A local proof of the dimensional Pr´ekopa’s theorem. Van Hoang Nguyen∗ April 17, 2014

Abstract The aim of this paper is to find an expression for second derivative of the function φ(t) defined by Z − 1 β−n −β , β 6= n, φ(t) = ϕ(t, x) dx V

where U ⊂ R and V ⊂ Rn are open bounded subsets, and ϕ : U × V → R+ is a C 2 −smooth function. As a consequence, this result gives us a direct proof of the dimensional Pr´ekopa’s theorem based on a local approach.

1

Introduction

The Pr´ekopa’s theorem [11] says that marginals of log-concave functions are log-concave, i.e, if ϕ : Rn+1 → R is convex, then the function φ defined by Z  −ϕ(t,x) φ(t) = − log e dx (1.1) Rn

is convex on R. By modifying ϕ if necessary, we can replace Rn+1 by any of its open convex subset Ω, and the integration in (1.1) is taken in the section Ω(t) = {x ∈ Rn : (t, x) ∈ Ω}. The Pr´ekopa’s theorem is a direct consequence of the Pr´ekopa-Leindler inequality which can be seen as the functional form of the Brunn-Minkowski inequality (see [6]). The ∗

School of Mathemacical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. Email: [email protected] Supported by a grant from the European Research Council 2010 Mathematics Subject Classification: 26A51. Key words and phrases: Pr´ekopa’s theorem, Dimensional Pr´ekopa’s theorem, L2 -method

1

van-

1 INTRODUCTION

2

Brunn-Minkowski inequality is known to be one of the most important tools in analysis and geometry. It states that if A, B are non-empty measurable subsets of Rn then 1

1

1

|A + B| n ≥ |A| n + |B| n , where A + B = {a + b : a ∈ A, b ∈ B} and | · | denotes the Lebesgue measure of the measurable set (See ([4, 6, 8, 9]) for the proofs and applications of the Brunn-Minkowski inequality). A new proof of the Pr´ekopa theorem is recently given in [1, 2]. In these papers, the authors proved a local formulation for the second derivative of the function φ above. By using the convexity of ϕ, they show that φ00 is nonnegative. This local approach was also used by D. Cordero-Erausquin (see [5]) to generalize a result of Berndtsson concerning the Pr´ekopa’s theorem for plurisubharmonic functions (see [3]). In this paper, we adapt the local approach given in [1, 2] to find an expression for the second derivative of the function φ defined by Z

−β

ϕ(t, x)

φ(t) =

1 − β−n

dx

,

β 6= n,

(1.2)

V

where U ⊂ R and V ⊂ Rn are open bounded subsets, boundary of V is C ∞ −smooth, and ϕ : U × V → R+ is a C 2 −smooth function on U × V . For this purpose, we denote for each t∈U ϕ(t, x)−β dx dµt = R ϕ(t, x)−β dx V the probability measure on V . We also denote the corresponding symmetric diffusion operation with the invariant measure µt by Lt u(x) = ∆u(x) − β

h∇x ϕ(t, x), ∇u(x)i , ϕ(t, x)

where u is any function in C 2 (V ). By using integration by parts, we have Z Z Z ∂u Lt u(x)v(x)dµt (x) = − h∇u(x), ∇v(x)idµt (x) + v(x) (x)dµt (x), ∂ν V V ∂V where ν(x) = (ν1 (x), · · · , νn (x)) is the outer normal vector to x ∈ ∂V . Since ∂V is C ∞ −smooth, then ν is C ∞ −smooth on ∂V and it can be extended to a C ∞ −smooth map on a neighbourhood of ∂V . Hence the second fundamental form II of ∂V at x ∈ ∂V is defined by IIx (X, Y ) =

n X

Xi Yj ∂i (νj )(x),

i,j=1

for any two vector fields X = (X1 , · · · , Xn ) and Y = (Y1 , . . . , Yn ) in ∂V .

