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arXiv · 2311.09453

Geometry of measures on smoothly stratified metric spaces

Jonathan C. Mattingly·Ezra Miller·Do Tran
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Abstract

Any measure $\mu$ on a CAT(k) space M that is stratified as a finite union of manifolds and has local exponential maps near the Fr\'echet mean $\bar\mu$ yields a continuous "tangential collapse" from the tangent cone of M at $\bar\mu$ to a vector space that preserves the Fr\'echet mean, restricts to an isometry on the "fluctuating cone" of directions in which the Fr\'echet mean can vary under perturbation of $\mu$, and preserves angles between arbitrary and fluctuating tangent vectors at the Fr\'echet mean.

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BibTeXRIS

Jonathan C. Mattingly, Ezra Miller, Do Tran. 2023-11-15. Geometry of measures on smoothly stratified metric spaces. https://arxiv.org/abs/2311.09453

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