On Isoperimetric Functions of Probability Measures Having Log-Concave Densities with Respect to the Standard Normal Law

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1 On Isoerimetric Functions of Probability Measures Having Log-Concave Densities with Resect to the Standard Normal Law Sergey G. Bobkov Abstract Isoerimetric inequalities are discussed for one-dimensional robability distributions having log-concave densities with resect to the standard Gaussian measure. Suose that a robability measure on R n has a log-concave density f with resect to the standard n-dimensional Gaussian measure n, that is, f.x/ D e jxj V.x/ ; x R n ; for some convex function V W R n!. ;. One may also say that is logconcave with resect to n. In this case, an imortant theorem due to D. Bakry and M. Ledoux [B-L] asserts that satisfies a Gaussian-tye isoerimetric inequality C.A/ ' ˆ..A// ; () relating the size".a/ of an arbitrary Borel subset A R n to its -erimeter C.A/ D lim inf "#0.A " /.A/ " (where A " stands for the Euclidean "-neighborhood of A). Here, ˆ denotes the inverse to the normal distribution function ˆ.x/ D.. ; x / with density '.x/ D e x (x R). In other words, the isoerimetric function of, I./ D inf.a/d C.A/; 0 < < (called also an isoerimetric rofile) dominates the isoerimetric function I./ D '.ˆ.// of the measure n, i.e., one has I./ I./ () S.G. Bobkov ( ) University of Minnesota, Minneaolis, MN 55455, USA bobkov@math.umn.edu Sringer Science+Business Media LLC 07 E. Carlen et al. (eds.), Convexity and Concentration, The IMA Volumes in Mathematics and its Alications 6, DOI 0.007/ _9 577

2 578 S.G. Bobkov for all. The original roof of () () givenin[b-l] is based on semi-grou arguments and a functional form roosed in [B]. As was shown by L. A. Caffarelli [C], all s under consideration reresent contractions of n, so the roof of () () may be reduced to the urely Gaussian case. An alternative localization aroach to the Bakry-Ledoux theorem was later roosed in [B3]; cf. also [B4] for an extension of () to a larger class of robability measures. Another aroach unifying a number of analytic and isoerimetric inequalities of Gaussian tye has been recently develoed by P. Ivanisvili and A. Volberg [I-V]. Recently, Rahaël Bouyrie raised the question of whether or not the inequality () is strict, even in dimension one, assuming that is symmetric and non-gaussian. Although we do not know the original motivation, this question seems to be rather interesting in itself and not so elementary. Here we give an affirmative answer, involving some arguments from [B3] which were used to rove () () in dimension one. Thus, we have: Theorem 3 Let be a symmetric robability measure on R which is log-concave with resect to the standard Gaussian measure.ifis not Gaussian, then its isoerimetric function satisfies I./ >I./ for all.0; /: Equivalently, the coincidence I. 0 / D I. 0 / for some 0 causes to be Gaussian. Of course, this is not true at all without the log-concavity hyothesis (with resect to ). For examle, consider the class of symmetric robability measures on R having log-concave densities f with resect to the linear Lebesgue measure (the class of log-concave measures). In this case, the isoerimetric functions have the form where F is the inverse to the distribution function J./ D I./ D f.f.//; (3) F.x/ D.. ; x / D Z x f.y/ dy restricted to the suort interval (cf. [B]). Here, J may be an arbitrary ositive concave function on.0; /, symmetric about the oint =. Hence, in this class it may easily haen that J./ I./ on.0; / with equality only at two oints 0 and 0 (or even for one oint 0 D =, only). Let us also mention that the roerty J I is another way to say that reresents a Lischitz transform of. Assuming that V is of class C in the reresentation (), we find from (3) that V 00.x/ D f 0.x/ 0 D.J 0.F.x// 0 f.x/ D J 00.F.x//f.x/ D J 00./J./; D F.x/:

3 On Isoerimetric Functions of Probability Measures Having Log-Concave Hence, in terms of the isoerimetric function, the log-concavity with resect to is equivalent to the relation J 00./J./ : For such functions (that are also symmetric about =), Theorem 3 may be stated as follows: If J 00./J./ D for some.0; /, then this equality holds true for all (in which case, necessarily J D I). It might be natural to try to rove Theorem 3 using this formulation. However, we refer to choose a different route, which allows one to avoid the C -assumtion on the density f, and which also suggests a ossible way to quantify the assertion of this theorem. To be more recise, we have: Theorem 4 Let be a robability measure suorted on the interval. a; a/ R with density e V.x/ '.x/, where V is an even, convex function, which is differentiable and increasing on.0; a/. Then the isoerimetric function of satisfies I./ ˆ.V 0.x// e V0.x/ V 0.x/ y '.y/; (4) where D.. ; x /, x. a;0/, and y D V 0.x/ C ˆ ˆ.V 0.x// : A similar bound also holds for >=,byusingi. / D I./. One can check that equality in (4) is attained for the family of robability measures D with densities '.x/ D Z e jxj '.x/; x R; (5) where is an arbitrary ositive arameter and Z D Z./ is a normalizing constant. We now turn to the roofs. As a first ste, let us verify Theorem 3 in the articular case of measures described in (5). Lemma 3 Given >0, we have I./ >I./ for all.0; /. Proof According to (3), the isoerimetric function of is given by I./ D ' ˆ ;./ where ˆ denotes the distribution function of. Therefore, we need to show that ˆ.y/ D ˆ.x/ ) '.y/ > '.x/ for all x; y R, where one may additionally assume that x 0 (using the symmetry). The increasing ma T.x/ D ˆ.ˆ.x// ushes forward to, so that ˆ.T.x// D ˆ.x/. After differentiation we have '.T.x//T 0.x/ D '.x/:

4 580 S.G. Bobkov Hence, it is sufficient to see that T 0.x/ <for all x <0. To this aim, first note that Z x Z 0 e y '.y/ dy D e = ˆ.x /; Z D e y '.y/ dy D e = ˆ. /: Hence, the distribution function of is described as ˆ.x/ D.. ; x / D ˆ.x / ˆ. / ; x 0; and, by the symmetry, ˆ.x/ D ˆ. x/ for x 0. It follows that T.x/ D ˆ. ˆ.x// C. D ˆ. /; x 0/; so, utting x D ˆ./, we get T 0.x/ D '.x/ I. ˆ.x// D I./ I. / : But the last ratio is smaller than, since <and since I./= is a decreasing function. The latter roerty is true for any ositive, strictly concave function I on.0; /, which follows from the reresentation I./ D I.0C/ C Z 0 I 0.s/ ds: (6) This roves the lemma. Lemma 4 Let be a symmetric robability measure, which is log-concave with resect to with density f D e V '. Suose that V is monotone in some neighborhood of a oint x R, and let D.. ; x /. Then I./ I./ for some >0: Proof We rove a stronger statement: Let a ositive finite measure have density f.y/ D e V.y/ '.y/ for some convex even function V W R!. ;, finite on the interval. a; a/. If a oint x. a;0/is such that.. a; x / ;..x;0 / 0< < ; (7) and if V is monotone in some neighborhood of x, then f.x/ I./ for some >0. To simlify this assertion, let l.y/ D c y be an affine function which is tangent to V.y/ at x, with necessarily >0in view of the monotonicity assumtion on V. We extend l from the negative half-axis. ;0/to.0; / to get an even function, and as a result we obtain a new ositive measure 0 with density

