**Abstract:**
We show that the saturation order of piecewise constant approximation in $L_p$ norm
on convex partitions with $N$ cells is $N^{-2/(d+1)}$, where $d$ is the number of variables. This order is achieved
for any $f\in W^2_p(\Omega)$ on a partition obtained by a simple algorithm involving an anisotropic
subdivision of a uniform partition. This improves considerably the approximation order $N^{-1/d}$
achievable on isotropic partitions. In addition we show that the saturation order of piecewise linear
approximation on convex partitions is $N^{-2/d}$, the same as on isotropic partitions.

**Preprint version:**
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