Shape spaces in terms of Wasserstein geometry
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911227812249600 |
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| author | Lessel, Bernadette |
| author_facet | Lessel, Bernadette |
| contents | For a Polish space $X$, we define the Shape space $\mathcal{S}_p(X)$ to be the Wasserstein space $W_p(X)$ modulo the action of a subgroup $G$ of the isometry group $ISO(X)$ of $X$, where the action is given by the pushforward of measures. The Wasserstein distance can then naturally be transformed into a \emph{Shape distance} on Shape space if $X$ and the action of $G$ are proper. This is shown for example to be the case for complete connected Riemannian manifolds with $G$ being equipped with the compact-open topology. Before finally proposing a notion for tangent spaces on the Shape space $\mathcal{S}_2(\mathbb{R}^n)$, it is shown that $\mathcal{S}_p(X)$ is Polish as well in case $X$ and the action of $G$ are indeed proper. Also, the metric geodesics in $\mathcal{S}_p(X)$ are put in relation to the ones in $W_p(X)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19998 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape spaces in terms of Wasserstein geometry Lessel, Bernadette Functional Analysis For a Polish space $X$, we define the Shape space $\mathcal{S}_p(X)$ to be the Wasserstein space $W_p(X)$ modulo the action of a subgroup $G$ of the isometry group $ISO(X)$ of $X$, where the action is given by the pushforward of measures. The Wasserstein distance can then naturally be transformed into a \emph{Shape distance} on Shape space if $X$ and the action of $G$ are proper. This is shown for example to be the case for complete connected Riemannian manifolds with $G$ being equipped with the compact-open topology. Before finally proposing a notion for tangent spaces on the Shape space $\mathcal{S}_2(\mathbb{R}^n)$, it is shown that $\mathcal{S}_p(X)$ is Polish as well in case $X$ and the action of $G$ are indeed proper. Also, the metric geodesics in $\mathcal{S}_p(X)$ are put in relation to the ones in $W_p(X)$. |
| title | Shape spaces in terms of Wasserstein geometry |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2510.19998 |