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The symmetric property (τ) for the Gaussian measure
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 2, pp. 357-370.

On dérive de l’inégalité de Poincaré la propriété (τ) symétrique pour la mesure Gaussienne. Si f: d est continue, minorée et paire, on a, en posant Hf(x)=inf y f(x+y)+1 2|y| 2   :

e-fdγdeHfdγd1.

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.

We give a proof, based on the Poincaré inequality, of the symmetric property (τ) for the Gaussian measure. If f: d is continuous, bounded from below and even, we define Hf(x)=inf y f(x+y)+1 2|y| 2 and we have

e-fdγdeHfdγd1.

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

Reçu le : 2006-12-27
Accepté le : 2008-03-02
Publié le : 2008-12-11
DOI : https://doi.org/10.5802/afst.1186
@article{AFST_2008_6_17_2_357_0,
     author = {Joseph Lehec},
     title = {The symmetric property~($\tau $) for the Gaussian measure},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {357--370},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 17},
     number = {2},
     year = {2008},
     doi = {10.5802/afst.1186},
     zbl = {pre05503159},
     mrnumber = {2487858},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2008_6_17_2_357_0/}
}
Joseph Lehec. The symmetric property ($\tau $) for the Gaussian measure. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 2, pp. 357-370. doi : 10.5802/afst.1186. https://afst.centre-mersenne.org/item/AFST_2008_6_17_2_357_0/

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