2026/07/20 by Manor Mendel
Mathematics · #math.MG
The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion O(1/ε) that carries a measure whose balls are controlled by the (1-ε)-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/ε2)) to O(1/ε).