International Congress of Mathematicians

On Some Inequalities for Gaussian Aleasures 817The formulation of the functional form of Gaussian isoperimetry was given byS.G. Bobkov [2].Theorem 4.1 For any locally Lipschitz function f : R" —¥ [0,1] and p = -y nwe haveH I fdp) < f VHf) 2 + |V/| 2 d M . (4.2)ii» JR»Theorem 4.1 easily implies the isoperimetric inequality (2.2) by approximatingthe indicator function I A by Lipschitz functions. On the other hand if we apply(2.2) to the set A = {(x,y) £ R" x R : $(y) < f(x)} in R" +1 we get (4.2). It isalso not hard to derive the logarithmic Sobolev inequality (4.1) as a limit case ofBobkov's inequality (cf. [1]): one should use (4.2) for / = eg 2 (with g bounded)and let e tend to 0 (I(t) ~ ty?2log(l/t) as t -t 0+).The crucial point of the inequality (4.2) is its tensorization property. To stateit precisely let us say that a measure p on R" satisfies Bobkov 's inequality if theinequality (4.2) holds for all locally Lipschitz functions / : R" —¥ [0,1]. Easyargumentshows that if pi are measures on R m , i = 1,2, that satisfy Bobkov'sinequality then the measure pi ® p 2 also satisfies Bobkov's inequality.The inequality (4.2) was proved by Bobkov in an elementary way, based onthe following "two-point" inequality:i( a -^) < y nap+( a -^) 2 +y m 2 +( V)2 (43)valid for all a,b £ [0,1]. In fact the inequality (4.3) is equivalent to Bobkov'sinequality for p = |óAi + ^öi and the discrete gradient instead of Vf. Using thetensorization property and the central limit theorem Bobkov deduces (in the similarway as Gross in his proof of (4.1)) (4.2) from (4.3).Using the co-area formula and Theorem 4.1 F. Barthe and Al. Alaurey [1]gave interesting characterization of all absolutely continuous measures that satisfyBobkov's inequality.Theorem 4.2 Let c > 0 and p be a Borei probability measure on the Riemannianmanifold M, absolutely continuous with respect to the Riemannian volume.Then the following properties are equivalent(i) For every measurable A c M, pfi(A) > cI(p(Aj);(ii) For every locally Lipschitz function f : M —¥ [0,1]H I fdp) < f J 1(f) 2 + \\Vf\ 2 dp.JM JM V c-Theorem 4.2 together with the tensorization property shows that if ls(pi) > ci,i = 1,2..., then also Is(^i ® ... ® p n ) > ci. In general it is not known how toestimate Is(^i ®...® p n ) in terms of ls(pi) even in the case when all pi's are equal(another important special case of this problem was solved in [3]) .5. S-inequality