1 INTRODUCTION

3

In the sequel, we denote by ∇f and ∇2 f the gradient and Hessian matrix of a function f , respectively. We also denote by k · kHS the Hilbert-Schmidt norm on the space of square matrices. When f is function of the variables t and x, we write ∇x f and ∇2x f for the gradient and Hessian matrix of f which are taken only on x, respectively. Our first main theorem of this paper is the following: Theorem 1.1. Suppose that V has C ∞ −smooth boundary, and ϕ is C ∞ −smooth up to boundary of U × V . Let φ be defined by (1.2) then  Z Z  h(∇2(t,x) ϕ)X, Xi β β2 φ00 (t) 1 2 2 2 = dµt + k∇ ukHS − (∆u) dµt φ(t) β−n V ϕ β−n V n !2 r Z Z r β ∂t ϕ |β − n| n + ∆u − sign(β − n) dµt dµt |β − n| V n |β − n| V ϕ Z β2 + II(∇u, ∇u)dµt , (1.3) β − n ∂V where u is the solution of the equation Z ∂t ϕ(t, ·) ∂t ϕ(t, x) Lt u = − dµt (x) ϕ(t, ·) V ϕ(t, x)

and

∂u(x) = 0, ∂ν(x)

x ∈ ∂V,

(1.4)

and X denotes the vector field (1, β∇u(x)) in Rn+1 . Since ∂u(x) = 0 for every x ∈ ∂V , hence ∇u(x) ∈ Tx (∂V ) (the tangent space to ∂V ∂ν(x) at x ∈ ∂V ). This implies that II(∇u, ∇u) is well-defined on ∂V . Theorem 1.1 is proved in the next section. We will need the following classical fact about the existence of the solution of the elliptic partial differential equation (see [7] and references therein): Lemma 1.2. If V has C ∞ −smooth boundary ∂V , and ϕ is C ∞ -smooth up to boundary of R ∞ V , then for any function f ∈ C (V ), V f (x)dµt (x) = 0 there exists a function u ∈ C ∞ (V ) such that Lt u = f and ∂u(x) = 0 on ∂V . ∂ν(x) Our second main theorem of this paper is the dimensional Pr´ekopa’s theorem which is considered as a direct consequence of Theorem 1.1 and stated in the following theorem. The first part of this theorem concerns the convex case, and the second part concerns the concave case. Theorem 1.3. Let Ω ⊂ Rn+1 be a convex open subset, and let ϕ : Ω → R+ be a C 2 −smooth function up to boundary of Ω. For t ∈ R, we define the section Ω(t) = {x ∈ Rn : (t, x) ∈ Ω}. Then the following assertions hold: (i) If ϕ is convex on Ω, and β > n, then the function φ defined by 1 Z − β−n , φ(t) = ϕ(t, x)−β dx Ω(t)

is convex on R.

2 PROOF OF MAIN THEOREMS

4

(ii) If ϕ is concave on Ω, and β > 0, then the function φ defined by Z φ(t) =

1  β+n , ϕ(t, x) dx

β

Ω(t)

is concave on R. Finally, we remark that the Pr´ekopa’s theorem can be deduced from Theorem 1.3 by letting β tend to infinity since   1 −β !− β−n  Z  Z ϕ(t, x) −ϕ(t,x) lim (β − n)  1+ dx − 1 = − log e dx β→∞ β Ω(t) Ω(t) + and 

 1 β ! β+n  Z  ϕ(t, x) −ϕ(t,x) dx − 1 = log e dx , 1− β Ω(t) Ω(t) +

Z lim (β + n) 

β→∞

where a+ = max{a, 0} denotes the positive part of a.

