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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.00775 |
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| _version_ | 1866915825664917504 |
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| author | Aussedat, Averil |
| author_facet | Aussedat, Averil |
| contents | We study the perturbation of a measure $μ\in \mathscr{P}(\mathbb{R})$ consisting in superposing two copies of $μ$, each slightly shifted by a small distance $\pm h$. The difference between $μ$ and its perturbation is measured with a Wasserstein distance. For any $μ$, this distance is bounded from above by $h$. We show that measures for which this critical rate is achieved when $h$ goes to 0 are characterized as the ones giving most of their mass to some particular porous sets. This is used to identify which measures $μ$ on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00775 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterization of measures on the real line that are critically unstable under small shifts Aussedat, Averil Optimization and Control 49Q22, 28A80, 26A30 We study the perturbation of a measure $μ\in \mathscr{P}(\mathbb{R})$ consisting in superposing two copies of $μ$, each slightly shifted by a small distance $\pm h$. The difference between $μ$ and its perturbation is measured with a Wasserstein distance. For any $μ$, this distance is bounded from above by $h$. We show that measures for which this critical rate is achieved when $h$ goes to 0 are characterized as the ones giving most of their mass to some particular porous sets. This is used to identify which measures $μ$ on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed. |
| title | Characterization of measures on the real line that are critically unstable under small shifts |
| topic | Optimization and Control 49Q22, 28A80, 26A30 |
| url | https://arxiv.org/abs/2603.00775 |