Zero Probability of the Cut Locus of a Fréchet Mean on a Riemannian Manifold
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916875920736256 |
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| author | Lytchak, Alexander Huckemann, Stephan F. |
| author_facet | Lytchak, Alexander Huckemann, Stephan F. |
| contents | We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fréchet functions and we discuss extensions of the statement to Fréchet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general metric spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_00747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zero Probability of the Cut Locus of a Fréchet Mean on a Riemannian Manifold Lytchak, Alexander Huckemann, Stephan F. Probability Differential Geometry 60D05, 53C20 We show that the cut locus of a Fréchet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fréchet functions and we discuss extensions of the statement to Fréchet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general metric spaces. |
| title | Zero Probability of the Cut Locus of a Fréchet Mean on a Riemannian Manifold |
| topic | Probability Differential Geometry 60D05, 53C20 |
| url | https://arxiv.org/abs/2508.00747 |