Blaschke–Santal´o inequality for many functions and ...Blaschke–Santal´o inequality for many functions and geodesic barycenters of measures ∗ Alexander V. Kolesnikov† and

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Blaschke–Santaló inequality for many functions and geodesic

barycenters of measures ∗

Alexander V. Kolesnikov†and Elisabeth M. Werner

‡

Abstract

Motivated by the geodesic barycenter problem from optimal transportation theory,we prove a natural generalization of the Blaschke–Santaló inequality and the affineisoperimetric inequalities for many sets and many functions. We derive from it anentropy bound for the total Kantorovich cost appearing in the barycenter problem.We also establish a “pointwise Prékopa–Leindler inequality” and show a monotonicityproperty of the multimarginal Blaschke–Santaó functional.

1 Introduction

The Blaschke–Santaló inequality, see [4, 36], states that every 0-symmetric convex bodyK in Rn satisfies

voln(K)voln(K◦) ≤ (voln(Bn2 ))2,

where K◦ = {y ∈ Rn : 〈x, y〉 ≤ 1 ∀x ∈ K} is the polar body of K, Bn2 = {x ∈ Rn : |x| ≤ 1}is the Euclidean unit ball and | · | denotes the Euclidean norm on Rn. The left-hand sideof this inequality is called the Mahler volume. The sharp lower bound for the Mahlervolume is still open in dimensions 4 and higher. The famous Mahler conjecture suggeststhat this functional is minimized by the couple (Bn1 , B

n∞). Partial results can be found in

e.g., [23, 27, 33, 34].Here we ask: What is a natural generalization of the bounds for the Mahler volume

for multiple sets? While this is not obvious from the geometric viewpoint, we suggest inthis paper a reasonable extension, which is naturally related to a functional counterpartof the Blaschke–Santaló inequality.

The functional Blaschke–Santaló inequality was discovered by K. Ball [6] and laterextended and generalized in [3], [20], [30]. In its simplest form it states that for every twomeasurable even functions V,W on Rn we have that

∫

e−V (x)dx

∫

e−W (y)dy ≤ (2π)n,

provided that V (x) +W (y) ≥ 〈x, y〉. Equality is attained if and only if V (x) = |x|2

2 +

c,W (y) = |y|2

2 − c, where c is a constant. An interesting link to optimal transportationtheory was discovered recently by M. Fathi [17]. He showed that for probability measures

∗Keywords: Blaschke–Santaló inequality, optimal transport, multimarginal Monge–Kantorovitch prob-

lem, geodesic barycenters of measures, Prekopa-Leindler inequality, Kähler–Einstein equation. 2020 Math-

ematics Subject Classification: 52A20, 52A40, 60B.†Supported by RFBR project 20-01-00432; Section 7 results have been obtained with support of RSF

grant No 19-71-30020. The article was prepared within the framework of the HSE University Basic Research

Program‡Partially supported by NSF grant DMS-1811146 and by a Simons Fellowship

1

http://arxiv.org/abs/2010.00135v3

µ = f · γ, ν = g · γ, where γ is the standard Gaussian measure, such that∫

xfdγ = 0, thefollowing inequality holds,

1

2W 22 (µ, ν) ≤ Entγ(µ) + Entγ(ν) (1.1)

and that this inequality is equivalent to the functional Blaschke–Santaló inequality. Here,W 22 (µ, ν) is the L

2 Kantorovitch distance (see Section 2 for the definition) and

Entγ(µ) =

∫

f log fdγ

is the relative entropy with respect to Gaussian measure. Inequality (1.1) is a remarkablestrengthening of the Talagrand transportation inequality and the starting point of ourpaper. We refer to, e.g., [5] for Talagrand’s inequality and it’s fundamental importancein probability theory. In this context, please also note a very recent result of N. Gozlanabout a transportational approach to the lower bound for the Blaschke–Santaló functional[26].

We would like to point out an important connection of the Blaschke–Santaló inequalityto the Kähler–Einstein equation. Fathi’s result establishes, in particular, that the func-tional f → 12W 22 (µ, ν)−Entγ(µ) is bounded from below. The minimum of this functionalsolves the so-called Kähler–Einstein equation. This was established by F. Santambrogio[35]. The form of the functional presented here was considered in [29]. The well-posednessof the Kähler–Einstein equation was proved by D. Cordero-Erausquin and B. Klartag [14].Generalization to the sphere and relations to the logarithmic Minkowski problem wereestablished in [28]. Other related transportation inequalities can be found in [19].

To analyze the case of k functions with k > 2 we consider the cost functional

c(x1, · · · , xn) =∑

i,j=1,i

1.1 The main results

In Section 2 we discuss some preliminary facts about Kantorovich duality theory for manyfunctions (Legendre-type transforms) and prove that our integral functional is boundedfor the case of quadratic cost (1.2). We also show that for k > 2 our functional has atrivial (zero) lower bound, unlike the case of two functions.

In Section 3 we verify the above conjecture in the unconditional case (see Section 3 forthe definition) and prove the following theorem.

