Isoperimetric minimizing movements and AC curves in $\text{PL}_q^p(\mathbb{R}^n)$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913160145928192 |
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| author | Aldrigo, Pietro |
| author_facet | Aldrigo, Pietro |
| contents | 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 $W_\infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Isoperimetric minimizing movements and AC curves in $\text{PL}_q^p(\mathbb{R}^n)$ Aldrigo, Pietro Metric Geometry 49Q22, 49Q20, 49J45 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 $W_\infty$. |
| title | Isoperimetric minimizing movements and AC curves in $\text{PL}_q^p(\mathbb{R}^n)$ |
| topic | Metric Geometry 49Q22, 49Q20, 49J45 |
| url | https://arxiv.org/abs/2605.25086 |