5 On Isoerimetric Functions of Probability Measures Having Log-Concave f 0.y/ D Ce jyj '.y/: Since l.x/ D V.x/ and l V everywhere on. a; a/, wehavef f 0, so that 0.. a; x / and 0..x;0 /. Therefore, in our stronger statement we are reduced to the class of densities of tye f D C', where C is an arbitrary ositive arameter. For such densities, we have.. ; x / D Cˆ.x/;..x;0 /D C ˆ.x/ ; f.x/ D C'.x/; and involving the assumtion (7), we get a constraint on C, namely, C C 0 D max ˆ.x/ ; ˆ.x/ : (8) Since C D C 0 is the worst situation in our conclusion, it remains to show that C 0 '.x/ '.ˆ.// I./ with C 0 defined in (8). Putting q D ˆ.x/, this is the same as max q ; q I.q/ I./: If q, it holds true, since q I.q/ I./, which in turn follows from the fact that the function I is strictly concave (so that I./= is strictly decreasing). In case q, weuse q I.q/ I./; or equivalently, after the change 0 D, q0 D q, 0 q I 0 q0 I 0 : Here again 0 q 0 and we deal with the concave function QI. 0 / D I. 0 / on the interval.0; =/. Hence, QI. 0 /= 0 is strictly decreasing, which is seen from the general identity (6). Lemma 4 is roved. Proof of Theorem 3 If a robability measure on the line is log-concave with resect to, it has a density f.x/ D e V.x/ '.x/;

6 58 S.G. Bobkov for some convex even function V on the interval. a; a/, finite or not, and one may ut V Doutside that interval. Since V attains its minimum at zero, necessarily V.0/ < 0, as long as is non-gaussian. In articular, in this case I.=/ D f.0/ > '.0/ D I.=/: Moreover, let Œ x 0 ; x 0 be the longest interval, where V is constant, so that V.x 0 / D V.0/. Then similarly I./ D I.=/ > I.=/ for all Œ 0 ; 0, where 0 D.. a; x 0 /. In case 0 < < 0, D.. a; x /, the oint x necessarily belongs to the interval. a; x 0 /, where V is strictly decreasing. Therefore, one may aly Lemma 4 and combine it with Lemma 3, which then leads to the required assertion I./ I./ >I./. Theorem 3 is thus roved. Proof of Theorem 4 If an even, convex function V in the reresentation f D e V ' for the density of is differentiable and is increasing on.0; a/, the assumtion of Lemma 4 is fulfilled for all oints x 0 from the suorting interval of the measure. In this case, since the tangent affine function in the roof of Lemma 4 is given by l.y/ D V.x/CV 0.x/.y x/; necessarily D.x/ D V 0.x/. a < x <0/. Hence, we obtain that I./ I.x/./; D.. a; x /: (9) The exression I.x/./ may be written in a more exlicit form. Recall that, for 0< <=, where D ˆ. /, so that y ˆ./ D ˆ. / C ; Z D e = ˆ. /; I./ D Z e jyj '.y/ D Hence, (9) turns into (4), thus roving Theorem 4. ˆ. / e =Cy '.y/: Acknowledgements This research was artially suorted by the Alexander von Humboldt Foundation and NSF grant DMS-696.

7 On Isoerimetric Functions of Probability Measures Having Log-Concave References [B-L] D. Bakry, M. Ledoux. Lévy-Gromov s isoerimetric inequality for an infinite-dimensional diffusion generator. Invent. Math. 3 (996), no., [B] S. G. Bobkov. Extremal roerties of half-saces for log-concave distributions. Ann. Probab. 4 (996), no., [B] S. G. Bobkov. An isoerimetric inequality on the discrete cube, and an elementary roof of the isoerimetric inequality in Gauss sace. Ann. Probab. 5 (997), no., [B3] S. G. Bobkov. A localized roof of the isoerimetric Bakry-Ledoux inequality and some alications [in Russian]. Teor. Veroyatnost. Primenen. 47 (00), no., ; English translation: Theory Probab. Al. 47 (003), no., [B4] S. G. Bobkov. Perturbations in the Gaussian isoerimetric inequality. J. Math. Sciences (New York), vol. 66 (00), no. 3, Translated from: Problems in Math. Analysis, 45 (00), 3 4. [C] L. A. Caffarelli. Monotonicity roerties of otimal transortation and the FKG and related inequalities. Commun. Math. Phys. 4 (000), [I-V] P. Ivanisvili, A. Volberg. Isoerimetric functional inequalities via the maximum rincile: the exterior differential systems aroach. Prerint (05), arxiv: