A note on an $L^p$-Brunn-Minkowski inequality for convex measures in the unconditional case

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A note on an L^p-Brunn-Minkowski inequality for convex measures in the unconditional case

Arnaud Marsiglietti

To cite this version:

Arnaud Marsiglietti. A note on anL^p-Brunn-Minkowski inequality for convex measures in the unconditional case. 2014. �hal-01081764�

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A note on an L^p-Brunn-Minkowski inequality for convex measures in the unconditional case

Arnaud Marsiglietti

Institute for Mathematics and its Applications, University of Minnesota arnaud.marsiglietti@ima.umn.edu

This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.

Abstract

We consider a differentL^p-Minkowski combination of compact sets in Rⁿ than the one introduced by Firey and we prove anL^p-Brunn- Minkowski inequality,p∈[0,1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the functiont7→µ(t¹^pA), p∈(0,1], for unconditional convex mea- suresµand unconditional convex body A in Rⁿ. We also prove that the (B)-conjecture for all uniform measures is equivalent to the (B)- conjecture for all log-concave measures, completing recent works by Saroglou.

Keywords: Brunn-Minkowski-Firey theory, L^p-Minkowski combination, convex body, convex measure, (B)-conjecture

1 Introduction

The Brunn-Minkowski inequality is a fundamental inequality in Mathemat- ics, which states that for every convex subset A, B ⊂ Rⁿ and for every λ∈[0,1], one has

|(1−λ)A+λB|ⁿ¹ ≥(1−λ)|A|ⁿ¹ +λ|B|ⁿ¹, (1) where

A+B ={a+b;a∈A, b∈B}

denotes theMinkowski sumofA andB and where| · | denotes the Lebesgue measure. The book by Schneider [21] and the survey by Gardner [14] fa- mously reference the Brunn-Minkowski inequality and its consequences.

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Several extensions of the Brunn-Minkowski inequality have been devel- oped during the last decades by establishing functional versions (seee.g.[16], [9], [10], [24]), by considering different measures (see e.g. [3], [4]), by gen- eralizing the Minkowski sum (seee.g.[11], [12], [13], [18], [19]), among others.

In this paper, we will combine these extensions to prove an L^p-Brunn- Minkowski inequality for a large class of measures, including the log-concave measures.

Firstly, let us consider measures other than the Lebesgue measure. Fol- lowing Borell [3], [4], we say that a Borel measure µ in Rⁿ is s-concave, s∈[−∞,+∞], if the inequality

µ((1−λ)A+λB)≥M_s^λ(µ(A), µ(B))

holds for everyλ∈[0,1]and for every compact subset A, B⊂Rⁿsuch that µ(A)µ(B)>0, whereM_s^λ(a, b) denotes thes-mean of the non-negative real numbersa, b with weightλ, defined as

M_s^λ(a, b) = ((1−λ)a^s+λb^s)¹^s if s /∈ {−∞,0,+∞},

M_−∞^λ (a, b) = min(a, b), M₀^λ(a, b) =a¹⁻^λb^λ,M₊^λ_∞(a, b) = max(a, b). Hence the Brunn-Minkowski inequality tells us that the Lebesgue measure inRⁿ is

1

n-concave.

As a consequence of the Hölder inequality, one has M_p^λ(a, b)≤M_q^λ(a, b) for every p ≤ q. Thus every s-concave measure is −∞-concave. The −∞- concave measures are also called convex measures.

For s ≤ _n¹, Borell showed that every measure µ, which is absolutely continuous with respect to then-dimensional Lebesgue measure, iss-concave if and only if its density is an α-concave function, with

α= s

1−sn ∈[−1

n,+∞], (2)

where a functionf :Rⁿ→R₊ is said to beα-concave, withα∈[−∞,+∞], if the inequality

f((1−λ)x+λy)≥M_α^λ(f(x), f(y))

holds for everyx, y∈Rⁿ such thatf(x)f(y)>0and for every λ∈[0,1].

