# The symmetric property ($\tau$) 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.
• Volume: 17, Issue: 2, page 357-370
• ISSN: 0240-2963

top

## Abstract

top
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 $Hf\left(x\right)={inf}_{y}f\left(x+y\right)+\frac{1}{2}{|y|}^{2}$ and we have$\int {\mathrm{e}}^{-f}\phantom{\rule{0.166667em}{0ex}}d{\gamma }_{d}\int {\mathrm{e}}^{Hf}\phantom{\rule{0.166667em}{0ex}}d{\gamma }_{d}\le 1.$This property is equivalent to a certain functional form of the Blaschke-Santaló inequality, as explained in a paper by Artstein, Klartag and Milman.

## How to cite

top

Lehec, Joseph. "The symmetric property ($\tau$) for the Gaussian measure." Annales de la faculté des sciences de Toulouse Mathématiques 17.2 (2008): 357-370. <http://eudml.org/doc/10089>.

@article{Lehec2008,
abstract = {We give a proof, based on the Poincaré inequality, of the symmetric property ($\tau$) for the Gaussian measure. If $f\colon \mathbb\{R\}^d\rightarrow \mathbb\{R\}$ is continuous, bounded from below and even, we define $Hf(x)=\inf _\{y\} f(x+y)+ \frac\{1\}\{2\} |y|^2$ 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.},
affiliation = {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.},
author = {Lehec, Joseph},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {property ; Gaussian measure; Santalo inequality},
language = {eng},
month = {6},
number = {2},
pages = {357-370},
publisher = {Université Paul Sabatier, Toulouse},
title = {The symmetric property ($\tau$) for the Gaussian measure},
url = {http://eudml.org/doc/10089},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Lehec, Joseph
TI - The symmetric property ($\tau$) for the Gaussian measure
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 2
SP - 357
EP - 370
AB - We give a proof, based on the Poincaré inequality, of the symmetric property ($\tau$) for the Gaussian measure. If $f\colon \mathbb{R}^d\rightarrow \mathbb{R}$ is continuous, bounded from below and even, we define $Hf(x)=\inf _{y} f(x+y)+ \frac{1}{2} |y|^2$ 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.
LA - eng
KW - property ; Gaussian measure; Santalo inequality
UR - http://eudml.org/doc/10089
ER -

## References

top
1. Artstein-Avidan (S.), Klartag (B.), and Milman (V.).— The Santaló point of a function, and a functional form of Santaló inequality, Mathematika 51, p. 33-48 (2005). Zbl1121.52021MR2220210
2. Ball (K.).— Isometric problems in ${\ell }_{p}$ and sections of convex sets, PhD dissertation, University of Cambridge, (1986).
3. Ball (K.).— An elementary introduction to modern convex geometry, Flavors of geometry, edited by S. Levy, Cambridge University Press, (1997). Zbl0901.52002MR1491097
4. Berger (M.).— Geometry, vol. I-II, translated from the French by M. Cole and S. Levy, Universitext, Springer, (1987). Zbl0606.51001
5. Caffarelli (L.A.).— Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. math. phys. 214, no. 3, p. 547-563 (2000). Zbl0978.60107MR1800860
6. Cordero-Erausquin (D.), Fradelizi (M.), and Maurey (B.).— The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems, J. Funct. Anal. 214, p. 410-427 (2004). Zbl1073.60042MR2083308
7. Fradelizi (M.) and Meyer (M.).— Some functional forms of Blaschke-Santaló inequality, Math. Z., 256, no. 2, p. 379-395 (2007). Zbl1128.52007MR2289879
8. Klartag (B.).— Marginals of geometric inequalities, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1910, Springer, p. 133-166 (2007). Zbl1132.52015MR2349606
9. Ledoux (M.).— The concentration of measure phenomenon, Mathematical Surveys and Monographs, American Mathematical Society, (2001). Zbl0995.60002MR1849347
10. Lutwak (E.) and Zhang (G.).— Blaschke-Santaló inequalities, J. Diff. Geom. 47, no. 1, p. 1-16 (1997). Zbl0906.52003MR1601426
11. Maurey (B.).— Some deviation inequalities, Geom. Funct. Anal. 1, no. 2, p. 188-197 (1991). Zbl0756.60018MR1097258
12. Meyer (M.) and Pajor (A.).— On Santaló’s inequality, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1376, Springer, 1989, p. 261-263. Zbl0673.52010
13. Pisier (G.).— The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, Cambridge University Press, (1989). Zbl0698.46008MR1036275
14. Saint-Raymond (J.).— Sur le volume des corps convexes symétriques, in Séminaire d’initiation à l’Analyse, 20ème année, edited by G. Choquet, M. Rogalski and J. Saint-Raymond, Publ. Math. Univ. Pierre et Marie Curie, 1981. Zbl0531.52006
15. Santaló (L.A.).— Un invariante afin para los cuerpos convexos del espacio de $n$ dimensiones, Portugaliae Math. 8, p. 155-161 (1949). Zbl0038.35702

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.