Restoring Wasserstein Rigidity with a single point

TL;DR

This paper shows that adding a single point to a metric space can transform its Wasserstein space from flexible to rigid, revealing a simple yet powerful method to control geometric properties.

Contribution

The paper introduces a novel approach demonstrating how a single point addition induces rigidity in Wasserstein spaces, bridging flexibility and rigidity concepts.

Findings

Adding one point induces rigidity in Wasserstein spaces.

Flexibility of Wasserstein spaces can be controlled by minimal modifications.

Single-point addition has significant geometric implications.

Abstract

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