Secondly, let us consider a generalization of the notion of the Minkowski sum introduced by Firey, which leads to anL^p-Brunn-Minkowski theory. For convex bodiesAandB inRⁿ(i.e.compact convex sets containing the origin in the interior), theL^p-Minkowski combination,p∈[−∞,+∞], of A andB with weightλ∈[0,1]is defined by

(1−λ)·A⊕_pλ·B ={x∈Rⁿ;hx, ui ≤M_p^λ(h_A(u), h_B(u)),∀u∈Sⁿ⁻¹},

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whereh_A denotes the support functionofA defined by hA(u) = max

x∈Ahx, ui, u∈Sⁿ⁻¹. Notice that for everyp≤q,

(1−λ)·A⊕pλ·B ⊂(1−λ)·A⊕qλ·B.

The support function is an important tool in Convex Geometry, having the property to determine a convex body and to be linear with respect to Minkowski sum and dilation:

A={x∈Rⁿ;hx, ui ≤hA(u),∀u∈Sⁿ⁻¹}, hA+B =hA+hB, hµA=µhA, for every convex bodyA, B inRⁿ and every scalar µ≥0. Thus,

(1−λ)·A⊕₁λ·B= (1−λ)A+λB.

In this paper, we consider a different L^p-Minkowski combination. Be- fore giving the definition, let us recall that a function f : Rⁿ → R is unconditional if there exists a basis (a₁,· · ·, a_n) of Rⁿ (the canonical basis in the sequel) such that for every x = Pn

i=1xiai ∈ Rⁿ and for every ε = (ε₁,· · ·, εn) ∈ {−1,1}ⁿ, one has f(Pn

i=1εixiai) = f(x). For p= (p₁,· · ·, p_n)∈[−∞,+∞]ⁿ,a= (a₁,· · ·, a_n)∈(R₊)ⁿ, b= (b₁,· · ·, b_n)∈ (R₊)ⁿ andλ∈[0,1], let us denote

(1−λ)a+pλb= (M_p^λ₁(a₁, b₁),· · ·, M_p^λ_n(an, bn))∈(R₊)ⁿ.

For non-empty subsetsA, B⊂Rⁿ, forp∈[−∞,+∞]ⁿand forλ∈[0,1], we define theL^p-Minkowski combination ofAandB with weight λ, denoted by (1−λ)·A+pλ·B, to be the unconditional subset (i.e.the indicator function is unconditional) such that

((1−λ)·A+pλ·B)∩(R₊)ⁿ={(1−λ)a+pλb; a∈A∩(R₊)ⁿ, b∈B∩(R₊)ⁿ}.

This definition is consistent with the well known fact that an unconditional set (or function) is entirely determined on the positive octant(R₊)ⁿ. More- over, thisL^p-Minkowski combination coincides with the classical Minkowski sum when p = (1,· · · ,1) and A, B are unconditional convex subsets of Rⁿ (see Proposition 2.1 below).

Using an extension of the Brunn-Minkowski inequality discovered by Uhrin [24], we prove the following result:

Theorem 1.1. Let p = (p1,· · ·, pn) ∈ [0,1]ⁿ and let α ∈ R such that α≥ − Pn

i=1p⁻_i ¹₋1

. Let µ be an unconditional measure in Rⁿ that has an α-concave density function with respect to the Lebesgue measure. Then, for every unconditional convex bodyA, B in Rⁿ and for everyλ∈[0,1],

µ((1−λ)·A+pλ·B)≥M_γ^λ(µ(A), µ(B)), (3) where γ = Pn

i=1p⁻_i ¹+α⁻¹₋1

.

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The case of the Lebesgue measure and p = (0,· · · ,0) is treated by Saroglou [21], answering a conjecture by Böröczky, Lutwak, Yang and Zhang [5]

in the unconditional case.

Conjecture 1.2(log-Brunn-Minkowski inequality [5]). LetA, Bbe symmetric convex bodies inRⁿ and let λ∈[0,1]. Then,

|(1−λ)·A⊕₀λ·B| ≥ |A|¹⁻^λ|B|^λ. (4) Useful links have been discovered by Saroglou [21], [22] between Conjec- ture 1.2 and the (B)-conjecture.

Conjecture 1.3((B)-conjecture [17], [8]). Letµbe a symmetric log-concave measure in Rⁿ and let A be a symmetric convex subset of Rⁿ. Then the function t7→µ(e^tA) is log-concave on R.

