On the geometry of Wasserstein barycenter I

Bang-Xian Han; Deng-Yu Liu; Zhuo-Nan Zhu

REVIEW 2 major objections 5 minor 1 cited by

On the geometry of Wasserstein barycenter I

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Wasserstein barycenters of probability measures exist and stay regular on non-compact spaces with synthetic Ricci lower bounds, and the paper proves this through Jensen's inequality derived from EVI gradient flows, introducing a new…

desk verdict Strong paper with a genuine new proof strategy, but the RCD(K,∞) claims lean on an optimal-map property that is only cited to a finite-dimensional reference. read the letter →

arxiv 2412.01190 v2 pith:PVXJC3IA submitted 2024-12-02 math.MG math.FAmath.PR

Bang-Xian Han , Deng-Yu Liu , Zhuo-Nan Zhu This is my paper · ORCID

classification math.MGmath.FAmath.PR MSC 53C2351F9949Q22

keywords Wassersteinbarycentermetricmeasurespacecurvature-dimensionconditionRiccicurvatureoptimaltransportgradientflowJensen'sinequalityEVI

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that Wasserstein barycenters—the natural averaged probability measures in optimal transport—exist and are well behaved on non-compact, non-smooth spaces with synthetic lower Ricci bounds, and even on extended metric spaces such as abstract Wiener spaces, without any local compactness assumption. The route is to run the relative entropy along an EVI gradient flow in the Wasserstein space and integrate the flow inequality: this yields Jensen's inequality for barycenters, which then acts as an a priori estimate from which existence, uniqueness, and absolute continuity of barycenters follow. On spaces satisfying a new Barycenter-Curvature-Dimension condition, the same mechanism gives existence of barycenters, stability under measured Gromov–Hausdorff convergence, and geometric inequalities such as multi-marginal Brunn–Minkowski and functional Blaschke–Santalo type estimates. The result matters because it replaces compactness-based existence proofs with a dimension-free functional-analytic mechanism.

What carries the argument

The load-bearing object is the $\mathrm{EVI}_K$ gradient flow of the relative entropy in the Wasserstein space: a curve $(\mu_t)$ satisfying $\frac{1}{2}\frac{d}{dt}W_2^2(\mu_t,\nu) + \frac{K}{2}W_2^2(\mu_t,\nu) \le \mathrm{Ent}_m(\nu)-\mathrm{Ent}_m(\mu_t)$. The paper's integral characterization of this flow inequality allows one to integrate it against the measure $\Omega$; choosing the starting point to be the barycenter then yields Jensen's inequality directly. Jensen's inequality plays the role of an a priori estimate: it bounds the entropy of the barycenter, forces the barycenter to be absolutely continuous, and, with a weak Monge property, yields uniqueness. The same estimate through the $\mathrm{EVI}_{K,N}$ version gives the dimension-dependent refinement, and the Barycenter-Curvature-Dimension condition abstracts this barycenter Jensen inequality into a new curvature-dimension condition.

What would settle it

Exhibit two absolutely continuous probability measures at finite Wasserstein distance on an infinite-dimensional RCD(K,∞) space, such as an abstract Wiener space, that admit two distinct optimal transport plans; the uniqueness assertions of Theorems 5.9, 5.16(B), and 5.17(B) would then fail.

Watch

Extended reading notes

Core claim

The paper's central claim is that, on an extended metric measure space where every measure with finite distance to the domain of the relative entropy starts an $\mathrm{EVI}_K$ gradient flow of the entropy in the Wasserstein space—in particular on $\mathrm{RCD}(K,\infty)$ spaces and abstract Wiener spaces—any probability measure $\Omega$ over the Wasserstein space with finite variance and finite expected entropy has a Wasserstein barycenter, every barycenter is absolutely continuous with respect to the reference measure, and Jensen's inequality $\mathrm{Ent}_m(\bar\mu) \le \int \mathrm{Ent}_m\,d\Omega - \frac{K}{2}\int W_2^2(\bar\mu,\cdot)\,d\Omega$ holds. The same method, pushed through the finite-dimensional $\mathrm{EVI}_{K,N}$ gradient flow, yields a dimension-dependent Jensen inequality that seems new even on $\mathbb{R}^n$ and implies uniqueness and absolute continuity on $\mathrm{RCD}(K,N)$ spaces. A second claim is that a new Barycenter-Curvature-Dimension condition, defined by requiring exactly this barycenter Jensen inequality, is stable under measured Gromov–Hausdorff convergence and suffices for existence of barycenters, multi-marginal optimal transport maps, and Brunn–Minkowski and Blaschke–Santalo type inequalities.