2

Proof of main theorems

We begin this section by giving the proof of Theorem 1.1. Our proof is direct and similar the method used in [10]. Proof of Theorem 1.1: If β = 0, then (1.3) is evident since φ is a constant function. If β 6= 0, then (1.3) is equivalent to β − n φ00 (t) = β φ(t)

Z

h(∇2(t,x) ϕ)X, Xi ϕ

V

Z + sign (β − n) V

 Z  1 2 2 2 dµt + β k∇ ukHS − (∆u) dµt n V !2 r Z r |β − n| n ∂t ϕ ∆u − sign(β − n) dµt dµt n |β − n| V ϕ

Z II(∇u, ∇u)dµt .

+β

(2.1)

∂V

By a direct computation, we easily get   Z 2 β − n φ00 (t) ∂tt ϕ(t, x) ∂t ϕ(t, ·) = dµt (x) − (β + 1) Varµt β φ(t) ϕ(t, x) ϕ(t, ·) V Z 2 n ∂t ϕ(t, x) + dµt (x) , β−n V ϕ(t, x)

(2.2)

2 PROOF OF MAIN THEOREMS

5

R R where Varµt (f ) := V f 2 dµt − ( V f dµt )2 denotes the variance of any function f on V with respect to µt . Let u ∈ C ∞ (V ) be the solution of the equation (1.4). Since µt is a probability measure on V , then we have    Z Z  Z ∂t ϕ(t, ·) ∂t ϕ ∂t ϕ 2 Varµt = − (Lt u) dµt + 2 − dµt Lt u dµt . ϕ(t, ·) ϕ V V V ϕ R Using integration by parts and the fact V Lt u dµt = 0, we get  Z  Z Z Z ∂t ϕ ∂t ϕ h∇x (∂t ϕ), ∇ui ∂t ϕ h∇x ϕ, ∇ui − dµt Lt u dµt = − dµt + dµt . (2.3) ϕ ϕ ϕ V V ϕ V V ϕ It follows from integration by parts (see also the proof of Theorem 1 in [10]) that Z Z Z h(∇2x ϕ)∇u, ∇ui 2 2 2 (Lt u) dµt = k∇ ukHS dµt + β dµt ϕ V V V Z Z h∇x ϕ, ∇ui2 −β h(∇2 u)∇u, νidµt . dµt − 2 ϕ V ∂V From (2.3) and (2.4), we get an expression of Varµt (∂t ϕ/ϕ) as follows   Z Z ∂t ϕ(t, ·) h∇x (∂t ϕ), ∇ui ∂t ϕ h∇x ϕ, ∇ui Varµt dµt + 2 dµt = −2 ϕ(t, ·) ϕ ϕ V V ϕ Z Z h(∇2x ϕ)∇u, ∇ui 2 2 − k∇ ukHS dµt − β dµt ϕ V V Z Z h∇x ϕ, ∇ui2 h(∇2 u)∇u, νidµt . +β dµt + 2 ϕ V ∂V It follows from the definition of Lt that Z Z Z   h∇x ϕ, ∇ui2 2 2 2 β dµt = (Lt u) + (∆u) dµt − 2 ∆u Lt u dµt . ϕ2 V V V

(2.4)

(2.5)

(2.6)

Plugging (2.4) and (1.4) into (2.6), we obtain Z Z Z h∇x ϕ, ∇ui2 2 2 β(β + 1) dµt = k∇ ukHS dµt + (∆u)2 dµt 2 ϕ V V V Z Z h(∇2x ϕ)∇u, ∇ui ∂t ϕ +β dµt − 2 ∆u dµt ϕ V V ϕ Z  Z Z ∂t ϕ +2 ∆u dµt dµt − h(∇2 u)∇u, νidµt . (2.7) ϕ V V ∂V

2 PROOF OF MAIN THEOREMS

6

Moreover, using again integration by parts, we have Z Z ∂t ϕ h∇x ϕ, ∇ui 1 ∂t ϕ(t, x) dµt = − h∇x (ϕ(t, x)−β ), ∇u(x)i dx ϕ β V ϕ(t, x) V ϕ Z Z 1 1 h∇x (∂t ϕ), ∇ui ∂t ϕ h∇x ϕ, ∇ui dµt − dµt = β V ϕ β V ϕ ϕ Z 1 ∂t ϕ + ∆u dµt . β V ϕ Or, equivalent Z (β + 1) V