The (B)-conjecture was solved by Cordero-Erausquin, Fradelizi and Mau- rey [8] for the Gaussian measure and for the unconditional case. As a variant of the (B)-conjecture, one may study concavity properties of the function t 7→ µ(V(t)A) where V : R→ R₊ is a convex function. As a consequence of Theorem 1.1, we deduce concavity properties of the functiont7→µ(t¹^pA), p ∈(0,1], for every unconditional s-concave measure µ and every unconditional convex bodyA inRⁿ (see Proposition 2.4 below).

Saroglou [22] also proved that the log-Brunn-Minkowski inequality for the Lebesgue measure (inequality (4)) is equivalent to the log-Brunn-Minkowski inequality for all log-concave measures. We continue these kinds of equiva- lences by proving that the (B)-conjecture for all uniform measures is equivalent to the (B)-conjecture for all log-concave measures (see Proposition 3.1 below).

We also investigate functional versions of the (B)-conjecture, which may be read as follows:

Conjecture 1.4(Functional version of the (B)-conjecture). Letf, g:Rⁿ→ R₊ be even log-concave functions. Then the function

t7→

Z

Rⁿ

f(e⁻^tx)g(x) dx is log-concave on R.

We prove that Conjecture 1.4 is equivalent to Conjecture 1.3 (see Propo- sition 3.2 below).

Let us note that other developments in the use of the earlier mentioned extensions of the Brunn-Minkowski inequality have been recently made as well (seee.g. [2], [6], [7], [15]).

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The rest of the paper is organized as follows: In the next section, we prove Theorem 1.1 and we extend it to m sets, m ≥ 2. We also compare our L^p-Minkowski combination to the Firey combination and derive an L^p- Brunn-Minkowski inequality for the Firey combination. We then discuss the consequences of a variant of the (B)-conjecture, namely we deduce concavity properties of the function t 7→ µ(t¹^pA), p ∈ (0,1]. In Section 3, we prove that the (B)-conjecture for all uniform measures is equivalent to the (B)- conjecture for all log-concave measures, and we also prove that the (B)- conjecture is equivalent to its functional version Conjecture 1.4.

2 Proof of Theorem 1.1 and consequences

2.1 Proof of Theorem 1.1

Before proving Theorem 1.1, let us show that our L^p-Minkowski combination coincides with the classical Minkowski sum when p = (1,· · ·,1), for unconditional convex sets.

Proposition 2.1. Let A, B be unconditional convex subsets of Rⁿ and let λ∈[0,1]. Then,

(1−λ)·A+_e₁λ·B= (1−λ)A+λB, wheree1 = (1,· · · ,1).

Proof. Since the sets(1−λ)·A+_e₁λ·B and(1−λ)A+λBare unconditional, it is sufficient to prove that

((1−λ)·A+_e₁λ·B)∩(R₊)ⁿ= ((1−λ)A+λB)∩(R₊)ⁿ.

Letx∈((1−λ)A+λB)∩(R₊)ⁿ. There existsa= (a₁,· · · , an)∈A and b= (b1,· · · , bn)∈Bsuch thatx= (1−λ)a+λband for everyi∈ {1,· · · , n}, (1−λ)ai+λbi ∈R₊. Letε, η ∈ {−1,1}ⁿsuch that(ε₁a₁,· · ·, εnan)∈(R₊)ⁿ and (η₁b₁,· · · , ηnbn) ∈ (R₊)ⁿ. Notice that for every i ∈ {1,· · ·, n}, 0 ≤ (1−λ)ai +λbi ≤ (1−λ)εiai+ληibi. Since the sets A and B are convex and unconditional, it follows thatx∈(1−λ)(A∩(R₊)ⁿ) +λ(B∩(R₊)ⁿ) = ((1−λ)·A+_e₁λ·B)∩(R₊)ⁿ.

The other inclusion is clear due to the definition of the set(1−λ)·A+_e₁ λ·B.

Proof of Theorem 1.1. Let λ ∈ [0,1] and let A, B be unconditional convex bodies inRⁿ.

It has been shown by Uhrin [24] that iff, g, h: (R₊)ⁿ→R₊are bounded measurable functions such that for every x, y∈(R₊)ⁿ,h((1−λ)x+_pλy)≥ M_α^λ(f(x), g(y)), then

Z

(R₊)ⁿ

h(x) dx≥M_γ^λ Z

(R₊)ⁿ

f(x) dx, Z

(R₊)ⁿ

g(x) dx

! ,

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whereγ = Pn

i=1p⁻_i ¹+α⁻¹₋1

.

