  We prove that the error of the best nonlinear $L_p$-approximation by
piecewise constants on convex partitions is
$\mathcal{O}\big(N^{-\frac{2}{d+1}}\big)$, where $N$ the number of cells, for
all functions in the Sobolev space $W^2_q(\Omega)$ on a cube
$\Omega\subset\mathbb{R}^d$, $d\geqslant 2$, as soon as $\frac{2}{d+1} +
\frac{1}{p} - \frac{1}{q}\geqslant 0$. The approximation order
$\mathcal{O}\big(N^{-\frac{2}{d+1}}\big)$ is achieved on a polyhedral partition
obtained by anisotropic refinement of an adaptive dyadic partition. Further
estimates of the approximation order from the above and below are given for
various Sobolev and Sobolev-Slobodeckij spaces $W^r_q(\Omega)$ embedded in
$L_p(\Omega)$, some of which also improve the standard estimate
$\mathcal{O}\big(N^{-\frac 1d}\big)$ known to be optimal on isotropic
partitions.
