Abstract:Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. Owing to its many applications in analysis, probability, and geometry, the problem of extending this theorem to curved spaces has appeared repeatedly in the literature, going back to Villani [V09]. More recently, Milman [Mil18] formulated precise conjectures in this direction. In this work, we construct counterexamples to these conjectures.
| Comments: | 20 pages |
| Subjects: | Differential Geometry (math.DG); Probability (math.PR); Spectral Theory (math.SP) |
| MSC classes: | 53C21 (primary), 58J50, 35P15, 53C20 |
| Cite as: | arXiv:2605.24705 [math.DG] |
| (or arXiv:2605.24705v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24705 arXiv-issued DOI via DataCite (pending registration) |
From: Shrey Aryan [view email]
[v1]
Sat, 23 May 2026 19:12:03 UTC (25 KB)
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