Let us denote byφthe density function ofµand let us seth= 1₍₁₋_λ)_·_A+_p_λ_·_Bφ, f = 1Aφ andg= 1Bφ. By assumption, the functionφ is unconditional and α-concave, henceφis non-increasing in each coordinate on the octant(R₊)ⁿ. Then for everyx, y∈(R₊)ⁿ, one has

φ((1−λ)x+pλy)≥φ((1−λ)x+λy)≥M_α^λ(φ(x), φ(y)).

Hence,

h((1−λ)x+pλy)≥M_α^λ(f(x), g(y)).

Thus we may apply the result mentioned at the beginning of the proof to obtain that

Z

(R₊)ⁿ

h(x) dx≥M_γ^λ Z

(R₊)ⁿ

f(x) dx, Z

(R₊)ⁿ

g(x) dx

! ,

whereγ = Pn

i=1p⁻_i ¹+α⁻¹₋1

. In other words, one has

µ(((1−λ)·A+pλ·B)∩(R₊)ⁿ)≥M_γ^λ(µ(A∩(R₊)ⁿ), µ(B∩(R₊)ⁿ)).

Since the sets(1−λ)·A+pλ·B,A andB are unconditional, it follows that µ((1−λ)·A+pλ·B)≥M_γ^λ(µ(A), µ(B)).

Remark. One may similarly define theL^p-Minkowski combination λ1·A1+p· · ·+pλm·Am,

for m convex bodiesA₁, . . . , Am⊂Rⁿ,m≥2, whereλ₁, . . . , λm∈[0,1]are such thatPm

i=1λi = 1, by extending the definition of the p-mean M_p^λ to m non-negative numbers. By induction, one has under the same assumptions of Theorem 1.1 that

µ(λ₁·A₁+p· · ·+pλm·Am)≥M_γ^λ(µ(A₁),· · · , µ(Am)), (5) whereγ = Pn

i=1p⁻_i ¹+α⁻¹₋1

. Indeed, let m≥2 and let us assume that inequality (5) holds. Notice that

λ₁·A₁+p· · ·+pλm·Am+pλ_m+1·A_m+1 = Xm

i=1

λi

!

·Ae+pλ_m+1·A_m+1,

where

Ae:=

λ₁ Pm

i=1λi

·A₁+_p· · ·+_p λ_m Pm

i=1λi

·A_m

.

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Thus, µ

_m X

i=1

λi

!

·Ae+pλm+1·Am+1

!

≥

_m X

i=1

λi

!

µ(A)e^γ+λm+1µ(Am+1)^γ

!¹

γ

≥

m+1X

i=1

λiµ(Ai)^γ

!_γ¹ .

2.2 Consequences

The following result compares the L^p-Minkowski combinations⊕p and+p. Lemma 2.2. Let p ∈ [0,1] and set pe = (p,· · · , p) ∈ [0,1]ⁿ. For every unconditional convex bodyA, B in Rⁿ and for everyλ∈[0,1], one has

(1−λ)·A⊕_pλ·B ⊃(1−λ)·A+_p_eλ·B.

Proof. The casep = 0 is proved in [21]. Let p 6= 0. Since the sets(1−λ)· A⊕pλ·B and(1−λ)·A+_p_eλ·B are unconditional, it is sufficient to prove that

((1−λ)·A⊕pλ·B)∩(R₊)ⁿ⊃((1−λ)·A+_epλ·B)∩(R₊)ⁿ. Letu∈Sⁿ⁻¹∩(R₊)ⁿ and letx∈((1−λ)·A+_p_eλ·B)∩(R₊)ⁿ. One has,

hx, ui = Xn i=1

((1−λ)a^p_i +λb^p_i)¹^pui

= Xn i=1

((1−λ)(a_iu_i)^p+λ(b_iu_i)^p)¹^p

= k(1−λ)X+λYk

1 p 1 p

,

where X = ((a₁u₁)^p,· · ·,(anun)^p) and Y = ((b₁u₁)^p,· · · ,(bnun)^p). Notice thatkXk¹

p ≤hA(u)^p, kYk¹

p ≤ hB(u)^p and that k · k¹

p is a norm. It follows that

hx, ui ≤

(1−λ)kXk1

p +λkYk1 p

¹

p ≤((1−λ)hA(u)^p+λhB(u)^p)¹^p. Hence, x∈((1−λ)·A⊕pλ·B)∩(R₊)ⁿ.