Load-bearing premise

The uniqueness and absolute-continuity conclusions for infinite-dimensional RCD(K,∞) spaces rely on the weak Monge property—that any two absolutely continuous measures at finite Wasserstein distance admit a unique optimal transport map—yet the paper cites support for that property only in finite-dimensional RCD(K,N) spaces and gives no proof or reference for the infinite-dimensional case.

Editorial extensions

If this is right

On any $\mathrm{RCD}(K,\infty)$ space, every probability measure $\Omega$ over the Wasserstein space with finite variance and finite expected entropy has a barycenter, and every barycenter is absolutely continuous with respect to the reference measure (Theorems 5.3 and 5.8).
Under the weak Monge property—valid in finite-dimensional RCD spaces—the barycenter is unique; the dimension-dependent Jensen inequality gives a refined uniqueness statement on $\mathrm{RCD}(K,N)$ spaces whenever $\Omega$ gives positive mass to finite-entropy measures (Theorem 5.9 and Corollary 5.13).
The multi-marginal optimal transport problem with cost $c(x_1,\ldots,x_n)=\inf_y \sum_i \tfrac12 d^2(x_i,y)$ has a unique Monge solution on RCD spaces, and finite-support Wasserstein barycenters of absolutely continuous measures are unique and absolutely continuous without a finite-entropy condition (Theorems 5.16 and 5.17).
The $\mathrm{BCD}(K,\infty)$ condition is closed under measured Gromov–Hausdorff convergence and, under exponential volume growth or a probability reference measure, guarantees existence of Wasserstein barycenters (Theorems 6.6 and 6.7).
$\mathrm{BCD}(K,N)$ spaces satisfy a multi-marginal Brunn–Minkowski inequality, and $\mathrm{BCD}(1,\infty)$ spaces satisfy a functional Blaschke–Santalo type inequality (Propositions 6.8–6.10).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same EVI-to-Jensen mechanism should give existence of barycenters in any barycenter space whose entropy functional has a contractive gradient flow, potentially far beyond Ricci bounds—for example on Poisson configuration spaces over manifolds with Ricci lower bound, where the EVI is already known.
The Barycenter-Curvature-Dimension condition is defined relative to the reference measure's entropy; a natural test is whether $\mathrm{BCD}(K,\infty)$ plus geodesicity actually forces the standard $\mathrm{CD}(K,\infty)$ condition, or whether a non-geodesic BCD space exists that has no Lott–Sturm–Villani structure, which would show the new condition is strictly broader.
The dimension-dependent Jensen inequality is new even on Euclidean space; applying it to indicator-type densities may yield quantitative finite-$N$ refinements of Brunn–Minkowski inequalities for Wasserstein barycenter sets.
A possible resolution of the weak Monge gap would be to prove the infinite-dimensional $\mathrm{RCD}(K,\infty)$ optimal transport map theorem by finite-dimensional approximation; until then, the uniqueness half of Theorems 5.9, 5.16(B), and 5.17(B) rests on an unproved premise.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core idea here — deriving Wasserstein Jensen's inequality from EVI gradient flows rather than local compactness — is genuinely new and the proof is clean. The dimensional Jensen inequality (Theorem 5.11) appears new even on Euclidean space, and I don't see an obvious flaw in that argument. The BCD condition is a real addition to the Lott–Sturm–Villani toolbox, and the stability under mGH convergence is a nice result. For RCD(K,N) spaces, the multi-marginal existence and uniqueness theorems look solid.

