Comptes Rendus
Probability Theory/Mathematical Physics
A (one-dimensional) free Brunn–Minkowski inequality
Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 301-304.

We present a one-dimensional version of the functional form of the geometric Brunn–Minkowski inequality in free (non-commutative) probability theory. The proof relies on matrix approximation as used recently by Biane and Hiai et al. to establish free analogues of the logarithmic Sobolev and transportation cost inequalities for strictly convex potentials, that are recovered here from the Brunn–Minkowski inequality as in the classical case. The method is used to extend to the free setting the Otto–Villani theorem stating that the logarithmic Sobolev inequality implies the transportation cost inequality.

Nous présentons une version uni-dimensionnelle de la forme fonctionnelle de l'inégalité géométrique de Brunn–Minkowski en théorie des probabilités libres. L'argument s'appuie sur l'approximation matricielle déjà mise en œuvre récemment par Biane et Hiai et al. pour établir les analogues libres des inégalités de Sobolev logarithmique et de coût du transport pour des potentiels strictement convexes, qui sont ici déduits de l'inégalité de Brunn–Minkowski comme dans le cas classique. La méthode permet, de la même façon, d'étendre au cadre libre le théorème d'Otto–Villani assurant que l'inégalité de Sobolev logarithmique entraîne l'inégalité de transport.

Published online:
DOI: 10.1016/j.crma.2004.12.017

Michel Ledoux 1

1 Institut de mathématiques, université Paul-Sabatier, 31062 Toulouse, France
     author = {Michel Ledoux},
     title = {A (one-dimensional) free {Brunn{\textendash}Minkowski} inequality},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {301--304},
     publisher = {Elsevier},
     volume = {340},
     number = {4},
     year = {2005},
     doi = {10.1016/j.crma.2004.12.017},
     language = {en},
AU  - Michel Ledoux
TI  - A (one-dimensional) free Brunn–Minkowski inequality
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 301
EP  - 304
VL  - 340
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2004.12.017
LA  - en
ID  - CRMATH_2005__340_4_301_0
ER  - 
%0 Journal Article
%A Michel Ledoux
%T A (one-dimensional) free Brunn–Minkowski inequality
%J Comptes Rendus. Mathématique
%D 2005
%P 301-304
%V 340
%N 4
%I Elsevier
%R 10.1016/j.crma.2004.12.017
%G en
%F CRMATH_2005__340_4_301_0
Michel Ledoux. A (one-dimensional) free Brunn–Minkowski inequality. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 301-304. doi : 10.1016/j.crma.2004.12.017.

[1] G. Ben Arous; A. Guionnet Large deviations for Wigner's law and Voiculescu's noncommutative entropy, Probab. Theory Related Fields, Volume 108 (1997), pp. 517-542

[2] P. Biane Logarithmic Sobolev inequalities, matrix models and free entropy, Acta Math. Sinica, Volume 19 (2003), pp. 1-11

[3] P. Biane; R. Speicher Free diffusions, free entropy and free Fisher information, Ann. Inst. H. Poincaré, Volume 37 (2001), pp. 581-606

[4] P. Biane; D. Voiculescu A free probability analogue of the Wasserstein distance on the trace-state space, Geom. Funct. Anal., Volume 11 (2001), pp. 1125-1138

[5] S. Bobkov; I. Gentil; M. Ledoux Hypercontractivity of Hamilton–Jacobi equations, J. Math. Pures Appl., Volume 80 (2001), pp. 669-696

[6] F. Hiai; D. Petz The Semicircle Law, Free Random Variables and Entropy, Math. Surveys and Monographs, vol. 77, American Mathematical Society, 2000

[7] F. Hiai, D. Petz, Y. Ueda, Inequalities related to free entropy derived from random matrix approximation (2003)

[8] F. Hiai; D. Petz; Y. Ueda Free transportation cost inequalities via random matrix approximation, Probab. Theory Related Fields, Volume 130 (2004), pp. 199-221

[9] K. Johansson On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J., Volume 91 (1998), pp. 151-204

[10] M. Ledoux The Concentration of Measure Phenomenon, Math. Surveys and Monographs, vol. 89, American Mathematical Society, 2001

[11] M. Ledoux, Measure concentration, transportation cost, and functional inequalities, Summer School on Singular Phenomena and Scaling in Mathematical Models, Bonn, 2003

[12] F. Otto; C. Villani Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, J. Funct. Anal., Volume 173 (2000), pp. 361-400

[13] E.B. Saff; V. Totik Logarithmic Potentials with External Fields, Grundlehren Math. Wiss., vol. 316, Springer, 1997

[14] S. Szarek; D. Voiculescu Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality, Commun. Math. Phys., Volume 178 (1996), pp. 563-570

[15] C. Villani Topics in Optimal Transportation, Grad. Stud. Math., vol. 58, American Mathematical Society, 2003

[16] D. Voiculescu The analogues of entropy and of Fisher's information measure in free probability theory, I, Commun. Math. Phys., Volume 155 (1993), pp. 71-92

[17] D. Voiculescu The analogues of entropy and of Fisher's information measure in free probability theory, V. Noncommutative Hilbert transforms, Invent. Math., Volume 132 (1998), pp. 189-227

Cited by Sources:

Comments - Policy