo semi-norm for
$\dot{W}^{s,p}(\mathbb{R}^N)$ with a weak-$L^p(\mathbb{R}^{2N})$ quasi-norm and
setting $s = 1$. This provides a characterization of such $\dot{W}^{1,p}$
norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM)
formula. In this paper, we obtain an analog for the case $s = 0$. In
particular, we present a new formula for the $L^p$ norm of any function in
$L^p(\mathbb{R}^N)$, which involves only the measures of suitable level sets,
but no integration. This provides a characterization of the norm on
$L^p(\mathbb{R}^N)$, which complements a formula by Maz'ya and Shaposhnikova.
As a result, by interpolation, we obtain a new embedding of the
Triebel-Lizorkin space $F^s_{p,2}(\mathbb{R}^N)$ (i.e. the Bessel potential
space $(I-\Delta)^{-s/2} L^p(\mathbb{R}^N)$), as well as its homogeneous
counterpart $\dot{F}^s_{p,2}(\mathbb{R}^N)$, for $s \in (0,1)$, $p \in
(1,\infty)$.