The soft spot is the RCD(K,∞) case. The paper repeatedly asserts that RCD(K,∞) spaces satisfy the weak Monge property, citing [GRS16], but that reference is for finite-dimensional RCD(K,N) spaces. [CM17] also requires an essentially non-branching MCP(K,N) condition with finite N. As far as I know, the existence of a unique optimal map between arbitrary absolutely continuous probability measures with finite W2 distance is not established for general RCD(K,∞) spaces. This gap affects Theorems 5.9, 5.15, 5.16, and 5.17 as stated for RCD(K,∞). It does not affect Theorems 5.3 and 5.8, which are the heart of the paper and appear sound.

This is not a fatal flaw; it is a citation/proof gap in a set of applications. The authors should either prove the weak Monge property for RCD(K,∞) (or find a correct reference) or restrict those statements to spaces where it is known, e.g., by adding an explicit assumption. The central existence and Jensen results stand, and the BCD theory is unaffected.

I'd send this to a serious referee. The paper is long but well-organized; the referee should focus on the RCD(K,∞) claims and verify the EVI_K,N proof. For a reading group, it's worth a session: the EVI-to-Jensen trick is a good concept, and the gap is instructive.

Referee Report

2 major / 5 minor

Summary. The paper develops a theory of Wasserstein barycenters on non-compact, non-smooth, and extended metric measure spaces. Its central mechanism is to derive Jensen-type inequalities for the relative entropy from the existence of EVI_K (and EVI_K,N) gradient flows, then to use those inequalities as a priori estimates to obtain existence, absolute continuity, and uniqueness of barycenters without local compactness. The authors prove such Jensen inequalities for RCD(K,∞) and RCD(K,N) spaces, for abstract Wiener spaces, and for configuration spaces, and they introduce a new Barycenter-Curvature-Dimension (BCD) condition defined through barycentric Jensen inequalities. They prove stability of BCD(K,∞) under measured Gromov–Hausdorff convergence, prove existence of barycenters on BCD spaces under mild hypotheses, and derive applications including multi-marginal Brunn–Minkowski and functional Blaschke–Santaló type inequalities. The paper also treats the multi-marginal Monge problem with the barycenter cost for RCD spaces.

Significance. The EVI-to-Jensen argument is elegant and genuinely dimension-free; the existence and Jensen parts of the paper do not use fitted parameters and are essentially self-contained. The interpretation of Jensen's inequality as an a priori estimate is a valuable conceptual contribution, and the BCD condition together with its stability and geometric applications is natural and potentially influential. If all advertised results are correct, the paper unifies and extends several earlier barycenter results in Euclidean, Riemannian, Alexandrov, Wiener, and configuration-space settings. However, the advertised uniqueness and multi-marginal results for RCD(K,∞) rest on an infinite-dimensional weak Monge property that is not proved or correctly referenced; this gap affects a substantial part of the theorem set, even though the main existence/Jensen core survives.

major comments (2)

[§5.3, Proposition 5.15(B), Theorems 5.16(B), 5.17(B), and Theorem 5.9] The weak Monge property for RCD(K,∞) spaces is asserted without proof. The proof of Proposition 5.15 uses unique optimal transport maps between absolutely continuous measures in the RCD(K,∞) case and cites [CM17] and [GRS16], but both references treat finite-dimensional settings (essentially non-branching MCP(K,N) and RCD(K,N) with finite N). Theorems 5.16(B) and 5.17(B) inherit this gap, as does the claim in Theorem 5.9 that RCD(K,∞) spaces are important examples. Because Theorems 1.4, 1.10, and 1.11 advertise RCD(K,∞) uniqueness and multi-marginal results, this is load-bearing. Please either provide a proof or a valid reference for the weak Monge property in infinite-dimensional RCD(K,∞) spaces, or restrict the corresponding statements to finite-dimensional RCD(K,N) and state the infinite-dimensional cases as conditional assumptions.
[§5.3, proofs of Theorem 5.16(B) and Theorem 5.17(B)] The RCD(K,∞) parts of these proofs are only sketched as "by induction and a similar truncation argument". Even if the weak Monge property were available, the truncation argument would need to show that the reduced multi-marginal problem still has all marginals absolutely continuous, that the optimal plan remains optimal after truncation, and that the reduced marginals have finite entropy so that Proposition 5.15(B) applies. As written, this reduction is not demonstrated. The gap is not merely expository: it is used to remove the finite-entropy condition and to prove absolute continuity of the barycenter. Please supply the missing argument or state these results conditionally on the weak Monge property plus the truncation procedure.

