[Localité des cônes tangents centrés dans l’espace de Wasserstein]
The geometric tangent cone to a probability measure $\mu $ is a set of measure-valued applications that are almost geodesics. This is a nonlocal condition, typically lost when conditioning the measure on a given set. We show that if one removes the barycenter of any element of the tangent cone, then the resulting set of centered measure fields is characterized by a local condition. Precisely, centered tangent fields must be concentrated on a family of vector subspaces attached to any point, and these subspaces correspond to the normal spaces to some sets of “dimension $k$” on which the measure $\mu $ is concentrated.
Le cône tangent géométrique en une mesure $\mu $ est défini comme un ensemble d’applications à valeurs dans les mesures, chacune décrivant presque une géodésique. Cette propriété n’est pas locale et est généralement perdue quand la mesure est restreinte à un sous-ensemble. Nous montrons qu’il suffit de retirer le barycentre des éléments du cône tangent pour obtenir un ensemble de champs centrés lui-même caractérisé par une condition locale. Plus précisément, les champs tangents centrés sont concentrés sur une famille de sous-espaces vectoriels attachés en chaque point, correspondant aux espaces normaux à certains ensembles de « dimension $k$ » sur lesquels la mesure $\mu $ est concentrée.
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Keywords: Wasserstein spaces, tangent cone, DC functions
Mots-clés : Espaces de Wasserstein, cône tangent, fonctions DC
Averil Aussedat 1
CC-BY 4.0
Averil Aussedat. Locality of centered tangent cones in the Wasserstein space. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1575-1606. doi: 10.5802/jep.355
@article{JEP_2026__13__1575_0,
author = {Averil Aussedat},
title = {Locality of centered tangent cones in {the~Wasserstein} space},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1575--1606},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.355},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.355/}
}
TY - JOUR AU - Averil Aussedat TI - Locality of centered tangent cones in the Wasserstein space JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1575 EP - 1606 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.355/ DO - 10.5802/jep.355 LA - en ID - JEP_2026__13__1575_0 ER -
%0 Journal Article %A Averil Aussedat %T Locality of centered tangent cones in the Wasserstein space %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 1575-1606 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.355/ %R 10.5802/jep.355 %G en %F JEP_2026__13__1575_0
[AB06] - Infinite dimensional analysis, Springer-Verlag, Berlin, Heidelberg, 2006 | DOI | Zbl
[AGS05] - Gradient flows, Lectures in Math. ETH Zürich, Birkhäuser-Verlag, Basel, 2005 | DOI | Zbl
[AM16] - “On the differentiability of Lipschitz functions with respect to measures in the Euclidean space”, Geom. Funct. Anal. 26 (2016) no. 1, p. 1-66 | DOI | Zbl
[AS94] - “Ck-rectifiable sets”, J. reine angew. Math. 453 (1994), p. 1-20 | Zbl
[Aus25a] - “On the structure of the geometric tangent cone to the Wasserstein space”, J. Differential Equations 442 (2025), p. 113520 | DOI | Zbl
[Aus25b] - Optimal control problems and Hamilton-Jacobi-Bellman equations in some curved metric spaces, Ph. D. Thesis, INSA Rouen Normandie, 2025, available at https://theses.hal.science/tel-05133552/
[Bog07] - Measure theory, Springer, Berlin, Heidelberg, 2007 | DOI | Zbl
[Bre91] - “Polar factorization and monotone rearrangement of vector-valued functions”, Comm. Pure Appl. Math. 44 (1991) no. 4, p. 375-417 | DOI | Zbl
[Fed96] - Geometric measure theory, Classics in Math., Springer, Berlin, Heidelberg, 1996 | DOI | Zbl
[Gig08] - On the geometry of the space of probability measures endowed with the quadratic optimal transport distance, Ph. D. Thesis, Scuola Normale Superiore di Pisa, Pisa, 2008, https://ricerca.sns.it/retrieve/e3aacdfd-ef40-4c98-e053-3705fe0acb7e/Gigli_Nicola.pdf
[Gig11] - “On the inverse implication of Brenier-McCann theorems and the structure of $(P_2(M),W_2)$”, Methods Appl. Anal. 18 (2011) no. 2, p. 127-158 | DOI | Zbl
[Har59] - “On functions representable as a difference of convex functions”, Pacific J. Math. 9 (1959) no. 3, p. 707-713 | DOI | Zbl
[HU85] - “Generalized differentiability/duality and optimization for problems dealing with differences of convex functions”, in Convexity and duality in optimization (J. Ponstein, ed.), Springer, Berlin, Heidelberg, 1985, p. 37-70 | DOI | Zbl
[Jui11] - “On displacement interpolation of measures involved in Brenier’s theorem”, Proc. Amer. Math. Soc. 139 (2011) no. 10, p. 3623-3623 | DOI | Zbl
[Kha06] - Strange functions in real analysis, Pure and Applied Math., Chapman & Hall/CRC, Boca Raton, FL, 2006 no. 272 | Zbl
[Lot16] - “On tangent cones in Wasserstein space”, Proc. Amer. Math. Soc. 145 (2016) no. 7, p. 3127-3136 | DOI | Zbl
[Pav04] - “On the points of non-differentiability of convex functions”, Comment. Math. Univ. Carolinae 45 (2004) no. 4, p. 727-734 | Zbl
[Pre87] - “Geometry of measures in $\mathbb{R}^n$: Distribution, rectifiability, and densities”, Ann. of Math. (2) 125 (1987) no. 3, p. 537-643 | DOI | Zbl
[Roc69] - “Measurable dependence of convex sets and functions on parameters”, J. Math. Anal. Appl. 28 (1969) no. 1, p. 4-25 | DOI | Zbl
[Roc70] - Convex analysis, Princeton University Press, Princeton, NJ, 1970 | DOI | Zbl
[San15] - Optimal transport for applied mathematicians, Progress in nonlinear differential equations and their appl., vol. 87, Springer International Publishing, 2015 | DOI | Zbl
[SW14] - “Characterization of the tangent space of monotone transport plans in $\mathbb{R}\times \mathbb{R}$ with prescribed first projection”, 2014 | arXiv | Zbl
[Vil09] - Optimal transport, Grundlehren Math. Wissen., vol. 338, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009 | DOI | Zbl
[Zaj79] - “On the differentiation of convex functions in finite and infinite dimensional spaces”, Czechoslovak Math. J. 29 (1979) no. 3, p. 340-348 | DOI | Zbl
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