Locality of centred tangent cones in the Wasserstein space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911474985730048 |
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| author | Aussedat, Averil |
| author_facet | Aussedat, Averil |
| contents | The geometric tangent cone to a probability measure $μ$ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one removes the barycenter of any element of the tangent cone, then the resulting set of centred measure fields is characterized by a local condition. Precisely, centred tangent fields must be concentrated on a family of vector subspaces attached to any point, and these subspaces correspond to the normal spaces to some sets of ``dimension $k$'' on which the measure $μ$ is concentrated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10837 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locality of centred tangent cones in the Wasserstein space Aussedat, Averil Metric Geometry 28A15, 51FXX, 35R06 The geometric tangent cone to a probability measure $μ$ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one removes the barycenter of any element of the tangent cone, then the resulting set of centred measure fields is characterized by a local condition. Precisely, centred tangent fields must be concentrated on a family of vector subspaces attached to any point, and these subspaces correspond to the normal spaces to some sets of ``dimension $k$'' on which the measure $μ$ is concentrated. |
| title | Locality of centred tangent cones in the Wasserstein space |
| topic | Metric Geometry 28A15, 51FXX, 35R06 |
| url | https://arxiv.org/abs/2508.10837 |