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We show that independent and uniformly distributed sampling points are asgood as optimal sampling points for the approximation of functions from theSobolev space $W_p^s(\Omega)$ on bounded convex domains $\Omega\subset\mathbb{R}^d$ in the $L_q$-norm if $q<p$. More generally, we characterize thequality of arbitrary sampling points $P\subset \Omega$ via the$L_\gamma(\Omega)$-norm of the distance function $\rm{dist}(\cdot,P)$, where$\gamma=s(1/q-1/p)^{-1}$ if $q<p$ and $\gamma=\infty$ if $q\ge p$. Thisimproves upon previous characterizations based on the covering radius of $P$.

Random points are optimal for the approximation of Sobolev functions