On Gaussian Brunn-Minkowski inequalities

HAL Id: hal-00270555

https://hal.archives-ouvertes.fr/hal-00270555v2

Submitted on 30 Jul 2008

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de

On Gaussian Brunn-Minkowski inequalities

Franck Barthe, Nolwen Huet

To cite this version:

Franck Barthe, Nolwen Huet. On Gaussian Brunn-Minkowski inequalities. Studia Mathematica,

Instytut Matematyczny - Polska Akademii Nauk, 2009, 191 (3), pp.283–304. �hal-00270555v2�

On Gaussian Brunn-Minkowski inequalities

Franck Barthe and Nolwen Huet July 30, 2008

Abstract

In this paper, we are interested in Gaussian versions of the clas- sical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for

Borel or convex sets based on a previous work by Borell. Our method also allows us to have semigroup proofs of the geometric Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines.

2000 Mathematics Subject Classification: 60E15, 60G15, 52A40, 35K05.

Keywords: Brunn-Minkowski, Gaussian measure, Heat equation, Brascamp- Lieb inequalities.

1 Introduction

In this paper, we are interested in Gaussian versions of the classical Brunn- Minkowski inequality on the Lebesgue measure of sum-sets (see e.g. [19, 20]). On R

with its canonical Euclidean structure ( h· , ·i , | · | ) we consider the standard Gaussian measure γ

(dx) = (2π)

exp( −| x |

/2) dx, x ∈ R

. Given α, β ∈ R and sets A, B ⊂ R

, we recall that their Minkowski combination is defined by

αA + βB = { αa + βb; (a, b) ∈ A × B } .

Using symmetrization techniques, Ehrhard [15] proved a sharp lower bound on the Gaussian measure of a convex combination of convex sets. Namely:

if α, β ≥ 0 satisfy α + β = 1 and if A, B ⊂ R

are convex, then

Φ

◦ γ

(αA + βB) ≥ αΦ

◦ γ

(A) + βΦ

◦ γ

(B ),

where Φ is the cumulative distribution function of γ

. This inequality be- comes an equality when A and B are parallel half-spaces or the same convex set. Lata la [17] showed that the inequality remains valid when A is convex and B is an arbitrary Borel set. In the remarkable paper [9], Borell was able to remove the remaining convexity assumption. He actually derived a functional version of the inequality (in the spirit of the Pr´ekopa-Leindler inequality) by a wonderful interpolation technique based on the heat equa- tion. In a series of papers, Borell extended the inequality to more general combinations:

Theorem (Borell [11]). Let α

, . . . , α

> 0. The inequality Φ

◦ γ

P α

A

≥ X

α

Φ

◦ γ

(A

) (1) holds for all Borel sets A

, . . . , A

in R

if and only if

X α

≥ 1 and ∀ j, α

− X

i6=j

α

≤ 1.

Moreover, it holds for all convex sets A

, . . . , A

in R

if and only if X α

≥ 1.

Borell established the case m = 2 for Borel sets in [10] thanks to his semigroup argument. His proof in [11] of the general case relies on a tricky and somewhat complicated induction. Remark that a linear combination of Borel sets need not be a Borel set; however it is analytic or Suslin, hence universally measurable, see e.g. [16].

In this note we give a slight extension of the above statement (which can actually be derived directly from the theorem of Borell, as pointed out by the referee). More importantly we propose a streamlined version of the semigroup argument for m functions directly, which allows to take advantage of convexity type assumptions. This better understanding of the semigroup technique also allows to study more general situations. The main result is stated next. It involves the heat semigroup, for which we recall the definition: given a Borel nonnegative function f on R

, its evolute at time t ≥ 0 is the function P

f given by

P

f (x) = Z

f x + √ t y

γ

(dy) = E f(x + B

)

where B is an n-dimensional Brownian motion. By convention ∞ − ∞ =

−∞ so that inequalities like Inequality (1), or the one introduced in the next theorem, make sense.