∂t ϕ h∇x ϕ, ∇ui dµt = ϕ ϕ

Z V

h∇x (∂t ϕ), ∇ui dµt + ϕ

Z V

∂t ϕ ∆u dµt . ϕ

(2.8)

Plugging (2.5), (2.7), and (2.8) into (2.2), we obtain Z 2 Z Z ∂tt ϕ h∇x (∂t ϕ), ∇ui h(∇2x ϕ)∇u, ∇ui β − n φ00 (t) 2 = dµt + 2β dµt + β dµt β φ(t) ϕ ϕ ϕ V V V Z  Z Z Z ∂t ϕ 2 2 k∇ ukHS dµt − 2 ∆u +β dµt dµt − (∆u)2 dµt ϕ V V V V 2 Z Z n ∂t ϕ(t, x) + dµt (x) − β h(∇2 u)∇u, νidµt β−n ϕ(t, x) V ∂V Z Z Z 2 h∇x (∂t ϕ), ∇ui h(∇2x ϕ)∇u, ∇ui ∂tt ϕ 2 dµt + 2β dµt + β dµt = ϕ ϕ ϕ V V V  Z  Z 1 β−n 2 2 2 +β (∆u)2 dµt k∇ ukHS − (∆u) dµt + n n V V  2 Z Z Z ∂t ϕ n ∂t ϕ(t, x) −2 ∆u dµt dµt + dµt (x) β−n V ϕ V ϕ(t, x) V Z −β h(∇2 u)∇u, νidµt . (2.9) ∂V

To finish our proof, we need to treat the term on boundary in (2.9). Since then ∇u(x) ∈ Tx (∂V ) for every x ∈ ∂V , and h(∇2 u(x))∇u(x), ν(x)i = −IIx (∇u(x), ∇u(x)),

x ∈ ∂V.

∂u ∂ν

= 0 on ∂V ,

(2.10)

Combining (2.9) and (2.10), and denoting X(t, x) = (1, β∇u(x)) with (t, x) ∈ U × V , we get (2.1). Then Theorem 1.1 is completely proved. In the following, we use Theorem 1.1 to prove the dimensional Pr´ekopa’s theorem (Theorem 1.3). Proof of Theorem 1.3: By using an approximation argument, we can assume that Ω is bounded and ϕ is C ∞ -smooth up to boundary of Ω.

2 PROOF OF MAIN THEOREMS

7

Part (i): We first prove when Ω = U × V with U ⊂ R, and V ⊂ Rn has C ∞ -smooth boundary ∂V . Since β > n, then applying Theorem 1.1, we have  Z Z  h(∇2(t,x) ϕ)X, Xi 1 β − n φ00 (t) 2 2 2 k∇ ukHS − (∆u) dµt = dµt + β β φ(t) ϕ n V V !2 r r Z Z β−n n ∂t ϕ + ∆u − dµt dµt n β−n V ϕ V Z +β II(∇u, ∇u)dµt , (2.11) ∂V

