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Wasserstein Rigidity over $\mathbb{R}^n$ with smooth norms

Zoltán M. Balogh, Eric Ströher, Tamás Titkos, Dániel Virosztek

TL;DR

The paper studies isometric rigidity of $p$-Wasserstein spaces $\mathcal{W}_p(\mathbb{R}^n, d_N)$ where $d_N$ is the norm-induced metric. It proves that, when $N$ is strictly convex and $C^2$-smooth away from the origin, $\mathcal{W}_p(\mathbb{R}^n, d_N)$ is isometrically rigid for all $p\neq 2$, and it shows rigidity in the $p=2$ case for $N$ equal to an $\ell_q$ norm with $q>2$. The approach combines a metric characterization of Dirac masses, a dimension-upgrading rigidity argument, and a $C^2$-differentiability-based analysis via potential functions, culminating in an induction on the ambient dimension. These results extend rigidity phenomena beyond inner-product norms, highlighting smoothness of the norm as a key driver. The findings have implications for understanding the symmetry structure of Wasserstein spaces over normed vector spaces and identify sharp conditions under which the Wasserstein isometry group reduces to trivial actions.

Abstract

We study $p-$Wasserstein spaces $ \mathcal{W}_p(\mathbb{R}^n, d_N)$ over $\mathbb{R}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p \neq 2$. We also show that, even when $p=2$, we can recover the isometric rigidity of the Wasserstein space $\mathcal{W}_2(\mathbb{R}^n, d_N)$ when $N$ is an $l_q-$norm and $q>2$.

Wasserstein Rigidity over $\mathbb{R}^n$ with smooth norms

TL;DR

The paper studies isometric rigidity of -Wasserstein spaces where is the norm-induced metric. It proves that, when is strictly convex and -smooth away from the origin, is isometrically rigid for all , and it shows rigidity in the case for equal to an norm with . The approach combines a metric characterization of Dirac masses, a dimension-upgrading rigidity argument, and a -differentiability-based analysis via potential functions, culminating in an induction on the ambient dimension. These results extend rigidity phenomena beyond inner-product norms, highlighting smoothness of the norm as a key driver. The findings have implications for understanding the symmetry structure of Wasserstein spaces over normed vector spaces and identify sharp conditions under which the Wasserstein isometry group reduces to trivial actions.

Abstract

We study Wasserstein spaces over equipped with a norm metric . We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever . We also show that, even when , we can recover the isometric rigidity of the Wasserstein space when is an norm and .

Paper Structure

This paper contains 7 sections, 18 theorems, 136 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. Metric characterization of Dirac masses
  3. Upgrading rigidity
  4. Isometric rigidity using $C^2$-differentiability
  5. First case: $p<2$
  6. Second case: $p>2$
  7. Special case: $p=2$, $N=l_q$, $q>2$

Key Result

Theorem 1.1

If $N: \mathbb{R}^n \to \mathbb{R}_{+}$ is a strictly convex norm that is $C^{2}$-smooth, then the Wasserstein space $\mathcal{W}_p(\mathbb{R}^n,d_N)$ is isometrically rigid for all $p\in [1, \infty)$, $p \neq 2$.

Figures (4)

  • Figure 1: In blue the subspace $L$, in green the unit ball of the $l_4$-norm, in red the surface $S=P_L^{-1}(\{0\})$. Created using Desmos.
  • Figure 2: An example for a measure $\mu \in \mathcal{F}$ (left) and a possible image measure $\Phi(\mu)$ (right).
  • Figure 3: The constructions $\mu'$ (up), $\nu_1'$ (down left) and $\nu_2'$ (down right).
  • Figure 4: In the Eucliden plane, there exists for $p=2$ shape-preserving isometries of $\mathcal{W}_2(\mathbb{R}^2, d_E)$ that send a measure supported on a line to a measure supported on a different line.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Example 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 25 more