Regular Ultrametric Skeletons

Abstract

The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion $O(1/\epsilon )$ that carries a measure whose balls are controlled by the $(1-\epsilon )$-power of the original measure on dilated concentric balls. We prove a two-sided version for arbitrary compact metric spaces: for every ball centered on the skeleton, the skeleton measure also has a lower bound in terms of the original measure on a smaller nonconcentric ball contained in it. We also give a short proof of the original skeleton theorem and improve the dilation of its control balls from $\exp (O(1/{\epsilon }^2))$ to $O(1/\epsilon )$.

14 pages, substantial revision