minor comments (5)

[Abstract and general presentation] There are typos, including "curvature-dimesion" in the abstract and "As a by product" in the introduction; these should be corrected.
[§4.2, Proposition 4.5] The proposition states that strict convexity holds if one marginal is absolutely continuous, but the proof requires optimal transport maps from the absolutely continuous measure μ to both ν1 and ν2. The stated hypothesis is broader than the proof supports. The later application in Theorem 5.9 only needs maps between absolutely continuous measures, so restating the proposition in that narrower form would make it correct.
[§5.3, cost function definition] The introduction defines the multi-marginal cost with a factor 1/2, while Section 5.3 uses c(x1,...,xn)=inf_y Σ d²(x_i,y) without the factor. The factor does not affect barycenters but does affect the numerical equality between the transport cost and the barycenter functional; the convention should be fixed consistently.
[§6.3, Theorem 6.7, Step 3] The text says the exponential volume growth condition holds "for all x0 ∈ X", but the definition used in Theorem 3.13 only assumes existence of some x0 and some c>0. Please make the base point and constant explicit in the proof.
[§3.1, Definition 3.1 and Theorem 5.8(A)] The paper defines the reference measure m as a probability measure, while standard RCD(K,∞) spaces are often stated for σ-finite reference measures. The tightness argument in Theorem 5.8(A) relies on a uniform entropy bound, which is used together with finite total mass of m; this normalization should be stated explicitly in the hypotheses of the theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EVI-to-Jensen derivation, the tightness-based existence proof, and the BCD applications are self-contained or externally grounded.

full rationale

The paper's central derivation is not circular. Theorem 5.2 proves that an EVI_K gradient flow implies Jensen's inequality for the functional E, using only the integral version of EVI and the definition of barycenter; this is an independent analytic argument, not an assumption of the conclusion. Theorem 5.3 then applies this to relative entropy on Wasserstein space, with the existence of EVI_K flows for RCD(K,∞) spaces taken from the external characterization of Ambrosio–Gigli–Savaré [AGS14, Theorem 5.1] and for abstract Wiener spaces from Ambrosio–Erbar–Savaré [AES16, Theorem 11.1]. The existence theorem 5.8 uses the estimate from Theorem 5.2 together with tightness via uniformly bounded entropy, again not assuming the barycenter existence it proves. The uniqueness theorem 5.9 explicitly assumes the weak Monge property, so its conclusion is conditional on that hypothesis rather than being smuggled in as a consequence. The BCD condition in Definition 6.1 is indeed defined by the Jensen inequality, but the paper does not present this definition as a derived theorem; it presents it as a new synthetic condition, and the subsequent stability and existence results are legitimate conditional statements about spaces satisfying that definition. The applications in Section 6.4 (multi-marginal Brunn–Minkowski, logarithmic Brunn–Minkowski, and Blaschke–Santaló-type inequalities) follow directly from the defining Jensen inequality together with standard entropy bounds, so they are consequences of the definition rather than circular restatements. The only notable weakness is that Proposition 5.15 and Theorem 5.16(B) cite [GRS16] for uniqueness of optimal transport maps in RCD(K,∞) spaces, while [GRS16] concerns finite-dimensional spaces; this is a possible gap or missing reference, but it is not circularity, since the cited result is external and not equivalent to the paper's own conclusion. No fitted parameters are renamed as predictions, and no load-bearing step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central results rest on standard optimal transport and gradient flow theorems from the literature. The only questionable input is the weak Monge property for RCD(K,∞), which is not established by the cited reference. No data-dependent free parameters appear.

