Approximation by piecewise constants on convex partitions

We show that the saturation order of piecewise constant approximation in LpLp norm on convex partitions with NN cells is N−2/(d+1)N−2/(d+1), where dd is the number of variables. This order is achieved for any f∈Wp2(Ω) on a partition obtained by a simple algorithm involving an anisotropic subdivision of a uniform partition. This improves considerably the approximation order N−1/dN−1/d achievable on isotropic partitions. In addition we show that the saturation order of piecewise linear approximation on convex partitions is N−2/dN−2/d, the same as on isotropic partitions.