Generalized Barycenter Problems

• Generalized barycenter problems are defined as finding averages of probability measures within nonlinear metric and optimal transport frameworks.
• The introduction of Barycenter–Curvature–Dimension (BCD) conditions and EVI(K) gradient flows ensures existence, uniqueness, and convexity even in non-smooth, infinite-dimensional spaces.
• These methods yield new geometric inequalities—such as the Wasserstein Jensen, multi-marginal Brunn–Minkowski, and Blaschke–Santaló-type results—with broad applications in optimal transport and data analysis.

Generalized barycenter problems concern finding “averages” (barycenters) of collections of probability measures or more general objects under nonlinear metric, geometric, or optimal transport structures. Traditionally, barycenter problems were confined to Euclidean or Riemannian contexts; recent work has developed existence, uniqueness, and regularity theory extending to non-smooth, non-compact, and infinite-dimensional metric measure spaces. Central innovations include the development of curvature-dimension conditions adapted to barycenter theory, the construction of gradient flows for convex functionals on Wasserstein space, and new synthetic frameworks for geometric inequalities.

1. Barycenter–Curvature–Dimension (BCD) Conditions

A foundational contribution is the introduction of the Barycenter–Curvature–Dimension (BCD) condition, which generalizes the classical curvature–dimension bounds (notably those of Lott–Sturm–Villani and Ambrosio–Gigli–Savare) from the displacement convexity of the entropy along Wasserstein geodesics to convexity properties at the Wasserstein barycenter. Formally, for an extended metric measure space $(X,d,m)$, a dimension-free BCD condition requires that for any finitely supported $\Omega \in \mathcal{P}_2(P(X))$, there exists a barycenter $p$ satisfying the Jensen-type inequality: $$\mathrm{Ent}_m(p) \leq \int_{P(X)} \mathrm{Ent}_m(\mu) \, d\Omega(\mu) - \mathrm{Var}(\Omega),$$ where $\mathrm{Ent}_m$ is the relative entropy with respect to $m$, and $\mathrm{Var}(\Omega)$ is a variance-type deficit. The finite-dimensional BCD($K,N$) condition introduces distortion coefficients and incorporates $K$-Ricci-type lower curvature bounds and effective dimension $N$, again expressed via Jensen-type inequalities at the barycenter with entropy weights: $$W_2(p,p)\, \int_{P(X)} \mathrm{S}_{K/N}(W_2(\mu,\mu))\, U_N(p)\, d\Omega(\mu) \leq U_N(p) \int_{P(X)} \mathrm{t}_{K/N}(W_2(\mu,\mu))\, d\Omega(\mu),$$ where $U_N(p) = \exp(-\mathrm{Ent}_m(p)/N)$. Unlike CD($K,N$), the BCD condition references the barycenter as a nonlinear mean, and thus is stable and meaningful even outside globally geodesic settings.

2. Existence and Uniqueness Theory in General Metric Spaces

The principal existence result establishes that the barycenter map $\Omega \mapsto \text{bary}(\Omega)$ is well-defined even on extended metric measure spaces lacking local compactness, provided either (A) the underlying space is an RCD($K,\infty$) space or (B) the measure $m$ is a probability measure and each candidate $\mu$ is the starting point for an evolution variational inequality (EVI($K$)) gradient flow for $\mathrm{Ent}_m$. Technically, this combines the concentration of mass property and the stability of entropy convexity under weak convergence.

Uniqueness of the Wasserstein barycenter is obtained under a “weak Monge property”: if one marginal is absolutely continuous, all optimal transport maps from the barycenter are unique. In this setting, strict convexity of the Wasserstein squared distance functional translates into strict convexity (up to equivalence classes) along generalized geodesics, yielding uniqueness of the barycenter. This extends classical results of Agueh–Carlier (Euclidean) and analogous results in Riemannian and Alexandrov spaces.

3. Jensen, Brunn–Minkowski, and Blaschke–Santaló–Type Inequalities

The BCD framework systematically produces new geometric inequalities for barycenters:

• Wasserstein Jensen Inequality: For any displacement $k$-convex functional $\mathcal{F}$ on $P(X)$, the barycenter $\bar{\mu}$ of $\Omega$ satisfies

$$\mathcal{F}(\bar{\mu}) \leq \int_{P(X)}\mathcal{F}(\mu)\, d\Omega(\mu) - \frac{k}{2}\int_{P(X)} W_2^2(\bar{\mu},\mu)\, d\Omega(\mu).$$

This is a nonlinear generalization of the classical Jensen’s inequality to Wasserstein space.

• Multi-marginal Brunn–Minkowski Inequality: In BCD($0,N$) spaces, given sets $E_1,\ldots,E_n$, the set of barycenters $E$ satisfies

$m(E) \geq \prod_{i=1}^n m(E_i)^{r_i\cdot N}$

for convex weights $r_i$, generalizing the classical Euclidean Brunn–Minkowski to multi-marginal and measure-theoretic settings.

• Functional Blaschke–Santaló-Type Inequality: For measurable functions $f_1,\ldots,f_k$ satisfying $$\prod_{i=1}^k e^{f_i(x_i)} \leq \exp\left\{-\frac{1}{2}\inf_x\sum_{i=1}^k d^2(x,x_i)\right\}$$, there is a corresponding integral inequality reminiscent of the Blaschke–Santaló inequality, extended to non-smooth, non-Euclidean background spaces.

These inequalities are systematically derived via the properties of barycenter convexity established by BCD and validated via gradient flows (EVI$K$) and measure-theoretic properties of $P(X)$.

4. Stability, Infinite-Dimensional, and Non-Compact Extensions

The BCD condition and barycenter theory extend to a broad class of spaces, including non-compact metric measure spaces, abstract Wiener spaces, and configuration spaces over Riemannian backgrounds, as well as spaces only equipped with measured-Gromov–Hausdorff convergence structures. Stability of the BCD property under measured-Gromov–Hausdorff convergence is established, implying that barycenter existence, uniqueness, and the associated inequalities persist in limits of approximating spaces or in settings that lack global geodesicity. Examples include non-smooth limit spaces arising as measured Gromov–Hausdorff limits of Riemannian manifolds, configuration spaces of measures, and certain infinite-dimensional spaces pertinent to stochastic analysis.

5. Comparison to Classical Settings and Extensions

Prior barycenter analysis was largely confined to the following regimes:

• Euclidean spaces: Existence and uniqueness rely on linear structure and uniform convexity; displacement convexity of the entropy is classical (Agueh–Carlier).
• Riemannian manifolds: Exploits local geometry, geodesic convexity, and standard curvature lower bounds.
• Alexandrov spaces: Synthetic curvature bounds (via comparison triangles) enable generalization to spaces with sectional curvature bounds.

The present BCD approach subsumes all these by working at the level of convexity at the barycenter, which can be verified in spaces without global geodesics or local compactness. Furthermore, the EVI$K$ gradient flow method, rather than geodesic or displacement convexity, is a non-linear a priori device drawing analogies with the theory of nonlinear PDEs in geometric analysis.

6. Implications and Future Directions

The generalized barycenter theory and BCD conditions facilitate:

• Extension of functional and geometric inequalities (e.g., Poincaré, logarithmic Sobolev, Brunn–Minkowski, Blaschke–Santaló) to new classes of spaces, including those arising in stochastic analysis and infinite-dimensional geometry.
• Synthetic Ricci-type curvature lower bounds at the barycenter level, supporting analysis in spaces where classical displacement convexity or geodesic completeness may fail.
• Robust tools for measure-valued data summarization, statistical aggregation, and geometric functional analysis on non-smooth and high-dimensional spaces.
• The development of convergence results and stability under various natural limits (e.g., measured-Gromov–Hausdorff convergence) for both the barycenter mapping and associated convexity properties.

By providing a comprehensive theoretical structure for barycenters on extended metric measure spaces—anchored in synthetic curvature, gradient flows, and nonlinear analysis—the BCD approach significantly broadens the landscape for geometric analysis and optimal transport. This enables further advances in the paper of measure spaces with synthetic lower curvature bounds, and endows applications in data science, stochastic analysis, and the geometry of metric spaces with powerful new analytic and geometric tools.