Geometric mean quantization via adaptive approximation

Let $ν$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $ν(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_ν(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometri