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 (2008)

  • Volume: 17, Issue: 2, page 357-370
  • ISSN: 0240-2963

Abstract

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

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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

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  8. Klartag (B.).— Marginals of geometric inequalities, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1910, Springer, p. 133-166 (2007). Zbl1132.52015MR2349606
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  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

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