Theorem 1. Let I

⊂ { 1, . . . , m } , α

, . . . , α

> 0. The following asser- tions are equivalent:

1. The parameter α satisfies

X α

≥ 1 and ∀ j / ∈ I

, α

− X

i6=j

α

≤ 1. (2)

2. For all Borel sets A

, . . . , A

in R

such that A

is convex when i ∈ I

,

Φ

◦ γ

P α

A

≥ X

α

Φ

◦ γ(A

)

3. For all Borel functions h, f

, . . . , f

from R

to [0, 1] such that Φ

◦ f

is concave when i ∈ I

, if

∀ x

, . . . , x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

), then

Φ

Z

h dγ

≥ X α

Φ

Z f

dγ

.

4. For all Borel functions h, f

, . . . , f

from R

to [0, 1] such that Φ

◦ f

is concave when i ∈ I

, if

∀ x

, . . . , x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

), then for all t ≥ 0

∀ x

, . . . , x

∈ R

, Φ

◦ P

h P α

x

≥ X

α

Φ

◦ P

f

(x

).

Remark. Condition (2) can be rephrased as

X α

≥ max 1, max { 2α

− 1; j 6∈ I

} .

Actually the condition will come up in our argument in the following geo- metric form: there exist vectors u

, . . . , u

∈ R

such that for all i ∈ I

,

| u

| ≤ 1, for all i 6∈ I

, | u

| = 1, and | P

α

u

| = 1.

In the next section we show that the condition on α implies the fourth (and formally strongest) assumption in the latter theorem, when restricted to smooth enough functions. The third section completes the proof of the theorem. In the final section we discuss related problems.

Before going further, let us introduce some notation.

• We consider functions depending on a time variable t and a space variable x. The time derivative is denoted by ∂

, while the gradi- ent, Hessian, and Laplacian in x are denoted by ∇

, Hess

, and ∆

, omitting the index x when there is no ambiguity.

• The unit Euclidean (closed) ball and sphere of R

are denoted respec- tively by B

and S

.

• For A ⊂ R

, we set A

= A + ε B

. The notation A

means (A

)

.

2 Functional and semigroup approach

As already mentioned we follow Borell’s semigroup approach of the Gaussian Brunn-Minkowski inequalities (see [9] and [10]): for parameters α verifying (2), the plan is two show the functional version of the inequality (the third assertion of Theorem 1), by means of the heat semigroup. Note that the fourth assertion implies the third one when choosing t = 1 and x

= 0 in the last equation. So our aim is to establish the fourth assumption. More precisely, given Borel functions h, f

, . . . , f

from R

taking into (0, 1), we define C on [0, T ] × ( R

)

by

C(t, x) = C(t, x

, . . . , x

) = Φ

◦ P

h P α

x

− X

α

Φ

◦ P

f

(x

).

Since P

f = f the assumption

∀ x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

) (3) translates as C(0, . ) ≥ 0. Our task is to prove

C(0, . ) ≥ 0 = ⇒ ∀ t ≥ 0, C (t, . ) ≥ 0.

2.1 Preliminaries

When the functions h and f

are smooth enough, the time evolution of P

h and P

f

is described by the heat equation. This yields a differential equation satisfied by C. Our problem boils down to determine whether this evolution equation preserves nonnegative functions. This is clearly related to the maximum principle for parabolic equations (see e.g. [13]). We will use the following lemma.

Lemma 1. Assume that C is twice differentiable. If







Hess(C) ≥ 0

∇ C = 0 C ≤ 0

= ⇒ ∂

C ≥ 0 (4)

and if for some T > 0

lim inf

|x|→∞

0≤

inf

C(x, t)

≥ 0, (5)

then

C(0, . ) ≥ 0 = ⇒ ∀ t ∈ [0, T ], C(t, . ) ≥ 0.

Proof. For ε > 0, set C

(t, x) = C(t, x) + εt on [0, T ] × ( R

)

. If C

< 0 at some point, then C

reaches its minimum at a point (t

, x

) where ∇ C = 0, Hess(C) ≥ 0, C < 0, and ∂

C + ε ≤ 0 (= 0 if t

< T ). By the hypotheses, it implies ∂

C ≥ 0 which is in contradiction with ∂

C ≤ − ε. So for all ε > 0 and T > 0, C

is non-negative on [0, T ] × ( R

)

, thus C is non-negative everywhere.

Property (5) is true under mild assumptions on h and f

which are related to the initial condition C(0, . ) ≥ 0 in the large:

Lemma 2. If there exist a

, . . . , a

∈ R such that

• lim sup

|x|→∞

f

(x) ≤ Φ(a

)

• h ≥ Φ P α

a

then for all T > 0,

lim inf

|x|→∞

0≤

inf

C(x, t)

≥ 0.

