Comptes Rendus
no. 8
Functional analysis/Geometry
One more proof of the Alexandrov–Fenchel inequality
[Une autre preuve de l'inégalité d'Alexandrov–Fenchel]

Présenté par : Gilles Pisier

Dario Cordero-Erausquin1 ; Bo'az Klartag2 ; Quentin Merigot3 ; Filippo Santambrogio4
1 Institut de mathématiques de Jussieu, Sorbonne Université, 4, place Jussieu, 75252 Paris cedex 05, France
2 Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
3 Laboratoire de mathématiques d'Orsay, Universié Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
4 Institut Camille-Jordan, Université Claude-Bernard – Lyon-1, 43, boulevard du 11-Novembre-1918, 69622 Villeurbanne cedex, France
Comptes Rendus. Mathématique, Volume 357 (2019) no. 8, pp. 676-680.

Nous donnons une preuve courte des inégalités d'Alexandrov–Fenchel qui repose sur des propriétés algébriques élémentaires ou de convexité des volumes mixtes de polytopes.

We present a short proof of the Alexandrov–Fenchel inequalities, which mixes elementary algebraic properties and convexity properties of mixed volumes of polytopes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.07.004

Dario Cordero-Erausquin 1 ; Bo'az Klartag 2 ; Quentin Merigot 3 ; Filippo Santambrogio 4

1 Institut de mathématiques de Jussieu, Sorbonne Université, 4, place Jussieu, 75252 Paris cedex 05, France
2 Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
3 Laboratoire de mathématiques d'Orsay, Universié Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
4 Institut Camille-Jordan, Université Claude-Bernard – Lyon-1, 43, boulevard du 11-Novembre-1918, 69622 Villeurbanne cedex, France 
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Dario Cordero-Erausquin; Bo'az Klartag; Quentin Merigot; Filippo Santambrogio. One more proof of the Alexandrov–Fenchel inequality. Comptes Rendus. Mathématique, Volume 357 (2019) no. 8, pp. 676-680. doi : 10.1016/j.crma.2019.07.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.07.004/

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  • Daniel Hug; Paul A. Reichert Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies, Advances in Geometry, Volume 25 (2025) no. 1, p. 13 | DOI:10.1515/advgeom-2024-0030
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  • Swee Hong Chan; Igor Pak Equality cases of the Alexandrov–Fenchel inequality are not in the polynomial hierarchy, Forum of Mathematics, Pi, Volume 12 (2024) | DOI:10.1017/fmp.2024.20
  • Daniel Hug; Paul A. Reichert The support of mixed area measures involving a new class of convex bodies, Journal of Functional Analysis, Volume 287 (2024) no. 11, p. 110622 | DOI:10.1016/j.jfa.2024.110622
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  • James Melbourne; Gerardo Palafox-Castillo A discrete complement of Lyapunov’s inequality and its information theoretic consequences, The Annals of Applied Probability, Volume 33 (2023) no. 6A | DOI:10.1214/22-aap1919
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