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arXiv · 2607.18525

Regular Ultrametric Skeletons

Manor Mendel
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Abstract

The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion $O(1/\varepsilon)$ that carries a measure whose balls are controlled by the $(1-\varepsilon)$-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 $\exp(O(1/\varepsilon^2))$ to $O(1/\varepsilon)$.

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Manor Mendel. 2026-07-20. Regular Ultrametric Skeletons. https://arxiv.org/abs/2607.18525

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