We obtain new characterizations of the Sobolev spaces $\dot W^{ 1,p}(\mathbb{ R}^N)$ and the bounded variation space $\dot{ BV}(\mathbb{ R}^N)$. The characterizations are in terms of the functionals $\nu_{ \gamma} (E_{ \lambda,\gamma/p}[u])$ where \[ E_{ \lambda,\gamma/p}[u]= \Big\{ (x,y )\in \mathbb{ R}^N \times \mathbb{ R}^N \colon x \neq y, \, \frac{ |u(x)-u(y)|}{ |x-y|^{ 1+\gamma/p}}>\lambda\Big\} \] and the measure $\nu_{ \gamma}$ is given by $\mathrm{ d} \nu_\gamma(x,y)=|x-y|^{ \gamma-N} \mathrm{ d} x \mathrm{ d} y$. We provide characterizations which involve the $L^{ p,\infty}$-quasi-norms $\sup_{ \lambda>0} \lambda \, \nu_{ \gamma} (E_{ \lambda,\gamma/p}[u]) ^{ 1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $\lambda\to\infty$ when $\gamma>0$ and the limit for $\lambda\to 0^+$ when $\gamma<0$. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For $p>1$ the characterizations hold for all $\gamma \neq 0$. For $p=1$ the upper bounds for the $L^{ 1,\infty}$ quasi-norms fail in the range $\gamma\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{ 1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $\gamma+1$. For $\gamma=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $\nu_0(E_{ \lambda,0}[u])$.
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