Theorem 3.1 Let fi : Rn → R+, 1 ≤ i ≤ k, be unconditional functions satisfying

k∏

i=1

fi(xi) ≤ ρ(

k∑

i,j=1i

In particular, if ρ(t) = e−t

k−1 , then, if∑k

i=1,i

Proposition 2.1. Let Vi, 1 ≤ i ≤ k, be a family of Borel functions on Rn. Then thefunctional

S(V1, · · · , Vk) =k∏

i=1

∫

e−Vi(xi) dxi

is bounded on the set

Ln,k ={

(V1, · · · , Vk) : Vi is even ∀i ∈ {1, · · · , k},k∑

i=1

Vi(xi) ≥k∑

i,j=1,i

Finally, we show that S is bounded. We observe that for every j 6= m

Vm(xm) ≥ supxi,i 6=m

(

k∑

i,j=1,i

At the end of this section we recall basic facts on duality relations for the transportationcost appearing in the theory of barycenters of measures. Recall that for a given familyof probability measures µ1, · · · , µk and weights λi ∈ [0, 1] satisfying

∑ki=1 λi = 1 its

barycenter µ is the minimum point of the functional

F(ν) = 12

k∑

i=1

λiW22 (µi, ν).

Here,

W 22 (ν1, ν2) = inf

{∫

|x− y|2dP : P ∈ P(Rn × Rn), P (·,Rn) = ν1, P (Rn, ·) = ν2}

is the L2 Kantorovitch distance of probability measures ν1, ν2. It is well-known that thebarycenter problem is closely related to the multimarginal (maximization) Kantorovichproblem with the cost function

k∑

i,j=1,i 6=j

λiλj〈xi, xj〉

and marginals µi. Let π be the solution to this problem, i.e. a measure that gives amaximum to the functional

P →∫ k∑

i,j,i 6=j

λiλj〈xi, xj〉dP (2.1)

among the measures on (Rn)k having µ1, . . . µk as marginals.

The following facts are collected from [1] and [24].

Theorem 2.4. [1], [24] Assume that µi are absolutely continuous measures with finitesecond moments and λi ∈ (0, 1) are numbers satisfying

∑ki=1 λi = 1. Then the following

facts hold.

1. There exists a unique absolutely continuous solution µ to the barycenter problem anda unique solution π to the problem (2.1).

2. The measure µ is the push-forward measure of π under the mapping T (x1, · · · , xk) =∑k

i=1 λixi and the following relation holds:

k∑

i=1

λiW22 (µi, µ) =

∫ k∑

i=1

|xi − T (x)|2dπ.

3. The optimal transportation mappings ∇Φi of µ onto µi satisfy

k∑

i=1

λi∇Φi(x) = x

for µ-a.e. x. and π is supported on the set (∇Φ1(x), · · · ,∇Φk(x)).

7

4. There exists a unique (up to addition of constants and sets of measure zero) dualsolution vi to the problem (2.1), i.e. a k-tuple of functions satisfying

k∑

i=1

vi(xi) ≥k∑

i,j,i 6=j

λiλj〈xi, xj〉

with equality π-a.e. The following relation holds between vi and Φi:

Φ∗i (xi) = λi|xi|22

+vi(xi)

λi+ Ci (2.2)

for µi-almost all xi.

Remark 2.5. The results of item 1. are obtained in Section 3 of [1], item 2. is containedin Proposition 4.2 [1], item 3. corresponds to Proposition 3.8 of [1]. Formula (2.2) needssome explanations. It corresponds to formula (4.8) in [1], but in the presentation in [1]there is no direct link to the optimal transportation of the barycenter µ onto µi. Let usgive some informal explanations.By the Kantorovich duality π is concentrated of the zero set of the positive function

k∑

i=1

vi(xi)−k∑

i,j,i 6=j

λiλj〈xi, xj〉.

Thus for π-a.e. (x1, · · · , xk) and all 1 ≤ i ≤ k one has ∇vi(xi) =∑k

j 6=i λiλjxj. Equiva-lently,

λixi +∇vi(xi)λi

=

k∑

j=1

λjxj, (2.3)

π-a.e. It remains to note that µ is the image of π under T =∑k

j=1 λjxj and µi is theprojection of π onto the i-th factor. Thus relation (2.3) immediately implies that µ is the

image of µi under the mapping xi → λixi + ∇vi(xi)λi . Since the latter is the gradient of theconvex function λi

|xi|2

2 +vi(xi)λi

, we conclude by uniqueness of the optimal transportaion

mapping that ∇Φ∗i = λixi +∇vi(xi)λi

.

3 The unconditional case

In this section we verify our conjecture (inequality part) for the unconditional functions.A function f : Rn → R is called unconditional, if

f(ε1x1, · · · , εnxn) = f(x1, x2, . . . , xn),

for every (ε1, · · · , εn) ∈ {−1, 1}n and every (x1, · · · , xn) ∈ Rn. We show that the conjec-ture is true in the unconditional case.

Theorem 3.1. Let fi : Rn → R+, 1 ≤ i ≤ k, be measurable unconditional functions

satisfyingk∏

i=1

fi(xi) ≤ ρ

k∑

i,j=1,i

where ρ is a positive non-increasing function such that∫

Rρ

1k (t2)dt

below. We get

(

k∏

i=1

∫

Rn+

fi(xi)dxi

)1k ≤

∫

Rn

supt= 1

k

∑ki=1 ti

k∏

i=1

(

f1k

i (eti)e

1k

∑nm=1(ti)m

)

dt

≤∫

Rn

supt= 1

k

∑ki=1 ti

[

ρ1k

(

k∑

i,j=1,i

and equality holds for λ 6= 0, if and only if there exists a ∈ R and a positive definite matrixA such that V (x) = 〈Ax, x〉 + a, for every x ∈ Rn. For λ = 0, equality holds trivially.