Proof. Let δ > 0. By continuity of Φ

, there exists ε > 0 such that Φ

Φ(a

) + 2ε

≤ a

+ δ P α

.

Let r > 0 be such that γ

(r B

) = 1 − ε. Then, for 0 ≤ t ≤ T , P

f

(x

) =

Z

rBⁿ

f

(x

+ √

t y) γ

(dy) + Z

(rBⁿ)^∁

f

(x

+ √

t y) γ

(dy)

≤ (1 − ε) sup

xi+r√ tBⁿ

f

+ ε sup f

≤ sup

xi+r√ TBⁿ

f

+ ε

≤ Φ(a

) + 2ε for | x

| large enough.

Moreover P

h ≥ Φ P α

a

so for | x | large enough and for 0 ≤ t ≤ T , it holds C(t, x) ≥ − δ. As δ > 0 was arbitrary, the proof is complete.

Checking Property (4) of Lemma 1 requires the following lemma:

Lemma 3. Let d ≥ 2, α

, . . . , α

> 0. Let k be an integer with 0 ≤ k ≤ m and

ϕ : ( S

)

× ( B

)

→ R

(v

, . . . , v

) 7→ | P

α

v

| . Then the image of ϕ is the interval

J :=

"

max n

0 o

∪ n

α

− X

i6=j

α

, 1 ≤ j ≤ k o , X

α

# .

Proof. As ϕ is continuous on a compact connected set, Im(ϕ) = [min ϕ, max ϕ].

Plainly | P

α

v

| ≤ P

α

, with equality if v

= · · · = v

is a unit vector. So max ϕ = P

i

α

. For all j ≤ k, since | v

| = 1, the triangle inequality gives

X α

v

≥ α

| v

| − X

i6=j

α

| v

| ≥ α

− X

i6=j

α

.

Hence Im(ϕ) ⊂ J and these two segments have the same upper bound.

Next we deal with the lower bound. Let us consider a point (v

, . . . , v

)

where ϕ achieves its minimum, and differentiate:

For j ≤ k, v

lies in the unit sphere. Applying Lagrange multipliers theorem to ϕ

with respect to v

gives a real number λ

such that,

α

X

i

α

v

= λ

v

. (6)

For j > k, the j-th variable lives in B

. If | v

| < 1 the minimum is achieved at an interior point and the full gradient on ϕ

with respect to the j-th variable is zero. Hence P

i

α

v

= 0. On the other hand if at the minimum | v

| = 1, differentiating in the j-th variable only along the unit sphere gives again the existence of λ

∈ R such that (6) is verified.

Eventually, we face 2 cases:

1. Either P

α

v

= 0 and min ϕ = 0. In this case, the triangle inequality gives 0 = | P

α

v

| ≥ α

− P

i6=j

α

whenever j ≤ k.

2. Or the v

’s are colinear unit vectors and there exists a partition S

∪ S

= { 1, . . . , m } and a unit vector v such that

min ϕ =

X

S+

α

v − X

S−

α

v = X

S+

α

− X

S−

α

> 0.

Assume that S

contains 2 indices j and ℓ. Let e

and e

be 2 or- thonormal vectors of R

and let us denote by R(θ) the rotation in the plane Vect(e

, e

) of angle θ. The length of the vector α

R(θ)e

+ α

e

is a decreasing and continuous function of θ ∈ [0, π]. Denote by U (θ) the rotation in the plane Vect(e

, e

) which maps this vector to | α

R(θ)e

+ α

e

| e

. Then

α

U (θ)R(θ)e

+ α

U (θ)e

+ X

S+\{j,ℓ}

α

e

− X

S−

α

e

= λ(θ)e

, where λ(0) = P

S+

α

− P

S−

α

= min ϕ > 0 and λ is continuous and decreasing in θ ∈ [0, π]. This contradicts the minimality of min ϕ. So S

contains a single index j and

min ϕ =

α

v − X

i6=j

α

v

= α

− X

i6=j

α

> 0.

Note that necessarily j ≤ k, otherwise one could get a shorter vector by replacing v

= v by (1 − ε)v. Besides, the condition α

− P

i6=j

α

> 0

ensures that α

> α

for ℓ 6 = j. This implies that for ℓ 6 = j , α

− X

i6=ℓ

α

≤ α

− α

< 0 < α

− X

i6=j

α

.

So min ϕ = max n

0 o

∪ n

α

− P

i6=j

α

, 1 ≤ j ≤ k o

as claimed.

2.2 Semigroup proof for smooth functions

We deal with smooth functions first, in order to ensure that P

f

and P

h verify the heat equation. This restrictive assumption will be removed in Section 3 where the proof of Theorem 1 is completed.

Theorem 2. Let f

, i = 1, . . . , m, and h be twice continuously differentiable functions from R

to (0, 1) satisfying the hypotheses of Lemma 2. Assume moreover that for f = f

or h,

∀ t > 0, ∀ x ∈ R

,

∇ f (x + √ t y)

e

2

2

−−−−→

|y|→∞

0.

Let α

, . . . , α

be positive real numbers such that X α

≥ 1 and ∀ j, α

− X

i6=j

α

≤ 1.

If

∀ x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

), then

∀ t ≥ 0, ∀ x

∈ R

, Φ

◦ P

h P α

x

≥ X

α

Φ

◦ P

f

(x

).

Proof. Let us recall that C is defined by C(t, x) = C(t, x

, . . . , x

) = H t, P

α

x

− X

α

F

(t, x

) where we have set

H(t, y) = Φ

◦ P

h(y) and F

(t, y) = Φ

◦ P

f

(y).

In what follows, we omit the variables and write H for H t, P α

x

and F

instead of F

(t, x

). With this simplified notation,

C = H − X

α

F

,

∇

C = α

( ∇ H − ∇ F

),

∇

∇

C = α

α

Hess(H) − δ

α

Hess(F

).

Moreover, one can use the property of heat kernel to derive a differential equation for F

and H. Indeed, for any f satisfying hypotheses of the theorem, we can perform an integration by parts so that it holds

∂

P

f = 1 2 ∆P

f.

Then we set F = Φ

◦ P

f and use the identity (1/Φ

(x))

= x/Φ

(x) to show

∂

F = ∂

P

f

Φ

(F ) = ∆P

f 2 Φ

(F ) ,

∇ F = ∇ P

f Φ

(F ) ,

∆F = ∆P

f

Φ

(F ) + F |∇ P

f |

(Φ

(F ))

. We put all together to get

∂

F = 1

2 ∆F − F |∇ F |

and to deduce the following differential equation for C:

∂

C = 1

2 ( S + P ) where the second order part is

S = ∆H − X α

∆F

and the terms of lower order are P = −

H |∇ H |

− X

α

F

|∇ F

|

.

We will conclude using Lemma 1. So we need to check Condition (4). First we note that P is non-negative when ∇ C = 0 and C ≤ 0, regardless of α.

Indeed, ∇ C = 0 implies that ∇ F

= ∇ H for all i. So P = − |∇ H |

C which is non-negative if C ≤ 0.

It remains to deal with the second order part. It is enough to express S as E C for some elliptic operator E , since then Hess(C) ≥ 0 implies S ≥ 0.

Such a second order operator can be written as E = ∇

A ∇ where A is a symmetric matrix nm × nm. Moreover E is elliptic if and only if A is positive semi-definite. In view of the structure of the problem, it is natural to look for matrices of the following block form

A = B ⊗ I

= (b

I

)

,

where I

is the identity n × n matrix and B is a positive semi-definite matrix of size m. Denoting x

= (x

, . . . , x

),

E C =

m

X

i,j=1

b

X

k=1

∂

∂x

∂x

C

!

=

m

X

i,j=1

b

α

α

∆H − δ

α

∆F

= h α , Bα i ∆H −

m

X

i=1

b

α

∆F

.

Hence there exists an elliptic operator E of the above form such that E C = S = ∆H − P

i=1

α

∆F

if there exits a positive semi-definite matrix B of size m such that

h α , Bα i = h e

, Be

i = · · · = h e

, Be

i = 1

where (e

)

is the canonical basis of R

. Now a positive semi-definite matrix B can be decomposed into B = V

V where V is a square matrix of size m.

Calling v

, . . . , v

∈ R

the columns of V , we can translate the latter into conditions on vectors v

. Actually, we are looking for vectors v

, . . . , v

∈ R

with

| v

| = · · · = | v

| =

X α

v

= 1.

By Lemma 3 for k = m, this is possible exactly when α satisfies the claimed condition:

X α

≥ 1 and ∀ j, α

− X

i6=j

α

≤ 1.

The following corollary will be useful in the next section.

Corollary 1. Let f be a function on R

taking values in (0, 1) and vanishing at infinity, i.e. lim

f (x) = 0. Assume also that

∀ t > 0, ∀ x ∈ R

,

∇ f (x + √ t y)

e

2

2

−−−−→

|y|→∞

0.

If Φ

◦ f is concave, then Φ

◦ P

f is concave for all t ≥ 0.

Proof. Let 1 > ε > 0 and α

> 0 with P

α

= 1. Choosing h = ε+(1 − ε)f ≥ f and f

= f for i ≥ 1, one can check that the latter theorem applies. Hence for all t ≥ 0 and x

∈ R

:

Φ

◦ P

(ε + (1 − ε)f ) P α

x

≥ X

α

Φ

◦ P

f (x

).

Letting ε go to 0, we get by monotone convergence that Φ

◦ P

f is concave.

2.3 Φ

-concave functions

When some of the f

’s are Φ

-concave, the conditions on the parameters can be relaxed. Such functions allow to approximate characteristic functions of convex sets. They will be useful in Section 3.

Theorem 3. Let I

⊂ { 1, . . . , m } . Let f

, i = 1, . . . , m, and h be twice continuously differentiable functions from R

to (0, 1) satisfying the hypothe- ses of Lemma 2. Assume also that for f = f

or h,

∀ t > 0, ∀ x ∈ R

,

∇ f (x + √ t y)

e

2

2

−−−−→

|y|→∞

0.

Assume moreover that Φ

◦ f

is concave, decreasing towards −∞ at infinity for all i ∈ I

.

Let α

, . . . , α

be positive numbers satisfying X α

≥ 1 and ∀ j / ∈ I

, α

− X

i6=j

α

≤ 1.

If

∀ x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

), then

∀ t ≥ 0, ∀ x

∈ R

, Φ

◦ P

h P α

x

≥ X

α

Φ

◦ P

f

(x

).

Proof. As in the proof of Theorem 2, we try to apply Lemma 1 to the equation satisfied by C:

∂

C(t, x) = 1

2 ( S + P ).

We have already shown that P is non-negative when ∇ C = 0 and C ≤ 0, for any α

, . . . , α

. We would like to prove that the conditions on α in the theorem imply that S is non-negative whenever Hess(C) ≥ 0.

By Corollary 1, for all i ∈ I

the function F

is concave, hence ∆F

≤ 0.

So we are done if we can write

S = E C − X

i∈Iconv

λ

∆F

,

for some elliptic operator E and some λ

≥ 0 . As in the proof of the previous theorem, we are looking for operators of the form E = ∇

A ∇ with A = B ⊗ I

= (b

I

)

where B is a symmetric positive semi-definite matrix m × m. Hence our task is to find B ≥ 0 and λ

≥ 0 such that λ

= 0 when i / ∈ I

and

∆H − X

α

∆F

= h α , Bα i ∆H − X

i

(b

α

+ λ

)∆F

.

When i ∈ I

, we can find λ

≥ 0 such that b

α

+λ

= α

whenever b

≤ 1.

Consequently, the problem reduces to finding a positive semi-definite matrix B of size m × m such that







h e

, Be

i ≤ 1, ∀ i ∈ I

h e

, Be

i = 1, ∀ i / ∈ I

h α , Bα i = 1

where (e

)

is the canonical basis of R

. Equivalently, do there exist v

, . . . , v

∈ R

such that







| v

| ≤ 1, ∀ i ∈ I

| v

| = 1, ∀ i / ∈ I

| P

α

v

| = 1

?

We conclude with Lemma 3.

3 Back to sets

This sections explains how to complete the proof of Theorem 1. The main issue is to get rid of the smoothness assumptions made so far. The plan of the argument is summed up in the next figure. The key point is that the conditions on α do not depend on n.

conditions on α

a

qyjjjjjjjjj

jjjjjjjjj

inequality with P

f

for smooth functions on R

b

RR RR RR R

RR RR RR RR RR

inequality with P

f

for Borel functions on R

d

emRRRRRRRR

RRRRRRRR

inequality for sets A

⊂ R

c

2:n

nn nn nn nn

nn nn nn nn n

If we can prove the above implications, we will have shown that assertion 1 ⇐⇒ assertion 2 ⇐⇒ assertion 4

in Theorem 1. Moreover, it is clear that assertion 4 = ⇒ assertion 3. To complete the picture, we can for instance prove assertion 3 = ⇒ assertion 1 in the same way we do below for the fourth implication.

a- “Conditions on α

⇒ inequality with P

f

for smooth functions on R

”:

This implication is nothing else than Theorem 3. Equivalently, the first as- sertion in Theorem 1 implies the fourth one restricted to “smooth” functions (i.e. verifying all the assumptions of the first paragraph of Theorem 3).

b- “Inequality with P

f

for smooth functions on R

⇒ inequality for sets A

⊂ R

”: For arbitrary α, let us prove that the fourth assertion in Theo- rem 1 restricted to smooth functions (in the above-mentioned sense) implies the second assertion of the theorem, involving sets. Let A

, . . . , A

be Borel sets in R

with A

convex when i ∈ I

. By inner regularity of the mea- sure, we can assume that they are compact. Let ε > 0 and b > a be fixed.

Then,

• for i / ∈ I

: there exists a smooth function f

such that f

= Φ(b) on A

, f

= Φ(a) off A

, and 0 < Φ(a) ≤ f

≤ Φ(b) < 1.

• for i ∈ I

: there exists a smooth function f

such that F

= Φ

◦ f

is concave, F

= b on A

, F

≤ a off A

, and F

≤ b on R

.

For instance, take a point x

in A

and define the gauge of A

with respect to x

by

ρ(x) = inf

λ > 0, x

+ 1

λ (x − x

) ∈ A

.

We know that ρ is convex since A

is convex (see for instance [20]).Then set F ˜

(x) = b + c

1 − max ρ(x) , 1

where c > 0 is chosen large enough to insure that ˜ F

≤ a off A

. Now, we can take a smooth function g with compact support small enough and of integral 1, such that f

= Φ ˜ F

∗ g

is a smooth Φ

- concave function satisfying the required conditions.

• for h: set

a

= max

u

= a or b u 6 = (b, . . . , b)

X α

u

and b

= X α

b.

Again, we can choose a smooth function h such that h = Φ(b

) on P α

A

, h = Φ(a

) off P

α

A

, and 0 < Φ(a

) ≤ h ≤ Φ(b

) < 1.

From these definitions, the functions h and f

are “smooth” and satisfy

∀ x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

).

By our hypothesis, the inequality remains valid with P

h and P

f

for all t > 0. Choosing t = 1, x

= 0 yields

Φ

Z

h dγ

≥ X α

Φ

Z

f

dγ

.

Remark here that the functions depends actually of a (respectively a

), b (respectively b

), and ε, possibly in a precise way with a procedure like described above for f

. We could then write h(a

, b

, ε, .) and f

(a, b, ε, .).

Letting first a → −∞ so that a

→ −∞ , we get by dominated conver- gence

Φ

Z

h( −∞ , b

, ε, .) dγ

≥ X α

Φ

Z

f

( −∞ , b, ε, .) dγ

.

Now let (b, ε) tend to ( ∞ , 0). Notice that f

( −∞ , ∞ , 0, .) and h( −∞ , ∞ , 0, .) are characteristic functions. Eventually we obtain, again by dominated convergence, that

Φ

◦ γ

P α

A

≥ X

α

Φ

◦ γ

(A

).

c- “Inequality for sets A

⊂ R

⇒ inequality with P

f

for Borel functions on R

”. Here we assume that the second assumption of Theorem 1 is valid for all Borel sets in R

and we derive the fourth assumption of the theorem for functions defined on R

.

For any Borel function f on R

taking values in [0, 1], t > 0, and x ∈ R

, we define

B

= n (u, y)

u ≤ Φ

◦ f x + √ t y o

⊂ R × R

. Then it holds

γ

B

= P

f(x).

Let h, f

, . . . , f

be Borel functions on R

with values in [0, 1], such that Φ

◦ f

is concave when i ∈ I

. Assume that

∀ x

∈ R

, Φ

◦ h P α

x

≥ X

α

Φ

◦ f

(x

).

Then for (u

, y

) in B

, we get X α

u

≤ X

α

Φ

◦ f

(x

+ √

t y

) ≤ Φ

◦ h P

α

(x

+ √ t y

) which means that

X α

B

⊂ B

.

The same argument shows that B

is convex if Φ

◦ f is concave. Thus, the result for sets in R

implies that

Φ

◦ P

h P α

x

≥ Φ

◦ γ

P

α

B

≥ X

α

Φ

◦ P

f

(x

).

d- “Inequality with P

f

for Borel functions on R

⇒ conditions on α

”: We will prove the contraposed assertion: if the conditions on α

are violated, then there exists Borel functions h and f

such that Φ

◦ f

is concave for i ∈ I

, which verify for all x

the relation Φ

◦ h( P

α

x

) ≥ P

Φ

◦ f

(x

)

but for which this inequality is not preserved by P

for some t. Actually since P

f (0) = R

f dγ, it will be enough to exhibit functions such that Φ

Z h dγ

< X α

Φ

Z f

dγ

.

Let f : R

→ (0, 1) be an even Borel function such that f(0) > 1

2 , Z

f dγ < 1

2 , and F = Φ

◦ f is concave.

For instance, we may take f(x) = Φ 1 − | ax |

for a large enough. Note that for 0 ≤ t ≤ 1,

F (tx) ≥ tF (x) + (1 − t)F (0) ≥ tF (x). (7) Assume first that P

α

< 1. Then by concavity and the latter bound, we get for all x

,

Φ

◦ f P

i

α

x

= F P

i

α

x

≥ X

i

α

P

j

α

F P

j

α

x

≥ X

i

α

F (x

) = X

i

α

Φ

◦ f (x

).

However since 1 > P

α

and Φ

R f dγ

< 0, it holds Φ

Z f dγ

< X

i

α

Φ

Z

f dγ

.

Assume now that there exists j / ∈ I

such that α

− P

i6=j

α

> 1.

Then using (7) and concavity again, we obtain for all x

, α

F (x

) ≥

1 + P

i6=j

α

F α

x

1 + P

i6=j

α

!

≥ F

α

x

− P

i6=j

α

x

+ X

i6=j

α

F (x

).

Let g = 1 − f. Since − F = − Φ

◦ f = Φ

◦ (1 − f ) = Φ

◦ g and f is even we may rewrite the latter as

Φ

◦ g

α

x

+ P

i6=j

α

( − x

)

≥ α

Φ

◦ g(x

) + X

i6=j

α

Φ

◦ f( − x

).

However, since Φ

( R

g dγ) = − Φ

( R

f dγ) > 0 and α

− P

i6=j

α

> 1 it also holds

Φ

Z

g dγ

< α

Φ

Z

g dγ

+ X

i6=j

α

Φ

Z

f dγ

.

Therefore the proof is complete.

4 Further remarks

4.1 Brascamp-Lieb type inequalities

In the previous papers [7, 8], Borell already used his semigroup approach to derive variants of the Pr´ekopa-Leindler inequality. The later is a functional counterpart to the Brunn-Minkowski inequality for the Lebesgue measure and reads as follows: if λ ∈ (0, 1) and f, g, h : R

→ R

are Borel functions such that for all x, y ∈ R

,

h λx + (1 − λ)y

≥ f(x)

g(y)

then R

h ≥ R

f

R g

where the integrals are with respect to Lebesgue’s measure. Borell actually showed the following stronger fact: for all t > 0 and all x, y ∈ R

P

h λx + (1 − λ)y

≥ P

f(x)

P

g(y)

.

Setting H(t, · ) = log P

h and defining F, G similarity, it is proved that C(t, x, y) := H t, λx + (1 − λ)y

− λF (t, x) + (1 − λ)G(t, y) satisfies a positivity-preserving evolution equation. The argument is simpler than for Ehrhard’s inequality since the evolution equation of individual functions is simpler: 2∂

H = ∆H + |∇ H |

.

The Brascamp-Lieb [12, 18] inequality is a powerful extension of H¨older’s

inequality. The so-called reverse Brascamp-Lieb inequality, first proved in

[2, 3], appears as an extension of the Pr´ekopa-Leindler inequality. In the pa-

per [4], it was noted that Borell’s semigroup method could be used to derive

the geometric reverse Brascamp-Lieb inequality (which in some sense is a

generic case, see [6]) for functions of one variable. This observation was also

motivated by a proof of the Brascamp-Lieb inequalities based on semigroup

techniques (Carlen Lieb and Loss [14] for functions of one variable, and Ben-

nett Carbery Christ and Tao [6] for general functions). In this subsection,

we take advantage of our streamlined presentation of Borell’s method, and quickly reprove the reverse Brascamp-Lieb inequality in geometric form, but for functions of several variables. More surprisingly we will recover the Brascamp-Lieb from inequalities which are preserved by the Heat flow. The result is not new (the inequality for the law of the semigroup appears in the preprint [5]), but it is interesting to have semigroup proofs of the direct and of the reverse inequalities which follow exactly the same lines. Recall that the transportation argument developed in [3] was providing the direct and the reverse inequality simultaneously.

The setting of the geometric inequalities is as follows: for i = 1, . . . , m let c

> 0 and let B

: R

→ R

be linear maps such that B

B

= I

and

m

X

i=1

c

B

B

= I

. (8)

These hypotheses were put forward by Ball in connection with volume es- timates in convex geometry [1]. Note that B

is an isometric embedding of R

into R

and that B

B

is the orthogonal projection from R

to E

= Im(B

). The Brascamp-Lieb inequality asserts that for all Borel func- tions f

: R

→ R

it holds

Z

R^N m

Y

i=1

f

(B

x)

dx ≤

m

Y

i=1

Z

Rⁿⁱ

f

.

The reverse inequality ensures that Z

R^N

sup (

Y

i=1

f

(x

)

; x

∈ R

with X

c

B

x

= x )

dx ≥

m

Y

i=1

Z

Rⁿⁱ

f

.

Following [4], we will deduce the later from the following result.

Theorem 4. If h : R

→ R

and f

: R

→ R

satisfy

∀ x

∈ R

, h X

i=1

c

B

x

≥

m

Y

i=1

f

(x

)

then

∀ x

∈ R

, P

h X

i=1

c

B

x

≥

m

Y

i=1

P

f

(x

)

.

The reverse inequality is obtained as t → + ∞ since for f on R

, P

f(x) is equivalent to (2πt)

R

R^d

f . To see it, note that:

P

f (x) = (2πt)

Z

R^d

f (y) exp

| x − y |

2t

dy.

Note also that taking traces in the decomposition of the identity map yields P

i

c

n

= N .

In order to recover the Brascamp-Lieb inequality, we will show the fol- lowing theorem.

Theorem 5. If h : R

→ R

and f

: R

→ R

satisfy

∀ x ∈ R

, h(x) ≤

m

Y

i=1

f

(B

x)

, then

∀ x ∈ R

, P

h(x) ≤

m

Y

i=1

P

f

(B

x)

.

Again, the limit t → + ∞ yields the Brascamp-Lieb inequality when choosing h(x) = Q

i=1

f

(B

x)

. We sketch the proofs the the above two statements, omitting the truncation arguments needed to ensure Condition (5).

Proof of Theorem 4. Set H(t, · ) = log P

h( · ) and F

(t, · ) = log P

f

( · ). As said above, the functions H and F

satisfy the equation 2∂

U = ∆U + |∇ U |

. Set for (t, x

, . . . , x

) ∈ R

× R

× · · · × R

C(t, x

, . . . , x

) := H t,

m

X

i=1

c

B

x

−

m

X

i=1

c

F

(t, x

).

By hypothesis C(0, · ) ≥ 0 and we want to prove that C(t, · ) is non-negative as well. As before, we are done if we can show that the three conditions C ≤ 0, ∇ C = 0, and Hess(C) ≥ 0 imply that ∂

C ≥ 0. Actually one can see that the condition C ≤ 0 will not be used in the following. Omitting variables,

2∂

C =

∆H − X c

∆F

+

|∇ H |

− X

c

|∇ F

|

=: S + P .