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The isometry group of Wasserstein spaces: the Hilbertian case

Gehér, G. P., Titkos, T. and Virosztek, D. (2022) The isometry group of Wasserstein spaces: the Hilbertian case. Journal of the London Mathematical Society, 106 (4). pp. 3865-3894. ISSN 1469-7750

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To link to this item DOI: 10.1112/jlms.12676


Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(ℝn), we describe the isometry group Isom(Wp(E))for all parameters 0<p<∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0<p<1. This is a consequence of our more general result: we prove that W1(X)is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p>1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. As a byproduct of our methods, we also obtain the isometric rigidity of Wp(X)for all complete and separable ultrametric spaces X and parameters 0<p<∞.

Item Type:Article
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:107550
Uncontrolled Keywords:General Mathematics
Publisher:Oxford Journals


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