# Isometric study of Wasserstein spaces - the real line

G.P. Geher, T. Titkos, D. Virosztek, (n.d.).

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Abstract
Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein space W_p(R) for all p \in [1,\infty) \setminus {2}. We show that W_2(R) is also exceptional regarding the parameter p: W_p(R) is isometrically rigid if and only if p is not equal to 2. Regarding the underlying space, we prove that the exceptionality of p = 2 disappears if we replace R by the compact interval [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only if p is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass, and Isom(W_1([0,1])) cannot be embedded into Isom(W_1(R)).
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Date Published
2020-01-30
Page
32
IST-REx-ID

### Cite this

Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the real line.
Geher, G. P., Titkos, T., & Virosztek, D. (n.d.). Isometric study of Wasserstein spaces - the real line.
Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Isometric Study of Wasserstein Spaces - the Real Line,” n.d.
G. P. Geher, T. Titkos, and D. Virosztek, “Isometric study of Wasserstein spaces - the real line.” .
Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the real line.
Geher, Gyorgy Pal, et al. Isometric Study of Wasserstein Spaces - the Real Line.
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