Shape spaces in terms of Wasserstein geometry

2025/10/22 by Bernadette Lessel, Lessel, Bernadette · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2510.19998

openalex publication_date 2025/10/22 · openalex created_date 2025/10/25 · openalex updated_date 2026/07/28

Abstract

For a Polish space X, we define the Shape space 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 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 S₂(ℝⁿ), it is shown that S_p(X) is Polish as well in case X and the action of G are indeed proper. Also, the metric geodesics in S_p(X) are put in relation to the ones in W_p(X).

Citations

• Differentiable maps between Wasserstein spaces

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