[Sur la démonstration de l’inégalité de Blaschke–Santaló par calcul stochastique]
In 2024, Courtade, Fathi and Mikulincer gave a proof of the symmetrized Talagrand inequality based on stochastic calculus, in the spirit of Borell’s proof of the Prékopa–Leindler inequality. The symmetrized Talagrand inequality can be seen as a dual form of the functional Santaló inequality. The modest purpose of this note is to give a simplified version of the Courtade, Fathi and Mikulincer argument. Namely we first recall briefly Borell’s original argument, and we then explain a simple twist in his proof that allows to recover the functional Santaló inequality directly, rather than in its dual form.
En 2024, Courtade, Fathi et Mikulincer ont donné une preuve basée sur le calcul stochastique de l’inégalité de Talagrand améliorée pour les fonctions centrées, dans l’esprit de la preuve de Borell de l’inégalité de Prékopa–Leindler. L’inégalité de Talagrand centrée peut se voir comme une forme duale de l’inégalité de Santaló fonctionnelle. Le modeste objectif de cette note est de donner une version simplifiée de l’argument de Courtade, Fathi et Mikulincer. Plus précisément, on rappelle d’abord l’argument originel de Borell, avant d’en expliquer une variation qui permet de retrouver directement l’inégalité de Santaló fonctionnelle, sans passer par sa forme duale.
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Keywords: Functional inequalities, Blaschke–Santaló inequality, Brownian motion
Mots-clés : Inégalités fonctionnelles, inégalité de Blaschke–Santaló, mouvement brownien
Joseph Lehec 1
CC-BY 4.0
Joseph Lehec. On the stochastic proof of the Blaschke–Santaló inequality. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 599-604. doi: 10.5802/crmath.849
@article{CRMATH_2026__364_G3_599_0,
author = {Joseph Lehec},
title = {On the stochastic proof of the {Blaschke{\textendash}Santal\'o} inequality},
journal = {Comptes Rendus. Math\'ematique},
pages = {599--604},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.849},
language = {en},
}
[1] The Santaló point of a function, and a functional form of the Santaló inequality, Mathematika, Volume 51 (2004) no. 1–2, pp. 33-48 | DOI | Zbl | MR
[2] Isometric problems in $\ell _p$ and sections of convex sets, Ph. D. Thesis, University of Cambridge (UK) (1986)
[3] Positive Gaussian kernels also have Gaussian minimizers, Memoirs of the American Mathematical Society, 1359, American Mathematical Society, 2022 | DOI | Zbl
[4] Positivity improving operators and hypercontractivity, Math. Z., Volume 180 (1982), pp. 225-234 | DOI | Zbl | MR
[5] Diffusion equations and geometric inequalities, Potential Anal., Volume 12 (2000) no. 1, pp. 49-71 | DOI | Zbl | MR
[6] Duality and heat flow (2024) | arXiv | Zbl
[7] Stochastic proof of the sharp symmetrized Talagrand inequality, Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1779-1784 | DOI | Zbl | Numdam
[8] A sharp symmetrized form of Talagrand’s transport-entropy inequality for the Gaussian measure, Electron. Commun. Probab., Volume 23 (2018), 81, 9 pages | DOI | Zbl | MR
[9] Some functional forms of Blaschke–Santaló inequality, Math. Z., Volume 256 (2007) no. 2, pp. 379-395 | DOI | Zbl | MR
[10] The symmetric property $(\tau )$ for the Gaussian measure, Ann. Fac. Sci. Toulouse, Math. (6), Volume 17 (2008) no. 2, pp. 357-370 | DOI | Zbl | Numdam | MR
[11] A direct proof of the functional Santaló inequality, Comptes Rendus. Mathématique, Volume 347 (2009) no. 1–2, pp. 55-58 | DOI | Zbl | Numdam
[12] Partitions and functional Santaló inequalities, Arch. Math., Volume 92 (2009) no. 1, pp. 89-94 | DOI | Zbl
[13] Representation formula for the entropy and functional inequalities, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 3, pp. 885-899 | DOI | Zbl | Numdam | MR
[14] Short probabilistic proof of the Brascamp–Lieb and Barthe theorems, Can. Math. Bull., Volume 57 (2014) no. 3, pp. 585-597 | DOI | Zbl | MR
[15] Some deviation inequalities, Geom. Funct. Anal., Volume 1 (1991) no. 2, pp. 188-197 | DOI | Zbl | MR
[16] On the Blaschke–Santaló inequality, Arch. Math., Volume 55 (1990) no. 1, pp. 82-93 | DOI | Zbl
[17] Gaussian correlation via inverse Brascamp–Lieb (2025) | arXiv | Zbl
[18] The functional volume product under heat flow (2023) | arXiv | Zbl
[19] A generalized Legendre duality relation and Gaussian saturation (2024) | arXiv
[20] On improved Gaussian correlation inequalities for symmetrical $n$-rectangles extended to certain multivariate gamma distributions and some further probability inequalities, Far East J. Theor. Stat., Volume 69 (2025) no. 1, pp. 1-38 | DOI | Zbl
[21] Sur le volume des corps convexes symétriques, Séminaire d’initiation à l’analyse. 20ème année, 1980/1981 (Gustave Choquet; Marek Rogalski; Jean Saint-Raymond, eds.) (Publications Mathématiques de l’Université Pierre et Marie Curie), Volume 46, Université Pierre et Marie Curie, 1981 (Exposé no. 11, 25 pages) | MR | Zbl
[22] Un invariante afin para los cuerpos convexos del espacio de $n$ dimensiones, Port. Math., Volume 8 (1949) no. 4, pp. 155-161
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