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Restoring Wasserstein Rigidity with a single point

Zoltán M. Balogh, Eric Ströher, Dániel Virosztek

TL;DR

The paper investigates when Wasserstein spaces admit nontrivial isometries beyond those induced by base-space isometries. Building on known flexibility results, it shows that augmenting the base space by a single distant point restores rigidity in both the $1$-Wasserstein setting on $[0,1]\cup\{q\}$ and the $2$-Wasserstein setting on $\mathbb{R}^2\cup\{q\}$, with explicit constructions of how isometries must act. The approach combines slice decomposition by the mass at the added point, Kantorovich duality to characterize optimal transport within slices, and known rigidity/classification results for base-space Wasserstein spaces to rule out nontrivial global isometries. This work clarifies the fragility of Wasserstein-space flexibility and demonstrates a canonical mechanism—adding a single distant point—that enforces rigidity, with implications for understanding isometry groups in optimal transport geometry.

Abstract

We consider isometrically flexible Wasserstein spaces and demonstrate that adding a single point to the underlying metric space makes these Wasserstein spaces rigid.

Restoring Wasserstein Rigidity with a single point

TL;DR

The paper investigates when Wasserstein spaces admit nontrivial isometries beyond those induced by base-space isometries. Building on known flexibility results, it shows that augmenting the base space by a single distant point restores rigidity in both the -Wasserstein setting on and the -Wasserstein setting on , with explicit constructions of how isometries must act. The approach combines slice decomposition by the mass at the added point, Kantorovich duality to characterize optimal transport within slices, and known rigidity/classification results for base-space Wasserstein spaces to rule out nontrivial global isometries. This work clarifies the fragility of Wasserstein-space flexibility and demonstrates a canonical mechanism—adding a single distant point—that enforces rigidity, with implications for understanding isometry groups in optimal transport geometry.

Abstract

We consider isometrically flexible Wasserstein spaces and demonstrate that adding a single point to the underlying metric space makes these Wasserstein spaces rigid.
Paper Structure (5 sections, 9 theorems, 35 equations)

This paper contains 5 sections, 9 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Motivation and background results
  3. Main theorems
  4. The unit interval plus one point far away
  5. The $2$-Wasserstein space over the Euclidean plane plus one point

Key Result

Theorem 1.1

The $1$-Wasserstein space $\mathcal{W}_1([0,1]\cup\{q\}, d_1)$ is isometrically rigid. That is, for all $\Phi \in \mathrm{Isom}\left(\mathcal{W}_1([0,1]\cup\{q\}, d_1)\right)$ there is a $\psi \in \mathrm{Isom}\left([0,1]\cup\{q\}, d_1\right)$ such that $\Phi=\psi_{\#}.$

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 9 more