On Codecell Convexity of Optimal Multiresolution Scalar Quantizers for Continuous Sources

It has been shown by earlier results that for fixed rate multiresolution scalar quantizers and for mean squared error distortion measure, codecell convexity precludes optimality for certain discrete sources. However it was unknown whether the same phenomenon can occur for any continuous source. In this paper, examples of continuous sources (even with bounded continuous densities) are presented for which optimal fixed rate multiresolution scalar quantizers cannot have only convex codecells, proving that codecell convexity precludes optimality also for such regular sources.