Geometry of measures on smoothly stratified metric spaces

2023/11/15 by Jonathan C. Mattingly, Ezra Miller, Mattingly, Jonathan C. +3 · 1 citation
Mathematics · #28C99 #49J52 #53C23 #53C80 #57N80 #57R57 #58A35 #58K30 #58Z05 #60B05 #62G20 #62R07 #62R20 #62R30 #92B10 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Point processes and geometric inequalities #Primary: 60D05 #Probability (math.PR) #Secondary: 60F05 #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2311.09453

openalex publication_date 2023/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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

Cited by

• Zero Probability of the Cut Locus of a Fréchet Mean on a Riemannian Manifold

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