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Optimal approximation order of piecewise constants on convex partitions

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 $W^2_q(Ω)$ on a cube $Ω\subset\mathbb{R}^d$, $d\geqslant 2$, 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 $W^r_q(Ω)$ embedded in $L_p(Ω)$, some of which also improve the standard estimate $\mathcal{O}\big(N^{-\frac 1d}\big)$ known to be optimal on isotropic partitions.