assumptions (5)

domain assumption Weak Monge property for general RCD(K,∞) spaces
Invoked via [GRS16] in Theorem 5.9 and Proposition 5.15; the cited work is finite-dimensional, so this is an unsupported assumption for infinite-dimensional RCD(K,∞) spaces.
standard math EVI_K characterization of RCD(K,∞)
Theorem 3.13 from [AGS14, AGMR15], used in Theorems 5.3 and 5.8.
standard math EVI_{K,N} characterization of RCD(K,N)
Theorem 3.16 from [EKS15], used for Corollaries 5.12 and 5.13.
standard math Tightness of measures with uniformly bounded entropy
Used in Theorem 5.8, citing [AGMR15, Lemma 4.4] and [FSS10, Theorem 1.2].
standard math Stability of Wasserstein barycenters under narrow convergence
Used in Theorem 6.7, citing [LGL17, Theorem 3].

invented entities (2)

Barycenter-Curvature-Dimension (BCD) condition
purpose: New synthetic curvature-dimension condition for extended metric measure spaces, defined via a Jensen inequality for entropy along Wasserstein barycenters.
Definition 6.1; the paper shows RCD spaces satisfy it and proves stability, but no example outside the paper's framework is given that would independently test the condition.
Barycenter extended metric space
purpose: New class of extended metric spaces where every finitely supported measure with finite variance has a barycenter.
Definition 2.5; it is a definition used to formulate results, not a conjectured object with external predictions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the geometry of Wasserstein barycenter I." pith.science (2026). https://pith.science/paper/PVXJC3IA

@misc{pith2026241201190,
  author       = {Pith},
  title        = {Pith review of: On the geometry of Wasserstein barycenter I},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVXJC3IA}},
  note         = {Machine review of arXiv:2412.01190}
}

read the original abstract

We study the Wasserstein barycenter problem in the setting of non-compact, non-smooth extended metric measure spaces. We introduce a couple of new concepts and obtain the existence, uniqueness, absolute continuity of the Wasserstein barycenter, and prove Jensen's inequality in an abstract framework. This generalized several results on Euclidean space, Riemannian manifolds and Alexandrov spaces, to metric measure spaces satisfying Riemannian Curvature-Dimension condition \`a la Lott--Sturm--Villani, and some extended metric measure spaces including abstract Wiener spaces and configuration spaces over Riemannian manifolds. We also introduce a new curvature-dimesion condition, we call Barycenter-Curvature-Dimension condition. We prove its stability under measured-Gromov--Hausdorff convergence and prove the existence of the Wasserstein barycenter under this new condition. In addition, we get some geometric inequalities including a multi-marginal Brunn--Minkowski inequality and a functional Blaschke--Santal\'o type inequality.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Barycenter curvature-dimension condition for extended metric measure spaces
math.MG 2025-01 conditional novelty 4.0 of 10

A new barycenter-based curvature-dimension condition, BCD(K,∞), is defined and shown to follow from EVI gradient-flow theory on RCD, Wiener, and configuration spaces.

Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

1, 179–203

[ABA22] Jason M Altschuler and Enric Boix-Adsera, Wasserstein barycenters are np-hard to compute , SIAM Journal on Mathematics of Data Science 4 (2022), no. 1, 179–203. 4 [AC11] Martial Agueh and Guillaume Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal. 43 (2011), no. 2, 904–924. MR 2801182 4, 9, 14, 15, 19, 33 [AES16] Luigi Ambrosio, M...

work page 2022
[12]

©2023, pp. 125–159. MR 4615741 17 [Vil09] C´ edric Villani, Optimal transport. old and new , Grundlehren der Math- ematischen Wissenschaften, vol. 338, Springer-Verlag, Berlin,

work page 2023
[44]

19, 41, 43 [Li24] Zhenhao Li, The Globalization Theorem for CD(K,N) on locally finite Spaces, Annali di Matematica Pura ed Applicata (1923 -) 203 (2024), no

