The symmetric property ($\tau $) for the Gaussian measure

Joseph Lehec

doi:10.5802/afst.1186

The symmetric property (

τ

) for the Gaussian measure

Joseph Lehec ¹

¹ Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Cité Descartes - 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne la Vallée Cedex 2, France.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 2, pp. 357-370.

Abstract
Abstract

We give a proof, based on the Poincaré inequality, of the symmetric property ( $τ$ ) for the Gaussian measure. If $f : ℝ^{d} \to ℝ$ is continuous, bounded from below and even, we define $H f (x) = {inf}_{y} f (x + y) + \frac{1}{2} {| y |}^{2}$ and we have

\int e^{- f} d γ_{d} \int e^{H f} d γ_{d} \leq 1 .

This property is equivalent to a certain functional form of the Blaschke-Santaló inequality, as explained in a paper by Artstein, Klartag and Milman.

On dérive de l’inégalité de Poincaré la propriété ( $τ$ ) symétrique pour la mesure Gaussienne. Si $f : ℝ^{d} \to ℝ$ est continue, minorée et paire, on a, en posant $H f (x) = {inf}_{y} f (x + y) + \frac{1}{2} {| y |}^{2}$ :

\int e^{- f} d γ_{d} \int e^{H f} d γ_{d} \leq 1 .

Comme indiqué dans un article d’Artstein, Klartag et Milman, cette propriété est équivalente à l’une des versions fonctionnelles de l’inégalité de Blaschke-Santaló.

MR

DOI: 10.5802/afst.1186

Author's affiliations:

Joseph Lehec ¹

¹ Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Cité Descartes - 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne la Vallée Cedex 2, France.

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     title = {The symmetric property~($\tau $) for the {Gaussian} measure},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {357--370},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 17},
     number = {2},
     year = {2008},
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Joseph Lehec. The symmetric property ($\tau $) for the Gaussian measure. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 2, pp. 357-370. doi : 10.5802/afst.1186. https://afst.centre-mersenne.org/articles/10.5802/afst.1186/

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