Extremal Properties of Central Half-Spaces for Product Measures

We deal with the isoperimetric and the shift problem for subsets of measure 1/2 in product probability spaces. We prove that the canonical central half-spaces are extremal in particular cases: products of log-concave measures on the real line satisfying precise conditions and products of uniform measures on spheres, or balls. As a corollary, we improve the known log-Sobolev constants for Euclidean balls. We also give some new results about the related question of estimating the volume of sections of unit balls of lp-sums of Minkowski spaces.

[1]  E. Schmidt.  Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. I , 1948 .

[2]  P. Levy,et al.  Problèmes concrets d'analyse fonctionnelle , 1952 .

[3]  H. Hadwiger,et al.  Gitterperiodische Punktmengen und Isoperimetrie , 1972 .

[4]  C. Borell.  The Brunn-Minkowski inequality in Gauss space , 1975 .

[5]  L. Gross.  LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .

[6]  E. Lieb,et al.  Best Constants in Young's Inequality, Its Converse, and Its Generalization to More than Three Functions , 1976 .

[7]  M. Kanter,et al.  Unimodality and dominance for symmetric random vectors , 1977 .

[8]  V. Sudakov,et al.  Extremal properties of half-spaces for spherically invariant measures , 1978 .

[9]  J. Vaaler.  A geometric inequality with applications to linear forms , 1979 .

[10]  A. Pajor,et al.  Sections of the unit ball of Ipn , 1988 .

[11]  K. Ball.  Volumes of sections of cubes and related problems , 1989 .

[12]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[13]  A. Caetano.  Weyl numbers in sequence spaces and sections of unit balls , 1992 .

[14]  F. Barthe.  Mesures unimodales et sections des boules Bnp , 1995 .

[15]  Sergey G. Bobkov,et al.  Extremal properties of half-spaces for log-concave distributions , 1996 .

[16]  M. Ledoux,et al.  Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator , 1996 .

[17]  C. Houdré,et al.  Isoperimetric constants for product probability measures , 1997 .

[18]  S. Bobkov.  An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space , 1997 .

[19]  Franck Barthe,et al.  Inégalités de Brascamp-Lieb et convexité , 1997 .

[20]  M. Ledoux.  The geometry of Markov diffusion generators , 1998 .

[21]  F. Barthe.  On a reverse form of the Brascamp-Lieb inequality , 1997, math/9705210.

[22]  James Kuelbs,et al.  Some Shift Inequalities for Gaussian Measures , 1998 .

[23]  Krzysztof Oleszkiewicz,et al.  Gaussian measures of dilatations of convex symmetric sets , 1999 .

[24]  M. Ledoux.  Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[25]  The Size of Singular Component and Shift Inequalities , 1999 .

[26]  S. Bobkov,et al.  From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities , 2000 .

[27]  B. Maurey,et al.  Institute for Mathematical Physics Some Remarks on Isoperimetry of Gaussian Type Some Remarks on Isoperimetry of Gaussian Type , 2022 .

[28]  F. Barthe,et al.  Shift inequalities of Gaussian type and norms of barycentres , 2001 .