Remark. Theorem 3.2 (iii) is just a special case for λ = 12 of a more general statementproved in [12].

We then get the following Proposition.

Proposition 3.3. Let Vi : Rn → R∪∞, 1 ≤ i ≤ k, be convex unconditional functions and

let ρ is a positive non-increasing function such that∫

Rρ

1k (t2)dt

Proof. (i) We get immediately from Theorem 3.1 and inequality (3.3) that for λ ∈ [0, 12 ],

k∏

i=1

asλ(Vi) ≤ (2π)knλ(

k∏

i=1

∫

e−Vi

)k(1−2λ)

≤ (2π)knλ(∫

Rn

ρ1k

(k(k − 1)2

|u|2)

du

)k(1−2λ)

.

If ρ(t) = e−t

k−1 , then

(2π)knλ(∫

Rn

ρ1k

(k(k − 1)2

|u|2)

du

)k(1−2λ)

= (2π)kn2 =

(

asλ

( | · |2

))k

,

which shows the second part of (i).

(ii) We use Theorem 3.2 (iii) and (iv) and Theorem 3.1 and get that for λ ∈ [12 , 1],

k∏

i=1

asλ(Vi) =

k∏

i=1

as1−λ(V∗i ) ≤ (2π)kn(1−λ)

(

k∏

i=1

∫

e−V∗i

)k(2λ−1)

≤ (2π)kn(1−λ)(∫

Rn

ρ1k

(k(k − 1)2

|u|2)

du

)k(2λ−1)

.

The second part for ρ(t) = e−t

k−1 follows.

The equality characterizations follow immediately from the results in the next section.

Remark 3.4. (i) Please note that for λ = 0, inequalities (3.4) and (3.5) are just theinequalities of Theorem 3.1. For λ = 12 , we do not need that the Vi are unconditional andthe inequalities are just the inequalities of (3.3),

k∏

i=1

as 12(Vi) ≤ (2π)knλ.

See also Section 8 for more on as 12.

(ii) For λ > 1, we get an estimate from below with an absolute constant c, see [12],

k∏

i=1

asλ(Vi) ≥ cknλ(∫

Rn

ρ1k

(k(k − 1)2

|u|2)

du

)k(1−2λ)

.

4 Characterization of the equality cases

In the proof of Theorem 3.1 we have used the Prékopa–Leindler inequality which is aparticular case of the more general Brascamp–Lieb inequality (see [10], [7]). To analyzethe equality case we need the equality characterizations of the Prékopa–Leindler inequality.We could not find those in the literature, except in the case of two functions, establishedby Dubuc [16]. We therefore give a proof of the equality characterization.

Theorem 4.1 (Prékopa–Leindler). Let fi, 1 ≤ i ≤ k, and h be nonnegative integrable realfunctions on Rn such that for all xi, for all λi, 1 ≤ λ ≤ k, with

∑ki=1 λ1 = 1,

h

(

k∑

i=1

λixi

)

≥k∏

i=1

fλii (xi).

12

Thenk∏

i=1

(

∫

Rn

fidxi

)λi ≤∫

Rn

hdx. (4.1)

Equality holds in the Prekopa–Leindler inequality if and only if there exist vectors y1, · · · , yksuch that for all x

f1(x− y1)∫

Rnf1dx

=f2(x− y2)∫

Rnf2dx

= · · · fk(x− yk)∫Rnfkdx

= e−ψ(x), (4.2)

where ψ is a convex function such that∫

R⋉e−ψ(x)dx = 1 and

h(x) = supx=

∑ki=1 λi xi

k∏

i=1

fλii (xi) =k∏

i=1

(

∫

Rn

fidxi

)λie−ψ

(

x+∑k

i=1 λiyi

)

.

Proof. It is clear that equality holds in inequality (4.1), if the functions satisfy the condi-tion (4.2).The proof of the inequality is well known and can be found in e.g., [25, 36]. We give aproof of the inequality by induction on the number of functions. This allows to establishthe equality characterizations, as for two functions, those were established by Dubuc [16].We have

supx=

∑ki=1 λi xi

k∏

i=1

fλii (xi) = supλ1x1+(1−λ1) y

fλ11 (x1)g1−λ1(y),

where

g(y) = supy= 1

1−λ1

∑ki=2 λi xi

k∏

i=2

f

λi1−λ1i (xi).

Applying the Prékopa–Leindler inequality for two functions gives

∫

supx=

∑ki=1 λi xi

k∏

i=1

fλii (xi) ≥(∫

f1dx1

)λ1 (∫

gdy

)1−λ1

.

Applying the induction step, one gets

∫

gdy ≥k∏

i=2

(∫

fi(xi)dxi

)

λi1−λ1

.

This completes the proof of the inequality. The equality characterization follows from theequality characterization for two functions.

