Abstract:We define a complete metric structure $\mathfrak{d}_q^p$ on the family $\text{PL}_q^p(\mathbb{R}^n)$ of probability measures with densities in $L^p(\mathbb{R}^n)$ and finite $q$-moments. We establish the existence of generalized minimizing movements for the isoperimetric ratio and characterize $\mathfrak{d}_q^p$-absolutely continuous curves through weak solutions of the continuity equation with velocity fields satisfying a Sobolev-type condition. We also characterize absolutely continuous curves in the $\infty$-Wasserstein space and prove a Benamou--Brenier formula for $W_\infty$.
| Comments: | Key words: Wasserstein spaces, isoperimetric problem, minimizing movement schemes |
| Subjects: | Metric Geometry (math.MG) |
| MSC classes: | 49Q22, 49Q20, 49J45 |
| Cite as: | arXiv:2605.25086 [math.MG] |
| (or arXiv:2605.25086v1 [math.MG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25086 arXiv-issued DOI via DataCite (pending registration) |
From: Pietro Aldrigo Sig. [view email]
[v1]
Sun, 24 May 2026 13:59:31 UTC (33 KB)
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