From Lemma 2.2 and Theorem 1.1, one obtains the following result:

Corollary 2.3. Let p∈[0,1]. Letµbe an unconditional measure inRⁿthat has an α-concave density function, with α ≥ −_n^p. Then for every unconditional convex bodyA, B in Rⁿ and for everyλ∈[0,1],

µ((1−λ)·A⊕pλ·B)≥M_γ^λ(µ(A), µ(B)), (6) where γ =

n

p + _α¹₋1

.

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Remarks.

1. By takingα = 0 in Corollary 2.3 (corresponding to log-concave measures), one obtains

µ((1−λ)·A⊕₀λ·B)≥µ(A)¹⁻^λµ(B)^λ.

2. By takingα= +∞ in Corollary 2.3 (corresponding to _n¹-concave measures), one obtains that for everyp∈[0,1],

µ((1−λ)·A⊕pλ·B)ⁿ^p ≥(1−λ)µ(A)ⁿ^p +λµ(B)ⁿ^p.

Equivalently, for everyp∈[0,1], for every unconditional convex body A, B inRⁿ and for every unconditional convex setK⊂Rⁿ,

|((1−λ)·A⊕pλ·B)∩K|^pⁿ ≥(1−λ)|A∩K|^pⁿ +λ|B∩K|ⁿ^p. Let us recall that the function t7→µ(e^tA) is log-concave on Rfor every unconditional log-concave measureµ and every unconditional convex body AinRⁿ(see [8]). By adapting the argument of [20], Proof of Proposition 3.1 (see Proof of Corollary 2.5 below), it follows that the function t7→ µ(t¹^pA) is ^p_n-concave on R₊, for every p ∈ (0,1], for every unconditional s-concave measure µ, with s≥ 0, and for every unconditional convex body A in Rⁿ. However, no concavity properties are known for the function t 7→ µ(e^tA) whenµ is ans-concave measure with s <0. Instead, for these measures we prove concavity properties of the function t7→µ(t¹^pA).

Proposition 2.4. Letp∈(0,1], letµ be an unconditional measure that has an α-concave density function, with α ∈ [−^p_n,0) and let A be an unconditional convex body in Rⁿ. Then the function t 7→ µ(t¹^pA) is

n

p +_α¹₋1

- concave on R₊.

Proof. Lett₁, t₂ ∈R₊. By applying Corollary 2.3 to the sets t

1 p

1A and t

1 p

2A, one obtains

µ(((1−λ)t₁+λt₂)¹^pA) =µ((1−λ)·t

1 p

1A⊕pλ·t

1 p

2A)≥M_γ^λ(µ(t

1 p

1A), µ(t

1 p

2A)), where γ =

n

p +_α¹₋1

. Hence the function t 7→ µ(t¹^pA) is γ-concave on R₊.

As a consequence, we derive concavity properties for the function t 7→

µ(tA).

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Corollary 2.5. Letp∈(0,1], letµbe an unconditional measure that has an α-concave density function, with α∈[−_n^p,0), and let A be an unconditional convex body in Rⁿ. Then the function t 7→ µ(tA) is

1−p n +γ

-concave on R₊, where γ =

n

p +_α¹₋1

.

Proof. We adapt [20], Proof of Proposition 3.1. Let us denote by φ the density function of the measureµ and let us denote byF the functiont7→

µ(tA). From Proposition 2.4, the function t 7→ F(t¹^p) is γ-concave, hence the right derivative ofF, denoted byF₊^′ , exists everywhere and the function t7→ ¹_pt¹^p⁻¹F₊^′(t¹^p)F(t¹^p)^γ⁻¹ is non-increasing. Notice that

F(t) =tⁿ Z

A

φ(tx) dx,

and that t7→ φ(tx) is non-increasing, thus the function t 7→ _t1−p¹ F(t)^1−pⁿ is non-increasing. Since

F₊^′ (t)F(t)^1−pⁿ ^+γ⁻¹ =t¹⁻^pF₊^′ (t)F(t)^γ⁻¹· 1

t¹⁻^pF(t)^1−pⁿ ,

it follows that F₊^′(t)F(t)^1−pⁿ ^+γ⁻¹ is non-increasing as the product of two non-negative non-increasing functions. HenceF is

1−p n +γ

-concave.

