On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error

Let ? be an Ahlfors-David probability measure on Rq with support K?. For every n?1, let Cn(?) denote the collection of all n-optimal sets for respect to geometric mean error. We prove that, there exist constants d1,d2>0, such that each ?n?Cn(?) and arbitrary Voronoi partition {Pa(?n)}a??n ?n, we haved1n?1?mina??n??(Pa(?n))?maxa??n??(Pa(?n))?d2n?1. Moreover, a??n, set Pa(?n) contains a closed ball radius d3|Pa(?n)?K?| which is centered at a, where d3 constant |B| denotes diameter B?Rq. Some estimates geometrical size elements are established in more general context. These also allow us drop additional condition our previous work self-similar measures.