BORELL S GENERALIZED PRÉKOPA-LEINDLER INEQUALITY: A SIMPLE PROOF. Arnaud Marsiglietti. IMA Preprint Series #2461. (December 2015)

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1 BORELL S GENERALIZED PRÉKOPA-LEINDLER INEQUALITY: A SIMPLE PROOF By Arnaud Marsiglietti IMA Preprint Series #2461 December 215 INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY OF MINNESOTA 4 Lind Hall 27 Church Street S.E. Minneapolis, Minnesota Phone: Fax: URL:

2 Borell s generalized Préopa-Leindler inequality: A simple proof Arnaud Marsiglietti Abstract We present a simple proof of Christer Borell s general inequality in the Brunn-Minowsi theory. We then discuss applications of Borell s inequality to the log-brunn-minowsi inequality of Böröczy, Lutwa, Yang and Zhang. 21 Mathematics Subject Classification. Primary 28A75, 52A4. Keywords. Brunn-Minowsi, Convex body, log-brunn-minowsi inequality, mass transportation. 1 Introduction Let us denote by suppf the support of a function f. In [6] Christer Borell proved the following inequality see [6, Theorem 2.1], which we will call the Borell-Brunn-Minowsi inequality. Theorem 1 Borell-Brunn-Minowsi inequality. Let f, g, h : R n [, + be measurable functions. Let ϕ = ϕ 1,..., ϕ n : suppf suppg R n be a continuously differentiable function with positive partial derivatives, such that ϕ x, y = ϕ x, y for every x = x 1,..., x n suppf, y = y 1,..., y n suppg. Let Φ : [, + [, + [, + be a continuous function, homogeneous of degree 1 and increasing in each variable. If the inequality hϕx, yπ n ϕ =1 ρ + ϕ η ΦfxΠ n x y =1ρ, gyπ n =1η 1 holds for every x suppf, for every y suppg, for every ρ 1,..., ρ n > and for every η 1,..., η n >, then h Φ f, g. C. Borell proved a slightly more general statement, involving an arbitrary number of functions. For simplicity, we restrict ourselves to the statement of Theorem 1. Theorem 1 yields several important consequences. For example, applying Theorem 1 to indicators of compact sets i.e. f = 1 A, g = 1 B, h = 1 ϕa,b yields the following generalized Brunn-Minowsi inequality. Corollary 2 Generalized Brunn-Minowsi inequality. Let A, B be compact subsets of R n. Let ϕ = ϕ 1,..., ϕ n : A B R n be a continuously differentiable function with positive partial derivatives, such that ϕ x, y = ϕ x, y for every x = x 1,..., x n A, y = y 1,..., y n B. Let Φ : [, + [, + [, + be a continuous function, homogeneous of degree 1 and increasing in each variable. If the inequality Π n ϕ =1 ρ + ϕ η ΦΠ n x y =1ρ, Π n =1η Supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. 1

3 holds for every ρ 1,..., ρ n, η 1,..., η n >, then ϕa, B Φ A, B, where denotes Lebesgue measure and ϕa, B = {ϕx, y : x A, y B}. The classical Brunn-Minowsi inequality see e.g. [23], [13] follows from Corollary 2 by taing ϕx, y = x + y, x A, y B, and Φa, b = a 1/n + b 1/n n, a, b. Although the Brunn-Minowsi inequality goes bac to more than a century ago, it still attracts a lot of attention see e.g. [2], [11], [14], [18], [9], [1], [12], [15], [17]. Theorem 1 also allows us to recover the so-called Borell-Brascamp-Lieb inequality. Let us denote by M λ s a, b the s-mean of the real numbers a, b with weight λ [, 1], defined as M λ s a, b = 1 λa s + λb s 1 s if s / {,, + }, M λ a, b = mina, b, M λ a, b = a 1 λ b λ, M λ + a, b = maxa, b. We will need the following Hölder inequality see e.g. [16]. Lemma 3 Generalized Hölder inequality. Let α, β, γ R {+ } such that β + γ and = 1. Then, for every a, b, c, d and λ [, 1], β γ α M λ αac, bd M λ β a, bm λ γ c, d. Corollary 4 Borell-Brascamp-Lieb inequality. Let γ 1 n, λ [, 1] and f, g, h : Rn [, + be measurable functions. If the inequality holds for every x suppf, y suppg, then h1 λx + λy M λ γ fx, gy R n h M λ γ 1+γn f, R n Corollary 4 follows from Theorem 1 by taing ϕx, y = 1 λx + λy, x suppf, y suppg, and Φa, b = M λ γ a, b, a, b. Indeed, using Lemma 3, one obtains that for 1+γn every x suppf, y suppg, and for every ρ 1,..., ρ n, η 1,..., η n >, ϕ hϕx, yπ n =1 ρ + ϕ η = h1 λx + λyπ n x y =11 λρ + λη R n g. Mγ λ fx, gym λ 1 Π n =1ρ, Π n =1η n fxπ n =1ρ, gyπ n =1η M λ γ 1+γn = ΦfxΠ n =1ρ, gyπ n =1η. Corollary 4 was independently proved by Borell see [6, Theorem 3.1], and by Brascamp and Lieb [8]. Another important consequence of the Borell-Brunn-Minowsi inequality is obtained when considering ϕ to be nonlinear. Let us denote for p = p 1,..., p n [, + ] n, x = x 1,..., x n [, + ] n and y = y 1,..., y n [, + ] n, M λ px, y = M λ p 1 x 1, y 1,..., M λ p n x n, y n. Corollary 5 nonlinear extension of the Brunn-Minowsi inequality. Let p = p 1,..., p n [, 1] n, γ n i=1 p 1 i 1, λ [, 1], and f, g, h : [, + n [, + be measurable functions. If the inequality hmpx, λ y Mγ λ fx, gy holds for every x suppf, y suppg, then [,+ n h M λ n i=1 p 1 i +γ 1 1 [,+ n f, [,+ n g. 2

4 Corollary 5 follows from Theorem 1 by taing ϕx, y = Mpx, λ y, x suppf, y suppg, and Φa, b = M λ n a, b, a, b. Indeed, using Lemma 3, one obtains that for i=1 p 1 i +γ 1 1 every x suppf, y suppg, and for every ρ 1,..., ρ n, η 1,..., η n >, ϕ hϕx, yπ n =1 ρ + ϕ η = hm x y px, λ yπ n =1M λ p x 1 p, y 1 p M 1 x p 1 ρ, y p 1 η 1 p M λ γ fx, gyπ n =1M λ p ρ, η Mγ λ fx, gym λ n Π n i=1 p 1 i =1ρ 1, Π n =1η M λ n fxπ n i=1 p 1 i +γ 1 =1ρ 1, gyπ n =1η = ΦfxΠ n =1ρ, gyπ n =1η. In the particular case where p =,...,, Corollary 5 was rediscovered by Ball [1]. In the general case, Corollary 5 was rediscovered by Uhrin [24]. Notice that the condition on p in Corollary 5 is less restrictive in dimension 1. It reads as follows: Corollary 6 nonlinear extension of the Brunn-Minowsi inequality on the line. Let p 1, γ p, and λ [, 1]. Let f, g, h : [, + [, + be measurable functions such that for every x suppf, y suppg, Then, + hm λ p x, y M λ γ fx, gy. + h M λ f, 1 p + 1 γ 1 + g. A simple proof of Corollary 6 was recently given by Bobov et al. [4]. In section 2, we present a simple proof of Theorem 1, based on mass transportation. In section 3, we discuss applications of the above inequalities to the log-brunn-minowsi inequality of Böröczy, Lutwa, Yang and Zhang. We also prove an equivalence between the log-brunn- Minowsi inequality and its possible extensions to convex measures see section 3 for precise definitions. 2 A simple proof of the Borell-Brunn-Minowsi inequality In this section, we present a simple proof of Theorem 1. Proof of Theorem 1. The proof is done by induction on the dimension. To prove the theorem in dimension 1, we use a mass transportation argument. Step 1 : In dimension 1 First let us see that if f = or g =, then the result holds. Let us assume, without loss of generality, that g =. By taing ρ = 1, by letting η go to and by using continuity and homogeneity of Φ in the condition 1, one obtains hϕx, y ϕ x It follows that, for fixed y suppg, hzdz hzdz = ϕsuppf,y suppf Φfx, = fxφ1,. hϕx, y ϕ x dx 3 fφ1, = Φ f, g.

5 A similar argument shows that the result holds if f = + or g = +. Thus we assume thereafter that < f < + and < g < +. Let us show that one may assume that f = g = 1. Let us define, for x, y R and a, b, fx = f Φ f, x Φ1,, hx = h Φ ϕx, y = ϕφ f, x, Φ, gy Φ f, g gx = g Φ, f, g x, g x Φ, 1, f g, Φa, b = Φ a Φ f, g, b Φ f, g Let x supp f, y supp g, and let ρ, η >. One has, ϕ ϕ h ϕx, y ρ + x y η Φ fφ f, x Φ f, Φ f, ρ, gφ, g = Φ fx ρ, gy η.. gy Φ, g Φ f, g η Notice that the functions ϕ and Φ satisfy the same assumptions as the functions ϕ and Φ respectively, and that f = g = 1. If the result holds for functions of integral one, then hwdw Φ1, 1 = 1. The change of variable w = z/φ f, g leads us to hzdz Φ f, g. Assume now that f = g = 1. By standard approximation, one may assume that f and g are compactly supported positive Lipschitz functions relying on the fact that Φ is continuous and increasing in each coordinate, compare with [2, page 343]. Thus there exists a non-decreasing map T : suppf suppg such that for every x suppf, fx = gt xt x, see e.g. [3], [25]. Since T is non-decreasing and ϕ/ x, ϕ/ y >, the function Θ : suppf ϕsuppf, T suppf defined by Θx = ϕx, T x is bijective. Hence the change of variable z = Θx is admissible and one has, ϕ hzdz hϕx, T x suppf x + ϕ y T x dx Φfx, gt xt xdx suppf = Φfx, fxdx. Using homogeneity of Φ, one deduces that h Φ1, 1 fxdx = Φ f, g. Step 2 : Tensorization Let n be a positive integer and assume that Theorem 1 holds in R n. Let f, g, h, ϕ, Φ satisfying the assumptions of Theorem 1 in R n+1. Recall that the inequality hϕx, yπ n+1 =1 ϕ x ρ + ϕ y η 4 ΦfxΠ n+1 =1 ρ, gyπ n+1 =1 η, 2

6 holds for every x suppf, y suppg, and for every ρ 1,..., ρ n+1, η 1,..., η n+1 >. Let us define, for x n+1, y n+1, z n+1 R, F x n+1 = fx, x n+1 dx, R n Gy n+1 = gx, g n+1 dx, R n Hz n+1 = hx, z n+1 dx. R n Since f >, g >, the support of F and the support of G are nonempty. Let x n+1 suppf, y n+1 suppg, and let ρ n+1, η n+1 >. Let us define, for x, y, z R n, f xn+1 x = fx, x n+1 ρ n+1, g yn+1 y = gy, y n+1 η n+1, ϕx, y = ϕ 1 x 1, y 1,..., ϕ n x n, y n, ϕn+1 h ϕn+1 z = hz, ϕ n+1 x n+1, y n+1 ρ n+1 + ϕ n+1 η n+1. x n+1 y n+1 Let x suppf xn+1, y suppg yn+1, and let ρ 1,..., ρ n, η 1,..., η n >. One has h ϕn+1 ϕx, yπ n ϕ =1 ρ + ϕ η = hϕx, x n+1, y, y n+1 Π n+1 ϕ =1 ρ + ϕ η x y x y Φfx, x n+1 Π n+1 =1 ρ, gy, y n+1 Π n+1 =1 η = Φf xn+1 xπ n =1ρ, g yn+1 yπ n =1η, where the inequality follows from inequality 2. Hence, applying Theorem 1 in dimension n, one has h ϕn+1 xdx Φ f xn+1 xdx, g yn+1 xdx. R n R n R n This yields that for every x n+1 suppf, y n+1 suppg, and for every ρ n+1, η n+1 >, ϕn+1 Hϕ n+1 x n+1, y n+1 ρ n+1 + ϕ n+1 η n+1 ΦF x n+1, Gy n+1. x n+1 y n+1 Hence, applying Theorem 1 in dimension 1, one has Hxdx Φ F xdx, R This yields the desired inequality. R R Gxdx. 3 Applications to the log-brunn-minowsi inequality In this section, we discuss applications of the above inequalities to the log-brunn-minowsi inequality of Böröczy, Lutwa, Yang and Zhang [7]. Recall that a convex body in R n is a compact convex subset of R n with nonempty interior. Böröczy et al. conjectured the following inequality. Conjecture 7 log-brunn-minowsi inequality. Let K, L be symmetric convex bodies in R n and let λ [, 1]. Then, 1 λ K λ L K 1 λ L λ. Here, 1 λ K λ L = {x R n : x, u h K u 1 λ h L u λ, for all u S n 1 }, where S n 1 denotes the n-dimensional Euclidean unit sphere, h K denotes the support function of K, defined by h K u = max x K x, u, and stands for Lebesgue measure. Böröczy et al. [7] proved that Conjecture 7 holds in the plane. Using Corollary 5 with p =,...,, Saroglou [21] proved that Conjecture 7 holds for unconditional convex bodies 5

7 in R n a set K R n is unconditional if for every x 1,..., x n K and for every ε 1,..., ε n { 1, 1} n, one has ε 1 x 1,..., ε n x n K. Recall that a measure µ is s-concave, s [, + ], if the inequality µ1 λa + λb M λ s µa, µb holds for all compact sets A, B R n such that µaµb > and for every λ [, 1] see [5], [6]. The -concave measures are also called log-concave measures, and the -concave measures are also called convex measures. A function f : R n [, + is α-concave, α [, + ], if the inequality f1 λx + λy M λ αfx, fy holds for every x, y R n such that fxfy > and for every λ [, 1]. Saroglou [22] recently proved that if the log-brunn-minowsi inequality holds, then the inequality µ1 λ K λ L µk 1 λ µl λ holds for every symmetric log-concave measure µ, for all symmetric convex bodies K, L in R n and for every λ [, 1]. An extension of the log-brunn-minowsi inequality for convex measures was proposed by the author in [19], and reads as follows: Conjecture 8. Let p [, 1]. Let µ be a symmetric measure in R n that has an α-concave density function, with α p n. Then for every symmetric convex body K, L in Rn and for every λ [, 1], Here, µ1 λ K p λ L M λ 1µK, µl. 3 n p + α 1 1 λ K p λ L = {x R n : x, u M λ p h K u, h L u, for all u S n 1 }. In Conjecture 8, if α or p is equal to, then n/p + 1/α 1 is defined by continuity and is equal to. Notice that Conjecture 7 is a particular case of Conjecture 8 when taing µ to be Lebesgue measure and p =. By using Corollary 6, we will prove that Conjecture 7 implies Conjecture 8, when α 1, generalizing Saroglou s result discussed earlier. Theorem 9. If the log-brunn-minowsi inequality holds, then the inequality µ1 λ K p λ L M λ 1µK, µl n p + α 1 holds for every p [, 1], for every symmetric measure µ in R n that has an α-concave density function, with 1 α p n, for every symmetric convex body K, L in Rn and for every λ [, 1]. Proof. Let K, K 1 be symmetric convex bodies in R n and let λ, 1. Let us denote K λ = 1 λ K p λ K 1 and let us denote by ψ the density function of µ. Let us define, for t >, ht = K λ {ψ t}, ft = K {ψ t} and gt = K 1 {ψ t}. Notice that µk λ = ψxdx = K λ Similarly, one has µk = K λ + ψx dtdx = + ftdt, µk 1 = 6 K λ {ψ t} = + gtdt. + htdt.

8 Let t, s > such that the sets {ψ t} and {ψ s} are nonempty. Let us denote L = {ψ t}, L 1 = {ψ s} and L λ = {ψ M λ αt, s}. If x L and y L 1, then ψ1 λx + λy M λ αψx, ψy M λ αt, s. Hence, L λ 1 λl + λl 1 1 λ L p λ L 1, the last inclusion following from the fact that p 1. We deduce that K λ L λ 1 λ K p λ K 1 1 λ L p λ L 1 1 λ K L p λ K 1 L 1. Hence, hmαt, λ s = K λ L λ 1 λ K L p λ K 1 L 1 M λ p ft, gs, n the last inequality is valid for p and follows from the log-brunn-minowsi inequality by using homogeneity of Lebesgue measure see [7, beginning of section 3]. Thus we may apply Corollary 6 to conclude that µk λ = h M λ f, g = M λ n p + α 1 1 n p + α 1µK, µk 1 1. Since the log-brunn-minowsi inequality holds true in the plane, we deduce that Conjecture 8 holds true in the plane with the restriction α 1. Notice that Conjecture 8 holds true in the unconditional case as a consequence of Corollary 5 see [19]. References [1] K. Ball, Some remars on the geometry of convex sets, Geometric aspects of functional analysis 1986/87, , Lecture Notes in Math., 1317, Springer, Berlin, [2] F. Barthe, On a reverse form of the Brascamp-Lieb inequality, English summary Invent. Math , no. 2, [3] F. Barthe, Autour de l inégalité de Brunn-Minowsi, French [On the Brunn-Minowsi inequality], Ann. Fac. Sci. Toulouse Math , no. 2, [4] S. G. Bobov, A. Colesanti, I. Fragalà, Quermassintegrals of quasi-concave functions and generalized Preopa-Leindler inequalities, Manuscripta Mathematica , no. 1-2, pp [5] C. Borell, Convex measures on locally convex spaces, Ar. Mat , [6] C. Borell, Convex set functions in d-space, Period. Math. Hungar. 6:2 1975, [7] K. J. Böröczy, E. Lutwa, D. Yang, G. Zhang, The log-brunn-minowsi inequality, Adv. Math , [8] H. J. Brascamp, E. H. Lieb, On extensions of the Brunn-Minowsi and Préopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis , no. 4, [9] E. A. Carlen, F. Maggi, Stability for the Brunn-Minowsi and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range non-local isoperimetry, preprint, arxiv: [math.oc]. 7

9 [1] W. Chen, N. Dafnis, G. Paouris, Improved Hölder and reverse Hölder inequalities for Gaussian random vectors, Adv. Math , [11] M. A. Hernández Cifre, J. Yepes Nicolás, Refinements of the Brunn-Minowsi inequality, J. Convex Anal , no. 3, [12] A. Colesanti, E. Saorín Gómez, J. Yepes Nicolás, On a linear refinement of the Préopa- Leindler inequality, preprint, arxiv: [math.fa]. [13] R. J. Gardner, The Brunn-Minowsi inequality, Bull. Amer. Math. Soc. N.S. 39:3 22, [14] R. J. Gardner, D. Hug, and W. Weil, The Orlicz-Brunn-Minowsi theory: A general framewor, additions, and inequalities, J. Differential Geom. 97:3 214, [15] D. Ghilli, P. Salani, Quantitative Borell-Brascamp-Lieb inequalities for compactly supported power concave functions and some applications, preprint, arxiv: [math.ap]. [16] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge, at the University Press, d ed. [17] G. Livshyts, A. Marsiglietti, P. Nayar, A. Zvavitch, On the Brunn-Minowsi inequality for general measures with applications to new isoperimetric-type inequalities, preprint, arxiv: [math.pr]. [18] A. Marsiglietti, On the improvement of concavity of convex measures, Proc. Amer. Math. Soc. 215, doi: [19] A. Marsiglietti, A note on an L p -Brunn-Minowsi inequality for convex measures in the unconditional case, Pacific Journal of Mathematics , doi: 1.214/pjm [2] P. Nayar, T. Tocz, A note on a Brunn-Minowsi inequality for the Gaussian measure, Proc. Amer. Math. Soc , no. 11, , DOI 1.19/S [21] C. Saroglou, Remars on the conjectured log-brunn-minowsi inequality, Geom. Dedicata , [22] C. Saroglou, More on logarithmic sums of convex bodies, preprint, arxiv: [math.fa]. [23] R. Schneider, Convex bodies: the Brunn-Minowsi theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, [24] B. Uhrin, Curvilinear Extensions of the Brunn-Minowsi-Lusterni Inequality, Adv. Math., , no. 2, [25] C. Villani, Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338. Springer-Verlag, Berlin, 29. xxii+973 pp. ISBN: Arnaud Marsiglietti Institute for Mathematics and its Applications University of Minnesota 8

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