Wasserstein Geometry Overview

• Wasserstein geometry is the study of metric, analytic, and probabilistic structures on spaces of probability measures defined via optimal transport.
• It underpins diverse applications including statistical inference, data science algorithms, and geometric analysis in both smooth and singular spaces.
• Utilizing synthetic curvature-dimension conditions, the field ensures existence, uniqueness, and stability of measures and barycenters across various settings.

Wasserstein geometry is the study of the geometric, analytic, and probabilistic structure induced by the Wasserstein distance—defined via optimal transport—on spaces of probability measures over a metric or metric-measure space. The resulting spaces, called Wasserstein spaces, underpin a broad spectrum of developments in analysis, geometry, statistics, and data science by encoding both the geometry of the underlying space and the structure of probability measures. Central topics include the definition and properties of the Wasserstein distance, geometric and curvature properties, the role of barycenters, synthetic curvature-dimension conditions, and significant applications to statistics, configuration spaces, and random measures.

1. Foundational Structures: Wasserstein Distances, Spaces, and Barycenters

Let $(X, d, m)$ be an extended metric measure space, where $d: X \times X \to [0, \infty]$ is an extended distance and $m$ is a Radon measure of full support. The prototypical Wasserstein-2 distance between Borel probability measures $\mu, \nu$ on $X$ is defined as

$$W_2(\mu, \nu)^2 = \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^2 \, d\pi(x, y),$$

where $\Pi(\mu, \nu)$ denotes the couplings with marginals $\mu$ and $\nu$. The space $P_2(X)$ of probability measures with finite second moment is a complete metric space under $d: X \times X \to [0, \infty]$0 if $d: X \times X \to [0, \infty]$1 is complete.

Given a probability law $d: X \times X \to [0, \infty]$2 on $d: X \times X \to [0, \infty]$3 with finite variance, its Wasserstein barycenter is any $d: X \times X \to [0, \infty]$4 minimizing

$d: X \times X \to [0, \infty]$5

In the case of finite $d: X \times X \to [0, \infty]$6, the problem reduces to minimizing $d: X \times X \to [0, \infty]$7 over $d: X \times X \to [0, \infty]$8.

2. Curvature-Dimension and Synthetic Ricci Bounds

The interplay between Wasserstein geometry and curvature is encoded in the curvature-dimension condition $d: X \times X \to [0, \infty]$9. For $m$0, a geodesic metric-measure space satisfies $m$1 if for every Wasserstein geodesic $m$2 connecting $m$3, there holds the entropy convexity estimate: $m$4 where $m$5 for $m$6.

Ambrosio–Gigli–Savaré introduced the $m$7 condition, combining the $m$8 entropy convexity with quadraticity of the Cheeger energy. This structure rules out Finsler-type pathologies and ensures that the Wasserstein gradient flow of the entropy evolves according to the Evolution Variational Inequality (EVI), leading to robust analytical and geometric control.

3. Barycenter-Curvature-Dimension (BCD) Condition and Main Existence Results

The Barycenter-Curvature-Dimension condition, denoted $m$9, is formulated to extend curvature concepts to barycenters: a space satisfies $\mu, \nu$0 if for each finitely-supported barycentric problem, there exists a barycenter $\mu, \nu$1 satisfying

$\mu, \nu$2

In the infinite-dimensional case, the distortion term is quadratic; for finite $\mu, \nu$3, it uses the $\mu, \nu$4 coefficients from the classical curvature-dimension theory.

Key consequences include:

• Existence: Under either $\mu, \nu$5 or the presence of an entropy EVI-gradient flow, any measure $\mu, \nu$6 with finite variance admits a barycenter.
• Uniqueness and absolute continuity: If $\mu, \nu$7, any barycenter is absolutely continuous w.r.t. $\mu, \nu$8 and is unique in spaces with strict convexity of the squared Wasserstein distance, such as $\mu, \nu$9 spaces.
• Multi-marginal transport: Under suitable regularity and curvature conditions, the Monge multi-marginal problem with cost $X$0 admits a unique optimal map, coinciding with the Wasserstein barycenter.

4. Stability, Functional, and Geometric Inequalities

The $X$1 class is stable under measured Gromov–Hausdorff convergence, making it suitable for analysis in singular or infinite-dimensional limits, including Alexandrov spaces, abstract Wiener spaces, and RCD spaces.

Significant inequalities inherited from the barycentric structure include:

• Multi-marginal Brunn–Minkowski Inequality: For $X$2 spaces and subsets $X$3 with $X$4 summing to 1,

$X$5

where $X$6 is the set of barycenters from points in each $X$7.

• Functional Blaschke–Santaló Inequality: In $X$8 spaces, for $X$9 with $$W_2(\mu, \nu)^2 = \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^2 \, d\pi(x, y),$$0,

$$W_2(\mu, \nu)^2 = \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^2 \, d\pi(x, y),$$1

These inequalities generalize classical results and position the Wasserstein barycenter as a central concept in non-linear interpolation, underlying volumetric and information-theoretic inequalities.

5. Comparative and Synthetic Perspective

The barycenter theory unifies previously disparate results for Euclidean spaces, Riemannian manifolds, Alexandrov spaces, and spaces with synthetic Ricci curvature bounds. The synthetic $$W_2(\mu, \nu)^2 = \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^2 \, d\pi(x, y),$$2 framework is strictly weaker than classical $$W_2(\mu, \nu)^2 = \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^2 \, d\pi(x, y),$$3 yet sufficient for barycentric Jensen-type inequalities and is robust under non-smooth and limit constructions.

Applications span from geometric measure theory to analysis on configuration spaces and abstract Wiener spaces, where the existence, uniqueness, and regularity of barycenters yield controlled non-linear averages and interpolants in high or infinite dimensions.

6. Significance and Outlook

The Wasserstein geometric viewpoint, particularly via the barycenter- and curvature-dimension structure, has catalyzed theoretical advances and algorithmic applications. It provides a flexible, robust framework to study measure interpolations, functional inequalities, and curvature bounds in extended metric measure spaces. The introduction and stability of the BCD($$W_2(\mu, \nu)^2 = \inf_{\pi \in \Pi(\mu, \nu)} \int_{X \times X} d(x, y)^2 \, d\pi(x, y),$$4) condition enable new analysis in highly singular or non-smooth contexts, extending the reach of optimal transport-based geometry. The developed inequalities and regularity results further solidify Wasserstein barycenters and associated structures as fundamental analytic objects in modern geometric analysis (Han et al., 2024).