Theorem 4.2. Let fi : Rn → R+, 1 ≤ i ≤ k, be measurable unconditional functions

satisfyingk∏

i=1

fi(xi) ≤ ρ

k∑

i,j=1,i

if and only if for all 1 ≤ i ≤ k, fi is log-concave and for almost all x ∈ Rn,

fi(x) = ρ1k

(

k(k − 1)2

|x|2)

.

In particular, ρ is log-concave.

Proof. Suppose that all fi are log-concave and a.e. equal to ρ1k

(

k(k−1)2 |x|2

)

. Then it is

obvious that equality holds in (3.1). Moreover, as then ρ is also log-concave,

k∏

i=1

fi(xi) =k∏

i=1

ρ1k

(

k − 12

|xi|2)

≤ ρ(

k − 12

k∑

i=1

|xi|2)

≤ ρ

k∑

i,j=1,i

The last identity holds as yi + zi = x for all i. Therefore, with (4.4),

e−ψ(x) =f1(x− y0)∫

Rnf1dx

=f2(x− y0)∫

Rnf2dx

= · · · fk(x− y0)∫Rnfkdx

=ρ

1k

(

k(k−1)2 |x− y0|2

)

∫

Rnρ

1k

(

k(k−1)2 |x|2

)

dx

or

fi(x)∫

Rnfidx

=ρ

1k

(

k(k−1)2 |x|2

)

∫

Rnρ

1k

(

k(k−1)2 |x|2

)

dx,

for all i.

5 The Blaschke–Santaló inequality and affine isoperimetric

inequality for many sets

The classical Blaschke–Santaló inequality for symmetric sets can be stated in the followingway,

∫

Sn−1fndx

∫

Sn−1gndy ≤ n2 (voln(Bn2 ))2 =

(

voln−1(Sn−1)

)2,

where f, g are positive symmetric functions on Sn−1 satisfying

f(x)g(y) ≤ 1〈x, y〉+,

and where for a ∈ R, a+ = max{a, 0}. The latter inequality is satisfied, in particular, if

f(x) = rK(x), g(y) =1

hK(y)= rK◦(y),

where rK(x) = max{λ ≥ 0 : λx ∈ K} is the radial function of the convex body K,hK(y) = sup{〈x, y〉 : x ∈ K} is the support function of K and where for a 0-symmetricconvex body K,

K◦ = {y ∈ Rn : 〈y, x〉 ≤ 1∀x ∈ K}is the polar body of K. We can then write the above as follows,

voln(K1) voln(K2) ≤ (voln(Bn2 ))2,

provided〈x, y〉 ≤ 1, ∀x ∈ K1, ∀y ∈ K2. (5.1)

We now prove a Blaschke–Santaló inequality for multiple sets. We recall that a subset Kin Rn is unconditional if its characteristic function 1K is unconditional.

Theorem 5.1. Let Ki, 1 ≤ i ≤ k, be unconditional convex bodies in Rn such that

k∏

i=1

e− 1

2‖xi‖

2Ki ≤ ρ

k∑

i=1,i

where ρ is a positive non-increasing function such that∫

Rρ

1k (t2)dt

where µK the usual surface area measure on ∂K, the boundary of K, N(x) is the outerunit normal at x ∈ ∂K and κ(x) is the generalized Gauss curvature at x ∈ ∂K. Note thatas0(K) = n voln(K), and if K is C

2+, then as±∞(K) = n voln(K

◦).The Lp-affine isoperimetric inequalities state that for 0 ≤ p ≤ ∞,

asp(K)

asp(Bn2 )

≤(

voln(K)

voln(Bn2 )

)n−pn+p

(5.3)

and for −n < p ≤ 0,asp(K)

asp(Bn2 )

≥(

voln(K)

voln(Bn2 )

)n−pn+p

.

Equality holds trivially if p = 0. In both cases equality holds for p 6= 0 if and only if K isan ellipsoid. If −∞ ≤ p < −n and K is C2+, then

cnpn+p

(

voln(K)

voln(Bn2 )

)n−pn+p

≤ asp(K)asp(Bn2 )

. (5.4)

These inequalities were proved by Lutwak [32] for p > 1 and for all other p by Werner andYe [39]. The case p = 1 is the classical case.

Theorem 5.1 leads to a multi-set “affine” isoperimetric inequality.

Proposition 5.2. Let Ki, 1 ≤ i ≤ k, be unconditional convex bodies in Rn such that

k∏

i=1

e− 1

2‖xi‖

2Ki ≤ ρ

k∑

i=1,i

Proof. Let 0 ≤ p ≤ n. By the affine isoperimetric inequality and Theorem 5.1 we getk∏

i=1

asp(Ki) ≤ (asp(Bn2 ))kk∏

i=1

(

voln(Ki)

voln(Bn2 )

)n−pn+p

≤ (asp(Bn2 ))k(

1

(2π)n2

∫

Rn

ρ1k

(k(k − 1)2

|x|2)

dx

)k

The first inequality shows that for p = n,

k∏

i=1

asn(Ki) ≤ (asn(Bn2 ))k .

If ρ(t) = e−t

k−1 , then we have for all 0 ≤ p ≤ n,k∏

i=1

asp(Ki) ≤ (asp(Bn2 ))kk∏

i=1

(

voln(Ki)

voln(Bn2 )

)n−pn+p

≤ (asp(Bn2 ))k

The equality characterizations follow from Section 4 and the equality characterizations ofthe above affine isoperimetric inequalities.

Remark 5.3. (i) For p = n, the inequality is just the affine isoperimetric inequality (5.3).As as0(K) = nvoln(K), the inequalities of the theorem for p = 0 are just the inequalitiesof Theorem 5.1.

(ii) The corresponding inequalities for −∞ ≤ p < −n also hold, using (5.4).(iii) It was shown in [32, 37] that for all p 6= −n and for all invertible linear transforma-tions T : Rn → Rn

asp(T (K)) = |det(T )|n−pn+p asp(K). (5.7)

For p = 1, (5.7) even holds for affine T . Therefore we get that for all invertible lineartransformations Ti : R

n → Rn,k∏

i=1

asp(Ti(Ki)) =

k∏

i=1

|det(Ti)|n−pn+p asp(Ki).

Hence∏ki=1 asp(Ki) and thus inequality (5.5) are invariant under linear transformations

with determinant 1 and for p = 1 under affine transformations with determinant 1 andfor p = n invariant under invertible linear transformations.

A further multiple set version of the Blaschke–Santaló inequality is given in the nextproposition.

Proposition 5.4. Let Ki, 1 ≤ i ≤ k, be unconditional convex bodies in Rn with radialfunctions ri = rKi. Assume that for all xi = ((xi)1, · · · , (xi)n) ∈ Rn,

k∏

i=1

ri(xi) ≤1

(

∑nj=1

(

|(x1)j|1k · · · |(xk)j |

1k

)2)

k2

. (5.8)

Thenk∏

i=1

voln(Ki) ≤ (voln(Bn2 ))k .

Equality holds if and only if Ki is a Euclidean ball for all 1 ≤ i ≤ k.

18

Proof. Let m ∈ R, 1 ≤ m < n and put xi = eti . Set w = 1k∑k

i=1 ti. Then

k∏

i=1

rmi (eti)1{|eti |≤1} e

∑i,j(ti)j ≤

1{|ew|≤1} e∑

i,j(ti)j

(

∑nj=1 e

2wj)km

2

=1{|ew|≤1} e

k∑n

j=1(w)j

(

∑nj=1 e

2wj)km

2

. (5.9)

We now apply again the change of variables xi = eti , 1 ≤ i ≤ k, the Prékopa–Leindler

inequality and (5.9),

(

k∏

i=1

∫

Bn2 ∩Rn+

rmi dxi

)

1k

=

(

k∏

i=1

∫

Rn

rmi (eti)1{|eti |≤1} e

∑j(ti)j dti

)

1k

≤∫

Rn

supw= 1

k

∑ki=1 ti

[

k∏

i=1

rmk

i (eti)1{|eti |≤1} e

1k

∑i,j(ti)j

]

dw

≤∫

Rn

1{|ew|≤1} e∑n

j=1(w)j

(

∑nj=1 e

2wj)

m2

dw =

∫

Bn2 ∩Rn+

dx

|x|m .

Hence by symmetry(

k∏

i=1

∫

Bn2

rmi dxi

)

1k

≤∫

Bn2

dx

|x|m .

Next we observe that every radial function ri satisfies

ri(xi) = ri

(

xi

|xi|

)

1

|xi|.

For every 1 ≤ m < n, m ∈ R, we introduce the finite probability measure dµm =1Bn2

(u)∫Bn2

du|u|m

du|u|m . The inequality above can then be rewritten as follows,

k∏

i=1

∫

Bn2

rmi

(

xi

|xi|

)

dµm ≤ 1.

Since µm is rotational invariant, the above inequality can be rewritten as

k∏

i=1

∫

Sn−1rmi (θ) dσ(θ) ≤ σ(Sn−1)k, (5.10)

where σ is the (n − 1)-dimensional Hausdorff measure. Passing to the limit m → n andapplying the Fatou’s Lemma one gets that (5.10) holds for m = n. On the other hand,for m = n one has for all i

∫

Sn−1rni (θ) dσ(θ) = σ(S

n−1)voln(Ki)

voln(Bn2 ). (5.11)

From this we derive the desired estimate.Now we address the equality characterizations. If equality holds in the inequality, thenequality holds everywhere and we get with (5.10) for m = 1,

k∏

i=1

∫

Bn2 ∩Rn

ri dxi =k∏

i=1

∫

Sn−1ri(θ) dσ(θ) = σ(S

n−1)k. (5.12)

19

Equality holds in the Prékopa–Leindler inequality. Thus there exist a convex function ψand y1, · · · , yk such that for all xr1(x− y1)1Bn2 (x− y1)

∫

Rn∩Bn2r1dx

=r2(x− y2)1Bn2 (x− y2)

∫

Rn∩Bn2r2dx

= · · ·rk(x− yk)1Bn2 (x− yk)

∫

Rn∩Bn2rkdx

= e−ψ(x).

(5.13)As Ki is in particular 0-symmetric, we have that ri(x − yi) = 1‖x−yi‖Ki . We put Ri =∫

Bn2 ∩Rn ri dxi and thus we get for all i, j

Ri‖x− yi‖Ki1Bn2 (x− yj) = Rj‖x− yj‖Kj1Bn2 (x− yi).

We let x = yi in this inequality and get for all j 6= i,

0 = ‖yi − yj‖Kjor y1 = y2 = · · · = yk = y0. With z = x− y0, we get by (5.13) for all z ∈ Bn2 ,

r1(z)

R1=r2(z)

R2= · · · rk(z)

Rk= e−ψ(z+y0). (5.14)

With (5.12)

k∏

i=1

r1k

i (z) =

(

k∏

i=1

Ri

)

1k

e−1k

∑ki=1 ψ(z+y0) = σ(Sn−1)e−ψ(z+y0).

Now we use (5.8) and get that∏ki=1 r

1k

i (z) =1|z| . Hence we have for all i,

1

σ(Sn−1) |z| = e−ψ(z+y0) =

ri(z)

Ri,

which means that Ki =Ri

σ(Sn−1)Bn2 for all i.

6 Prékopa–Leindler and displacement convexity inequali-

ties: refinement of the transportational argument

In this section we recall the transportational arguments of F. Barthe [7] in his proof ofthe reverse Brascamp–Lieb inequality. We show that the use of barycenters gives certainrefinements of the Prekopa–Leindler inequality.

Let fi, 1 ≤ i ≤ k, be nonnegative integrable functions and λi ∈ [0, 1] be numbers suchthat

∑ki=1 λi = 1, and let dµ = p(x)dx be a probability measure. For every i, ∇Φi is the

optimal transportation mapping that pushes forward µ onto µi.

In what follows we apply the change of variables formula for the optimal transportationmapping. In that form it was established by R. McCann (see [38]),

p(x) =fi(∇Φi)∫

fidxidetD2aΦi(x),

where D2aΦi is the absolutely continuous part of the distributional Hessian D2Φi of Φi.

This formula holds almost everywhere with respect to Lebesgue measure. We will alsoapply below the following results

20

• The arithmetic-geometric mean inequality

k∏

i=1

(detAi)λi ≤ det

(

k∑

i=1

λiAi

)

,

where the Ai are symmetric nonnegative matrices, λi ≥ 0,∑k

i=1 λi = 1.

• The inequality between the distributional Hessian and its absolutely continuous part

D2aΦi ≤ D2Φi.

First, we get by the arithmetic-geometric mean inequality

p(x) =

k∏

i=1

(

fi(∇Φi)(x)∫

fidxidetD2aΦi(x)

)λi

≤k∏

i=1

(

fi(∇Φi)(x)∫

fidxi

)λi

det

(

k∑

i=1

λiD2aΦi(x)

)

≤ sup{yi:∑i λiyi=

∑i λi∇Φi(x)}

k∏

i=1

(

fi(yi)∫

fidxi

)λi

det

(

k∑

i=1

λiD2aΦi(x)

)

. (6.1)

In the proof of Barthe one fixes an arbitrary measure µ and integrates inequality (6.1).By the change of variables y =

∑

i λi∇Φi(x), we get the Prékopa–Leindler inequality

k∏

i=1

(∫

fidxi

)λi

≤∫

sup{yi:

∑i λiyi=y}

k∏

i=1

fλii (yi)dy.

If instead of an arbitrary measure µ, we apply this result to the barycenter of the µ′is, weobtain the following pointwise refinement of the Prékopa–Leindler inequality.

Theorem 6.1. (Pointwise Prékopa–Leindler inequality) Let µ be the barycenter ofthe µi with weights λi. Then it has a density p satisfying

k∏

i=1

(∫

fidxi

)λi

p(x) ≤ supx=

∑ki=1 λiyi

k∏

i=1

fλii (yi) (6.2)

Proof. By Theorem 2.4, 3. and the arithmetric-geometric mean inequality one has

k∏

i=1

(

detD2aΦi(x))λi

≤ det(

k∑

i=1

λiD2aΦi(x)

)

.

Since∑

i=1 λiΦi(x) =|x|2

2 for p(x) dx-almost all x, then

k∑

i=1

λiD2aΦi(x) ≤ D2

(

∑

i=1

λiΦi(x))

= I

p(x) dx-a.e. Using this inequality and inequality (6.1) one gets the result.

Let us rewrite (6.1) in terms of the standard Gaussian reference measure dγ = e−

|x|2

2

(2π)n2dx.

21

Corollary 6.2. Let fidxi = ρi · dγ be probability measures and dµ = ρ · dγ. Then µ-a.e.

ρ(x) e12

∑ki=1 λi|∇Φi(x)−x|

2 ≤k∏

i=1

ρλii (∇Φi). (6.3)

Proof. Applying the first inequality of (6.1) to fi = ρie−

|x|2

2

(2π)n2

and p = ρ e−

|x|2

2

(2π)n2, we get

ρ(x) e−|x|2

2 ≤k∏

i=1

ρλii (∇Φi)e−λi|∇Φi|

2

2 .

Also using Theorem 2.4, 3. we finally observe that

k∑

i=1

λi

( |∇Φi(x)|22

− |x|2

2

)

=

k∑

i=1

λi

( |∇Φi(x)|22

− |x|2

2− 〈∇Φi(x)− x, x〉

)

=1

2

k∑

i=1

λi|∇Φi(x)− x|2.

Integrating pointwise inequality (6.2) we get the Prékopa–Leindler inequality. Takinglogarithm of (6.3) and integrating we get the displacement convexity property of theGaussian entropy,

Entγ(µ) +1

2

k∑

i=1

λiW22 (µ, µi) ≤

k∑

i=1

λiEntγ(µi). (6.4)

This result was proved in [1].

Mimicking the arguments that were used in the proof of (6.1), leads to the following result.

Theorem 6.3. Let fi, 1 ≤ i ≤ k, be integrable functions satisfyingk∏

i=1

fλii (xi) ≤ g(

k∑

i=1

λixi

)

, (6.5)

where λi ∈ [0, 1],∑k

i=1 λi = 1 and g is a nonegative function. Then for ρdx-almost all x,

k∏

i=1

(∫

fidxi

)λi

ρ(x) ≤ g(x), (6.6)

where ρ(x)dx is the barycenter of the measures fi∫fidxi

dxi with weights λi.

Proof. Applying inequality (6.1) and the relation∑k

i=1 λi∇Φi(x) = x one immediatelygets

k∏

i=1

(

∫

fi(xi)dxi

)λiρ(x) ≤ sup

{yi:∑i λiyi=x}

k∏

i=1

fλii (yi) det(

k∑

i=1

λiD2Φi(x)

)

≤ g(x).

22

Remark 6.4. Assuming (6.5) and integrating (6.6) one gets the inequality

∏

i=1

(

∫

fidxi

)λi≤∫

g(x)dx,

which can be considered as a weak form of the Blaschke–Santaló functional inequality. Thisfollows, of course, directly from the Prekopa–Leindler inequality.

In particular, assuming that the functions Vi satisfy

∑

i=1

λiVi(xi) ≥1

2

∣

∣

∣

k∑

i=1

λixi

∣

∣

∣

2,

one gets(

k∏

i=1

∫

e−Vi(xi) dxi

)λiρ(x) ≤ e−

|x|2

2 .

Rewriting this inequality with respect to the Gaussian reference measure γ, one gets thefollowing equivalent formulation.

Corollary 6.5. Assume that the measurable functions Fi satisfy

k∑

i=1

λiFi(xi) ≤1

2

[

k∑

i=1

λi|xi|2 −∣

∣

∣

k∑

j=1

λjxj

∣

∣

∣

2]

.

Then(

k∏

i=1

∫

eFidγ)λi

p(x) ≤ 1,

where p · γ is the barycenter of eFi∫eFidγ

· γ.

7 Talagrand-type estimates for the barycenter functional

In this section we show that a weak form of the Blaschke–Santaló inequality is related todisplacement convexity property of the Gaussian entropy. The conjectured strong formof the Blaschke–Santaló inequality is equivalent to certain strong entropy-W2-bound, aparticular case of this bound for two functions was proved by M. Fathi in [17].

In what follows π denotes the solution to the multimarginal Kantorovich problem withmarginals µi. Note that

k∑

i=1

∣

∣xi −1

k

k∑

j=1

xi∣

∣

2=

1

k

k∑

i,j=1,i

Theorem 7.1. Assume that for 1 ≤ i ≤ k, µi = ρi · γ are probability measures and the ρiare unconditional. Then

F(µ) ≤ k − 1k2

k∑

i=1

∫

ρi log ρidγ =k − 1k2

k∑

i=1

Entγ(µi). (7.1)

Proof. By the Kantorovich duality (see e.g., [38]),

F(µ) = 12k2

∫

∑

i,j=1,i

Remark 7.2. This result is a generalization in the unconditional setting of a result ofM. Fathi [17] for two functions:Let ρ0, ρ1 be two Gaussian unconditional probability densities and ρ1/2 be the correspondingbarycenter. Then inequality 7.1 implies

1

2W 22 (ρ0 · γ, ρ1 · γ) = 2W 22 (ρ0 · γ, ρ1/2 · γ) =W 22 (ρ0 · γ, ρ1/2 · γ) +W 22 (ρ1 · γ, ρ1/2 · γ)

≤∫

ρ0 log ρ0dγ +

∫

ρ1 log ρ1dγ. (7.3)

This is a particular case of Fathi’s inequality.

Fathi has shown that in the class of symmetric functions inequality (7.3) is equivalentto a Blaschke–Santaló inequality involving two exponential functions. We follow his ap-proach in [17] to show that the inequality of Theorem 7.1 is also equivalent to a functionalBlaschke–Santaló for multiple exponential functions.

Indeed, letting ρ(t) = e−t

k−1 in Theorem 3.1, we get the following multifunctional Blaschke–Santaló inequality:Let fi : R

n → R+, 1 ≤ i ≤ k, be measurable unconditional functions such thatk∑

i=1

fi(xi) ≤ −1

k − 1

k∑

i,j=1,i

By the Kantorovitch duality, the left hand side of this inequality equals

−2kinfP

k∑

i,j=1,i

Theorem 7.6. Let µi = ρi ·γ be probability measures and fi(xi) be the solution to the dualmultimarginal problem with marginals µi and the cost function

12k

∑ki,j=1,i

Proposition 8.1. Let V be a strictly convex C2-function such that e−V , e−V∗are inte-

grable functions. Let ∇Ψ be the optimal transportation of e−V dx∫e−V dx

onto e−V ∗dx∫e−V ∗dx

. Then

BS(V ) ≤ J 2(Ψ) ≤ BS(Ψ).

Equivalently

∫

e−V dx

∫

e−V∗dx ≤

(

∫

e−12〈x,∇Ψ〉

√detD2Ψdx

)2≤∫

e−Ψdx

∫

e−Ψ∗dx.

Proof. The second inequality is just Theorem 3.2 (iii). To prove the first inequality, weapply the change of variables formula

e−V∫

e−V dx=e−V

∗(∇Ψ)

∫

e−V∗dx

detD2Ψ.

Then∫

e−12〈x,∇Ψ〉

√detD2Ψdx =

√

∫

e−V∗dx

∫

e−V dx

∫

eV ∗(∇Ψ)−V (x)−〈x,∇Ψ〉

2 dx.

The result follows from the inequality V ∗(∇Ψ) + V (x) ≥ 〈x,∇Ψ〉.

It can be easily seen from the proof that equality is attained if and only if V = Ψ+ a forsome constant a. Thus, within a certain appropriate class of functions, e.g., symmetric,

the maximum of the Blaschke–Santaló functional must satisfy that the measure e−Ψ∗

∫e−Ψ∗dx

is the push-forward measure of e−Ψ

∫e−Ψdx

under the mapping ∇Ψ. This means that Ψ solvesthe following Monge–Ampère equation

e−Ψ∫

e−Ψdx=e−Ψ

∗(∇Ψ)

∫

e−Ψ∗dx

detD2Ψ. (8.2)

It was shown in [13] that this equation admits the following family of solutions, providede−Ψ∫e−Ψdx

has logarithmic derivatives,

Ψ =〈Ax, x〉

2+ c,

where A is a positive definite matrix and c is a constant. These are exactly the maximizersof the Blaschke–Santaló functional.

Thus this observation suggests the following (so far heuristic) approach to the Blaschke–Santaló inequality. Let Ψ0 = V , and consider iterations Ψl, l ∈ N, where Ψl+1 is theoptimal transportation potential pushing forward e

−Ψldx∫e−Ψldx

onto e−Ψ∗

l dx∫e−Ψ∗

l dx. By Proposition

8.1, one gets an increasing sequence BS(Ψl), l ∈ N. From this one can try to extractconvergence of Ψl to a potential Ψ, which gives a maximum to the Blaschke–Santalófunctional. Then prove that Ψ solves (8.2), and by uniqueness deduce that Ψ is quadratic.

8.2 The multimarginal case

Next we generalize the previous result to the multimarginal case, k > 2.

28

Theorem 8.2. Assume that Vi(xi), 1 ≤ i ≤ k, are measurable functions satisfying

k∑

i=1

λiVi(xi) ≥ Ck∑

i 0 and λi ∈ (0, 1) with∑k

i=1 λi = 1.Let the tuple of functions λiUi(xi) be the solution to the dual multimarginal maximization

problem with marginals e−Vidxi∫e−Vidxi

and the cost function C∑k

i

Let us informally analyze the equality case. Clearly, in this case one has for almost all y,

∑

i=1

λiVi(∇Φi(y)) =∑

i=1

λiUi(∇Φi(y)).

Integrating over ρ dy we get that (λiVi) is a dual Kantorovich solution as well. Hence, byuniqueness of the dual solution

Vi = Ui + Ci,

k∑

i=1

Ci = 0.

In addition, one has for all i that

e−Ui(∇Φi)∫

e−UidxidetD2Φi = ρ,

or, equivalently,e−Ui

∫

e−Uidxi= ρ(∇Φ∗i ) detD2Φ∗i .

In particular, since (see Theorem 2.4)

Φ∗i (xi) = λi|xi|22

+Ui(xi)

C+ Ci,

every function Ui must satisfy

e−Ui∫

e−Uidxi= ρ

(∇Ui(xi)C

+ λixi

)

det

(

D2Ui

C+ λiI

)

. (8.3)

Thus, a maximizer of the Blaschke–Santaló inequality, if it exists, must satisfy the systemof equations (8.3), where every Ui is convex.

Remark 8.3. Equation (8.3) is an equation of the Kähler–Einstein type. We do notknow whether (8.3) admits a unique solution. The well posedness of the classical Kähler–Einstein equation

e−Φ∫

e−Φdx= ρ(∇Φ)detD2Φ

was proved under broad assumptions in [14].

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Alexander V. KolesnikovFaculty of Mathematics

National Research Institute Higher School of Economics

Moscow, Russian Federation

[email protected]

Elisabeth WernerDepartment of Mathematics Université de Lille 1

Case Western Reserve University UFR de Mathématique

Cleveland, Ohio 44106, U. S. A. 59655 Villeneuve d’Ascq, France

[email protected]

33

1 Introduction1.1 The main results

2 Integral bounds and facts about barycenters3 The unconditional case4 Characterization of the equality cases5 The Blaschke–Santaló inequality and affine isoperimetric inequality for many sets 6 Prékopa–Leindler and displacement convexity inequalities: refinement of the transportational argument7 Talagrand-type estimates for the barycenter functional8 Monotonicity of the Blaschke-Santaló functional8.1 The case of two functions8.2 The multimarginal case