PREPRINT 2006:7 Inequalities of the Brunn-Minkowski Type for Gaussian Measures CHRISTER BORELL Departent of Matheatical Sciences Division of Matheatics CHALMERS UNIVERSITY OF TECHNOLOGY GÖTEBORG UNIVERSITY Göteborg Sweden 2006
Preprint 2006:7 Inequalities of the Brunn-Minkowski Type for Gaussian Measures Christer Borell Departent of Matheatical Sciences Division of Matheatics Chalers University of Technology and Göteborg University SE-42 96 Göteborg, Sweden Göteborg, August 2006
Preprint 2006:7 ISSN 652-975 Mateatiska vetenskaper Göteborg 2006
INEQUALITIES OF THE BRUNN-MINKOWSKI TYPE FOR GAUSSIAN MEASURES Christer Borell School of Matheatical Sciences, Chalers University of Technology and Göteborg University, S-42 96 Göteborg, Sweden e-ail: borell@ath.chalers.se Key words: Gauss easure, linear cobination of sets, easurable set, convex set AMS Subject Classi cation 2000: 60B05, 60D05, 60E5, 60G5 Abstract Let be the standard Gauss easure on R n, let t = R t exp s2 =2ds= p 2; t ; and let 2 be an integer. Given positive real nubers ; :::; this paper gives a necessary and su cient condition such that the inequality A + ::: + A A + ::: + A is true for all Borel sets A ; :::; A in R n of positive -easure or all convex Borel sets A ; :::; A in R n of positive -easure; respectively. In particular, the paper exhibits inequalities of the Brunn-Minkowski type for which are true for all convex sets but not for all easurable sets. Introduction The ain purpose of this paper is to study inequalities of the Brunn-Minkowski type for Gaussian easures when the nuber of sets involved is ore than two. Besides, we will show that an inequality of the Brunn-Minkowski type for Gaussian easures, valid for all convex sets, in general, will not be true for all easurable sets. First, however, we will introduce ore precise de nitions.
2 Let be the standard Gaussian easure on R n ; that is dx = e jxj2 =2 dx p n 2 where and let v ux j x j= t n x 2 k if x = x ; :::; x n 2 R n t = Z t e s2 =2 p ds ; t : 2 Moreover, if A ; :::; A are subsets of R n and ; :::; 2 R ; the linear cobination A + ::: + A of the sets A ; :::; A equals fy; y = x + ::: + x where x i 2 A i ; i = ; :::; g : Recall fro the theory of analytic sets that a linear cobination of Borel subsets of R n is universally Borel easurable, that is -easurable with respect to every nite positive Borel easure on R n : Below BR n stands for the Borel eld of R n and R + eans the open interval ]0; [ : Now suppose 2 is a xed integer and denote by S the set of all = ; :::; 2 R + such that A + ::: + A A + ::: + A. where A ; :::; A 2 BR n : Here, if t ; :::; t 2 [ ; ] ; the su t i is de ned to be equal to if soe of the t i = and in other cases has its usual eaning. The ain concern in this paper is to nd an explicit for of the set S : In the special case = 2 the above proble was recently solved by the author who proved that S 2 = 2 R 2 +; + 2 and j 2 j.2 see [3]. By applying this result we will show that S = 2 R +; i ax; + 2 ax i :.3 i
Given ; :::; 2 R +, clearly equality occurs in. if A ; :::; A are parallel a ne half-spaces and if, in addition, +:::+ = equality occurs in. if A ; :::; A are equal and convex. We will also consider the inequality. restricted to convex Borel sets. To be ore precise let be as above and denote by C the set of all = ; :::; 2 R + such that the inequality. holds for arbitrary convex sets A ; :::; A 2 BR n : Clearly, S C : Moreover, it will be proved that C = 2 R +; i :.4 In particular, fro.3 and.4, S 6= C and consequently S is a proper subset of C : Fro this we conclude that an inequality of the Brunn- Minkowski type for valid for all convex sets, in general, will not extend to all easurable sets. As far as we know a siilar phenoenon has never been reported on before. Note that the relations.3 and.4 show that the sets S and C are independent of n. In Section 4 we will point out that all the results in Sections 2 and 3 extend to centred Gaussian easures on real, separable Fréchet spaces. Thus, in particular, the results apply to Wiener easure, the probability law of Brownian otion. Finally, in this section let us repeat ore on the history of the probles we face here. Ehrhard [4] proved in 983 that 2 R 2 +; + 2 = C 2 and in 996 the Ehrhard result was generalized by Lata a [5] who established. when = 2 and one of the Borel sets A and A 2 is convex. In particular, this result has the isoperietric inequality in Gauss space, independently due to Sudakov and Tsirelson [7] and the author [] as an iediate corollary. Moreover, in 2003 the author [2] proved that 2 R 2 +; + 2 = S 2 and, as already entioned above, the description of the class S 2 given by.2 goes back to y forth-coing paper [3] : 3 2 Characterization of S and C Theore 2. C = 2 R 2 +; i :
4 The proof of Theore 2. is based on the following Lea 2. 2 R +; i = S : PROOF It follows fro.2 that Lea 2. holds for = 2: If 3 and 2 R + satis es + ::: + = ; then where = + ::: + A + ::: + A = A + ::: + A + A : Now ; 2 S 2 and we get A + ::: + A A + ::: + A + A and the proof of Lea 2. can be copleted by induction on : PROOF OF THEOREM 2. We rst prove that 2 R 2 +; i C : To this end let 2 R + be such that = def + ::: +. Futherore, suppose A ; :::; A 2 BR n are convex and de ne so that C = A + ::: + A A + ::: + A = C:
Now since C is convex an inequality by Sudakov and Tsirelson [7] states that C C note that this inequality follows fro the relations =2C + =2C = C and =2; =2 2 S 2 ; see [3]. Hence A + ::: + A A + ::: + A and by Lea 2. the eber in the right-hand side does not fall below A + ::: + A : 5 Accordingly fro this 2 C : Next we clai that C 2 R +; i : To see this, let C 2 BR n be a convex syetric set such that 0 < C < 2 : Then, if 2 C, and we get C + ::: + C = + ::: + C + :::: + C C + ::: + C: Here, if + ::: + < it follows that or C + ::: + C 0 + ::: + C which is a contradiction. This proves Theore 2.. Theore 2.2 S = 2 R +; i ax; i= + 2 ax i i :
6 The proof of Theore 2.2 is based on two leas. Lea 2.2 For any I f; :::; g the set P I = 2 R +; i and j X I is contained in S : i X I c i j PROOF If I = or f; :::; g ; Lea 2. shows that P I S : Therefore assue I is a non-epty proper subset of f; :::; g and set 0 = I i and = I c i. Now since 0 ; > 0; A + ::: + A X = i X i 0 A i + A i 0 and the description of S 2 given in.2 iplies that I I c A + ::: + A 0 X I i 0 A i + X I c i A i : Here by Lea 2. the last expression does not fall below X i X 0 i A i + A i 0 which proves Lea 2.2. I = A + ::: + A I c Lea 2.3 2 R +; = t; t; :::; t and t S :
7 PROOF Let d denote the integer part of =2: If = t the -tuple t; :::; t 2 P f;:::;dg and t; :::; t S by Lea 2.2. In addition, if t ; then ta + ::: + ta = ta + ::: + A + ta t A + ::: + A + t A due to.2 and by induction on it follows that t; :::; t 2 S : This proves Lea 2.3. PROOF OF THEOREM 2.2 The relation.2 shows that Theore 2.2 is true if = 2 and there is no loss of generality in assuing that 3: We rst prove that S 2 R +; i ax; i= To this end let 2 S and rst assue ::: > : Then, if C is as in the proof of Theore 2., and we get or + 2 ax i i R n n C C + ::: + C + R n n C R n n C + ::: + C + R n n C C + ::: + C C since y = y for all 0 < y < : Thus 0 > + ::: + + C which is a contradiction and we conclude that ::: : In a siilar way it follows that X i if k = ; :::; k i2f;:::;k ;k+;:::;g :
8 and we get i i= + 2 ax i i: Finally since S C Theore 2. iplies that i ax; i= + 2 ax i i : 2. We next show that 2 R +; i ax; i= + 2 ax i i S : Therefore suppose 2 R + satis es 2. so that, in particular, Now, if k i2f;:::;k i2f;:::;k X ;k+;:::;g X ;k+;:::;g i if k = ; :::; : i k for soe k 2 f; :::; g Lea 2.2 proves that 2 S. On the other hand if X i k > i2f;:::;k ;k+;:::;g for every k 2 f; :::; g we proceed as follows. Since i i= 2 S if and only if i i= 2 S for a suitable perutation : f; :::; g! f; :::; g it can be assued that 2 ::: and, hence, and i > + i > : 2.2
9 Next we clai there exists a subset I of f; :::; j X X i i j : I f;:::; gni g such that In fact, if is odd and we get 0 2 + 4 3 + ::: + 2 j 2 + 4 + ::: + 3 ::: 2 j : On the other hand if is even and we get 0 2 4 3 ::: 2 3 j + 3 + ::: + 2 4 ::: 2 j : Consequently, for each positive integer 3 there exists a subset I of f; :::; g such that j X i X i j : 2.3 I f;:::; Now let A ; :::; A 2 BR n : Using the inequality 2. we get = and Lea 2.3 yields gni A + ::: A = A + ::: + A + A A + ::: + A + A : We now use 2.2, 2.3, and Lea 2.2 to obtain A + ::: + A A + ::: + A
0 and the inequality. follows at once. This copletes the proof of Theore 2.2. As a consequence of Lea 2.2 let us point out the following Corollary. 2 R +; i and 2 i S : PROOF Suppose 2 R +; + ::: + ; and 2 + ::: + 2. If [ =2 If;:::;g that is j " i i j> a.s. where " i i= denotes a sequence of independent rando variables such that P I P [" i = ] = P [" i = ] ; i = ; :::; the Jensen inequality forces v # ux t 2 i "j E " i i j > : [ Fro this contradiction we conclude that 2 P I and Lea 2.2 iplies that 2 S ; which proves Corollary.. If;:::;g 3 Other descriptions of the classes S and C
In this section FS denotes the set of all = ; :::; 2 R + such that the inequality Z Z Z f 0 d f d + :::: + f d 3. R n R n R n is valid for all Borel functions f k : R n! [0; ] ; k = 0; ; :::; ; satisfying the inequality f 0 x + ::: + x f x + ::: + f x 3.2 for all x ; :::; x 2 R n : Theore 3. S = FS : PROOF We rst prove that S FS Therefore let 2 S ; let f k : R n! [0; ] ; k = 0; ; :::; be Borel functions such that 3.2 holds for all x ; :::; x 2 R n ; and as in the Lata a paper [6] de ne Then B k = x; t 2 R n R; t f k x ; k = 0; ; :::; : B 0 B + ::: + B and as Z B k = f k d; k = 0; ; :::; R n it follows that 2 FS : Here note that the class S is independent of n: We next clai that FS S : To see this let 2 FS ; let A ; :::; A E be copact and de ne A 0 = A + ::: + A and f k = Ak ; k = 0; ; :::; : Then 3.2 is true and as 2 FS ; 3. is true which iplies.. Since any nite positive easure
2 on R n is inner regular with respect to copact sets it is obvious that 2 S : Suing up we have proved Theore 3.. Next let FC denote the set of all = ; :::; 2 R + such that the inequality 3. is valid for all Borel functions f k : R n! [0; ] ; k = 0; ; :::; ; satisfying 3.2 and such that the functions f k ; k = 0; ; :::; ; are concave. Here we say that a function g : R n! [ ; ] is concave if the set fx; t 2 R n R; t gxg is convex. The proof of the following theore is very siilar to the proof of Theore 3. and will not be included here. Theore 3. C = FC : 4 Extension to in nite diension A Borel probability easure on a real, locally convex Hausdor vector space E is said to be a centred Gaussian easure if the iage easure is a centred Gaussian easure on the real line for each bounded linear functional in E: For siplicity, we here restrict ourselves to a centred, non-degenerate Gaussian easure on a real, separable Fréchet space F. To say that is non-degenerate eans that is not the Dirac easure at the origin. Recall that a nite positive Borel easure on F is inner regular with respect to copact sets and oreover, if A is a convex Borel subset of F there are copact sets K n ; n 2 N; such that K K 2 :::; [ 0 K n is convex, and An[ 0 K n a -null set. Note also that the theory of analytic sets iplies that a linear cobination of Borel subsets of F is universally Borel easurable. We de ne the classes S ; C ; FS ; and FC as above with R n replaced by F and where now stands for a centred, non-degenerate Gaussian easure on F: The paper [3] shows that.2 still holds and it is obvious that the results in Sections 2 and 3 extend to this ore general situation. Note here that the Sudakov and Tsirelson inequality, which is iportant in the proof of Theore 2., extends to convex universally Borel easurable sets on F see [3]. The details are oitted here.
3 References [] Ch. Borell. The Brunn-Minkowski inequality in Gauss space. Inventiones ath. 30 975, 207-26 [2] Ch. Borell. The Ehrhard inequality, C. R. Acad. Sci. Paris, Ser. I 337 2003, 663-666 [3] Ch. Borell. Minkowski sus and Brownian exit tie. Ann. Fac. Sci. Toulouse to appear [4] A. Ehrhard. Syétrisation dans l espace de Gauss. Math. Scand. 53 983, 28-30 [5] R. A. Lata a. A note on Ehrhard s inequality. Studia Matheatica 8 996, 69-74. [6] R. A. Lata a. On soe inequalities for Gaussian easures. Proceedings of the ICM 2 2002, 83-822 [7] V. N. Sudakov and B. S. Tsirelson. Extreal properties of half-spaces for spherically-invariant easures. Zap. Nauch. Se L.O.M.I. 4 974 4-24 translated in J. Soviet Math. 9 978 9-8