where II denotes the second fundamental form of ∂V , and u is the C ∞ −smooth solution of the equation (1.4) with Lt = ∆ − βh∇x ϕ, ·i/ϕ, and X denotes the vector field (1, β∇u) in Rn+1 . We have IIx (∇u(x), ∇u(x)) ≥ 0, x ∈ ∂V because of the convexity of V . By CauchySchwartz inequality, we have 1 k∇2 uk2HS ≥ (∆u)2 . n As a consequence of the convexity of ϕ, we obtain ∇2 ϕ ≥ 0 in the sense of symmetric matrix. All the integrations on the right hand side of (2.11) hence are nonnegative. This implies that φ00 ≥ 0, or φ is convex. In the general case, there exists an increasing sequences of C ∞ -smooth open convex Ωk such that Ωk = {(t, x) : ρk (t, x) < 0}, S with ρk ∈ C ∞ (Rn+1 ), k = 1, 2 · · · are convex functions, and Ω = k Ωk . Hence, by using an approximation argument, we can assume that Ω = {(t, x) : ρ(t, x) < 0} with a C ∞ −smooth convex function ρ, and ϕ is defined in a neighborhood of Ω. Since the convexity is local, it is enough to prove that φ is convex in a neighborhood of each t. Fix t0 , choose a small enough neighborhood U of t0 such that (U × Rn ) ∩ Ω ⊂ U × V and ρ, ϕ are defined in U × V , where V is convex subset of Rn and has C ∞ −smooth boundary ∂V . Define ρ0 = max{ρ, 0}, then ρ0 is a convex function in U × V . With N > 0, we know that the function 1 Z − β−n −β φN (t) = (ϕ(t, x) + N ρ0 (t, x)) dx V

is convex in U . Moreover, φN (t) → φ(t) in U as N tends to infinity, then φ is convex in U . This finishes the proof of the convexity of φ.

REFERENCES

8

Part (ii): As explained in the proof of the part (i) above, it suffices to prove the part (ii) in the case Ω = U × V with U ⊂ R and V ⊂ Rn are bounded open convex subsets, and ∂V is C ∞ −smooth. Since β > 0, by applying Theorem 1.1 to −β instead of β, we have β + n φ00 (t) = β φ(t)

h(∇2(t,x) ϕ)X, Xi

Z

ϕ

V

r

Z − V

Z  k∇

dµt − β

β+n ∆u + n

2

V

r

n β+n

Z V

 1 2 − (∆u) dµt n !2

uk2HS

∂t ϕ dµt ϕ

dµt

Z −β

II(∇u, ∇u)dµt .

(2.12)

∂V

Using the arguments in the proof of part (i) and the concavity of ϕ, we get φ00 (t) ≤ 0 from (2.12), or φ is concave.

Acknowledgment The author would like to sincerely thank anonymous referee for many useful and valuable comments which improved the quality of this paper.

References [1] S. Artstein, K. Ball, F. Barthe, and A. Naor, Solution of Shannon’s problem on the monotonicity of entropy, J. Amer. Math. Soc., 17 (2004) 975-982. [2] K. Ball, F. Barthe, and A. Naor, Entropy jumps in the presence of a spectral gap, Duke Math. J., 119 (1) (2003) 41-63. [3] B. Berndtsson, Pr´ekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions, Math. Ann., 312 (1998) 785-792. [4] H. J. Brascamp, and E. H. Lieb, On extensions of the Brunn-Minkowski and Pr´ekopaLeindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal., 22 (4) (1976) 366-389. [5] D. Cordero-Erausquin, On Berndtsson’s generalization of Pr´ekopa’s theorem, Math. Z., 249 (2005) 401-410. [6] R. J. Gardner, The Brunn-Minkowski inequality, Bull. Am. Math. Soc., 39 (3) (2002) 355-405. [7] A. V. Kolesnikov, and E. Milman, Poincar´e and BrunnMinkowski inequalities on weighted Riemannian manifolds with boundary, arXiv:1310.2526 [math.DG].

REFERENCES

9

[8] M. Ledoux, The concentration of measure phenomenon, American Mathematical Society, Providence, RI, (2001). [9] B. Maurey, In´egalit´es de Brunn-Minkowski-Lusternik, et autres g´eom´etriques et fonctionnelles, S´eminaire Bourbaki, Novembre 2003.

in´egalit´es

[10] V. H. Nguyen, Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Pr´ekopa’s theorem, J. Funct. Anal., 266 (2014) 931-955. [11] A. Pr´ekopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged), 34 (1973) 335-343.