Remark. For everys-concave measureµand every convex subsetA⊂Rⁿ, the functiont7→µ(tA) iss-concave. Hence Corollary 2.5 is of value only if

1−p

n +γ ≥ _1+αn^α (see relation (2)). Notice that this condition is satisfied if α≥ −_n(1+p)^p . We thus obtain the following corollary:

Corollary 2.6. Let p ∈ (0,1], let µ be an unconditional measure that has an α-concave density function, with −_n(1+p)^p ≤ α < 0 and let K be an unconditional convex body in Rⁿ. Then, for every subsets A, B ∈ {µK;µ >

0} and everyλ∈[0,1], one has

µ((1−λ)A+λB)≥M^λ1−p

n +γ(µ(A), µ(B)), where γ =

n

p + _α¹₋1

.

In [20], the author investigated improvements of concavity properties of convex measures under additional assumptions, such as symmetries. Notice that Corollary 2.6 follows the same path and completes the results that can be found in [20]. Let us conclude this section by the following remark, which concerns the question of the improvement of concavity properties of convex measures.

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Remark. Letµbe a Borel measure that has a density function with respect to the Lebesgue measure inRⁿ. One may write the density function of µin the forme⁻^V, where V :Rⁿ → R is a measurable function. Let us assume thatV isC². Letγ ∈R\{0}. The functione⁻^V isγ-concave ifHess(γe⁻^γV), the Hessian ofγe⁻^γV, is non-positive (in the sense of symmetric matrices).

One has

Hess(γe⁻^γV) =−γ²∇ ·(∇Ve⁻^γV) =γ²e⁻^V(γ∇V ⊗ ∇V −Hess(V)), where∇V ⊗ ∇V =

∂V

∂xi

∂V

∂xj

1≤i,j≤n. Hence the matrixHess(γe⁻^γV)is non- positive if and only if the matrixγ∇V ⊗ ∇V −Hess(V) is non-positive.

Let us apply this remark to the Gaussian measure dγ_n(x) = 1

(2π)ⁿ² e⁻^|x|

2

2 dx, x∈Rⁿ.

HereV(x) = ^|^x₂^|²+c_n, wherec_n= ⁿ₂ log(2π). Thus,∇V⊗∇V = (x_ix_j)₁_≤_i,j_≤_n andHess(V) =Idthe Identity matrix. Notice that the eigenvalues ofγ∇V⊗

∇V −Hess(V) are −1 (with multiplicity (n−1)) and γ|x|²−1. Hence if γ|x|²−1 ≤ 0, then γ∇V ⊗ ∇V −Hess(V) is non-positive. One deduces that for every γ > 0, for every compact sets A, B ⊂ √¹γB₂ⁿ and for every λ∈[0,1], one has

γn((1−λ)A+λB)≥M^λ^γ

1+γn(γn(A), γn(B)), (7) whereB₂ⁿ denotes the Euclidean closed unit ball inRⁿ.

Since the Gaussian measure is a log-concave measure, inequality (7) is an improvement of the concavity of the Gaussian measure when restricted to compact sets A, B⊂ √¹γB₂ⁿ.

3 Equivalence between (B)-conjecture-type prob- lems

In the following proposition, we demonstrate that it is sufficient to prove the (B)-conjecture for all uniform measures in Rⁿ, for every n∈ N∗, to obtain the (B)-conjecture for all symmetric log-concave measures in Rⁿ, for every n∈N^∗. This completes recent works by Saroglou [21], [22].

In the following, we say that a measure µ satisfies the (B)-property if the functiont7→µ(e^tA) is log-concave onRfor every symmetric convex set A⊂Rⁿ.

Proposition 3.1. If every symmetric uniform measure in Rⁿ, for every n∈N^∗, satisfies the (B)-property, then every symmetric log-concave measure in Rⁿ, for every n∈N∗, satisfies the (B)-property.