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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.19998 |
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Table of 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 $W_p(X)$.