A (one-dimensional) free Brunn–Minkowski inequality

Ledoux, Michel

Comptes Rendus. Mathématique, Tome 340 (2005) no. 4, pp. 301-304.

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.

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.

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