MR 4370472 43 [LGL17] Thibaut Le Gouic and Jean-Michel Loubes, Existence and consistency of wasserstein barycenters , Probability Theory and Related Fields 168 (2017), 901–917. 19, 41, 43 [Li24] Zhenhao Li, The Globalization Theorem for CD(K,N) on locally finite Spaces, Annali di Matematica Pura ed Applicata (1923 -) 203 (2024), no. 1, 49–70. 17 [LN05] Ja...

arXiv 2017
[357]

I , Acta Math

3, 12, 18 [Stu06a] , On the geometry of metric measure spaces. I , Acta Math. 196 (2006), no. 1, 65–131. MR MR2237206 7, 15 [Stu06b] , On the geometry of metric measure spaces. II , Acta Math. 196 (2006), no. 1, 133–177. MR MR2237207 (2007k:53051b) 7, 8, 15, 16 [Stu23] , Metric measure spaces and synthetic Ricci bounds: fundamental concepts and recent dev...

work page 2006
[2001]

2, 7, 10, 39 [BK18] J´ er´ emie Bigot and Thierry Klein, Characterization of barycenters in the Wasserstein space by averaging optimal transport maps, ESAIM Probab. Stat. 22 (2018), 35–57. MR 3872127 4 46 [BR07] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas, Mea- sure theory, vol. 1, Springer,

work page 2018
[2007]

20 [BS10] Kathrin Bacher and Karl-Theodor Sturm, Localization and tensoriza- tion properties of the curvature-dimension condition for metric mea- sure spaces , J. Funct. Anal. 259 (2010), no. 1, 28–56. MR 2610378 (2011i:53050) 17 [CE10] Guillaume Carlier and Ivar Ekeland, Matching for teams, Economic the- ory 42 (2010), 397–418. 4, 33 [CEMS01] Dario Corde...

work page Pith review arXiv 2010
[2008]

MR 2401600 (2009h:49002) 27 [AGS14] , Metric measure spaces with Riemannian Ricci curvature bounded from below , Duke Math. J. 163 (2014), 1405–1490. 16, 17, 18, 27 [AKP24] Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin, Alexan- drov geometry—foundations , Graduate Studies in Mathematics, vol. 236, American Mathematical Society, Providence, RI,

work page 2014
[2009]

MR MR2459454 2, 14, 18, 39, 41 [Vil16] C´ edric Villani, Synthetic theory of Ricci curvature bounds, Jpn. J. Math. 11 (2016), no. 2, 219–263. MR 3544918 17 [vRS05] Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature , Comm. Pure Appl. Math. 58 (2005), no. 7, 923–940. MR 2142879 38 [Yok16] Tak...

work page 2016

Show all 13 references

[2018]

Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 301–340. MR 3966731 17 [AMS19] Luigi Ambrosio, Andrea Mondino, and Giuseppe Savar´ e, Nonlinear dif- fusion equations and curvature conditions in metric measure spaces , Mem. Amer. Math. Soc. 262 (2019), no....

2019
[2022]

7, 26 [EH15] Matthias Erbar and Martin Huesmann, Curvature bounds for configura- tion spaces, Calc. Var. Partial Differential Equations 54 (2015), no. 1, 397–430. MR 3385165 7, 26 [EKS15] Matthias Erbar, Kazumasa Kuwada, and Karl-Theodor Sturm, On the equivalence of the entrop...

2015
[2023]

1, 153–179

4 [McC97] Robert J McCann, A convexity principle for interacting gases, Advances in mathematics 128 (1997), no. 1, 153–179. 4, 7, 38 [Mil21] Emanuel Milman, The quasi curvature-dimension condition with ap- plications to sub-Riemannian manifolds , Comm. Pure Appl. Math. 74 (202...

1997 arXiv
[2024]

©2024. MR 4734965 7 [Amb18] Luigi Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces , Proceedings of the International Congress of Mathematicians—Rio de Janeiro

2024
[2025]

1, 175–193

8 [Jen06] Johan Ludwig William Valdemar Jensen, Sur les fonctions convexes et les in´ egalit´ es entre les valeurs moyennes, Acta mathematica 30 (1906), no. 1, 175–193. 3 48 [Jia17] Yin Jiang, Absolute continuity of Wasserstein barycenters over Alexan- drov spaces, Canad. J. M...

