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

Lehec, Joseph

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 ($\tau$) symétrique pour la mesure Gaussienne. Si $f:{ℝ}^d\to ℝ$ est continue, minorée et paire, on a, en posant $Hf\left(x\right)={inf}_{y}f(x+y)+\frac{1}{2}{\left|y\right|}^2$  :

$$\int {\mathrm{e}}^{-f}d{\gamma }_d\int {\mathrm{e}}^{Hf}d{\gamma }_d\le 1.$$

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.

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

$$\int {\mathrm{e}}^{-f}d{\gamma }_d\int {\mathrm{e}}^{Hf}d{\gamma }_d\le 1.$$

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

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