1906

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

1, 179–203

[ABA22] Jason M Altschuler and Enric Boix-Adsera, Wasserstein barycenters are np-hard to compute , SIAM Journal on Mathematics of Data Science 4 (2022), no. 1, 179–203. 4 [AC11] Martial Agueh and Guillaume Carlier, Barycenters in the Wasserstein space, SIAM J. Math. Anal. 43 (2011), no. 2, 904–924. MR 2801182 4, 9, 14, 15, 19, 33 [AES16] Luigi Ambrosio, M...

work page 2022

[2] [12]

©2023, pp. 125–159. MR 4615741 17 [Vil09] C´ edric Villani, Optimal transport. old and new , Grundlehren der Math- ematischen Wissenschaften, vol. 338, Springer-Verlag, Berlin,

work page 2023

[3] [44]

19, 41, 43 [Li24] Zhenhao Li, The Globalization Theorem for CD(K,N) on locally finite Spaces, Annali di Matematica Pura ed Applicata (1923 -) 203 (2024), no

MR 4370472 43 [LGL17] Thibaut Le Gouic and Jean-Michel Loubes, Existence and consistency of wasserstein barycenters , Probability Theory and Related Fields 168 (2017), 901–917. 19, 41, 43 [Li24] Zhenhao Li, The Globalization Theorem for CD(K,N) on locally finite Spaces, Annali di Matematica Pura ed Applicata (1923 -) 203 (2024), no. 1, 49–70. 17 [LN05] Ja...

arXiv 2017

[4] [357]

I , Acta Math

3, 12, 18 [Stu06a] , On the geometry of metric measure spaces. I , Acta Math. 196 (2006), no. 1, 65–131. MR MR2237206 7, 15 [Stu06b] , On the geometry of metric measure spaces. II , Acta Math. 196 (2006), no. 1, 133–177. MR MR2237207 (2007k:53051b) 7, 8, 15, 16 [Stu23] , Metric measure spaces and synthetic Ricci bounds: fundamental concepts and recent dev...

work page 2006

[5] [2001]

2, 7, 10, 39 [BK18] J´ er´ emie Bigot and Thierry Klein, Characterization of barycenters in the Wasserstein space by averaging optimal transport maps, ESAIM Probab. Stat. 22 (2018), 35–57. MR 3872127 4 46 [BR07] Vladimir Igorevich Bogachev and Maria Aparecida Soares Ruas, Mea- sure theory, vol. 1, Springer,

work page 2018

[6] [2007]

20 [BS10] Kathrin Bacher and Karl-Theodor Sturm, Localization and tensoriza- tion properties of the curvature-dimension condition for metric mea- sure spaces , J. Funct. Anal. 259 (2010), no. 1, 28–56. MR 2610378 (2011i:53050) 17 [CE10] Guillaume Carlier and Ivar Ekeland, Matching for teams, Economic the- ory 42 (2010), 397–418. 4, 33 [CEMS01] Dario Corde...

work page Pith review arXiv 2010

[7] [2008]

MR 2401600 (2009h:49002) 27 [AGS14] , Metric measure spaces with Riemannian Ricci curvature bounded from below , Duke Math. J. 163 (2014), 1405–1490. 16, 17, 18, 27 [AKP24] Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin, Alexan- drov geometry—foundations , Graduate Studies in Mathematics, vol. 236, American Mathematical Society, Providence, RI,

work page 2014

[8] [2009]

MR MR2459454 2, 14, 18, 39, 41 [Vil16] C´ edric Villani, Synthetic theory of Ricci curvature bounds, Jpn. J. Math. 11 (2016), no. 2, 219–263. MR 3544918 17 [vRS05] Max-K. von Renesse and Karl-Theodor Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature , Comm. Pure Appl. Math. 58 (2005), no. 7, 923–940. MR 2142879 38 [Yok16] Tak...

work page 2016