An extension of Berwald's inequality and its relation to Zhang's inequality

Journal Pre-proof An extension of Berwald’s inequality and its relation to Zhang’s inequality

David Alonso-Gutiérrez, Julio Bernués, Bernardo González Merino

PII:

S0022-247X(20)30037-8

DOI:

https://doi.org/10.1016/j.jmaa.2020.123875

Reference:

YJMAA 123875

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

2 August 2019

Please cite this article as: D. Alonso-Gutiérrez et al., An extension of Berwald’s inequality and its relation to Zhang’s inequality, J. Math. Anal. Appl. (2020), 123875, doi: https://doi.org/10.1016/j.jmaa.2020.123875.

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AN EXTENSION OF BERWALD’S INEQUALITY AND ITS RELATION TO ZHANG’S INEQUALITY ´ ´ AND BERNARDO GONZALEZ ´ DAVID ALONSO-GUTIERREZ, JULIO BERNUES, MERINO

Abstract. In this note prove the following Berwald-type inequality, showing that for any integrable log-concave function f : Rn → [0, ∞) and any concave function h : L → [0, ∞), where L = {(x, t) ∈ Rn × [0, ∞) : f (x) ≥ e−t f ∞ }, then  1  p 1 p −t  p→ h (x, t)e dtdx Γ(1 + p) L e−t dtdx L is decreasing in p ∈ (−1, ∞), extending the range of p where the monotonicity is known to hold true. As an application of this extension, we will provide a new proof of a functional form of Zhang’s reverse Petty projection inequality, recently obtained in [ABG].

1. Introduction and notation n

Let K ⊆ R be a convex body, i.e., a compact, convex set with non-empty interior, and let us denote by Kn the set of all convex bodies in Rn and by |K| the Lebesgue measure of K. We will also denote by K0n the set of convex bodies containing the origin. It is well known that, as a consequence of H¨older’s inequality, for any integrable function f : K → [0, ∞) the function   p1  1 p p→ f (x) dx |K| K is increasing in p ∈ (0, ∞). A famous inequality proved by Berwald [Ber, Satz 7] (see also [AAGJV, Theorem 7.2] for a translation into English) provides a reverse H¨older’s inequality for Lp norms (p > 0) of concave functions defined on convex bodies. It states that for any K ∈ Kn and any concave function f : K → [0, ∞), one has   p1  (1)

p→

p+n n

|K|

f (x)p dx

K

is decreasing in p ∈ (0, ∞). Date: January 14, 2020. The first and second authors are partially supported by MINECO Project MTM2016-77710-P, DGA E26 17R and IUMA. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Regi´ on de Murcia, Spain, by Fundaci´ on S´ eneca, Science and Technology Agency of the Regi´ on de Murcia. The third author is partially supported by Fundaci´ on S´ eneca project 19901/GERM/15, Spain, by MINECO Project MTM201563699-P Spain and by MICINN Project PGC2018-094215-B-I00 Spain. 1

2

´ AND B. GONZALEZ ´ D. ALONSO, J. BERNUES,

A function f : Rn → [0, ∞) is called log-concave if, for every x, y ∈ Rn , 0 < λ < 1, f (λx + (1 − λ)y) ≥ (f (x))λ (f (y))1−λ . Throughout the paper, we will denote by F(Rn ) the set of all integrable log-concave functions in Rn . In the context of log-concave functions, the following version of Berwald’s inequality (1) on epigraphs of convex functions was proved in [AAGJV, Lemma 3.3]: “Let f ∈ F(Rn ) and let h : L → [0, ∞) be a continuous concave non-identically null function, where L = {(x, t) ∈ Rn+1 : f (x) ≥ e−t f ∞ } is the the epigraph of − log ff∞ . Then, the function  p1   1 p −t h (x, t)e dtdx (2) p→ Γ(1 + p) L e−t dtdx L is decreasing in p ∈ (0, ∞).” When providing a new proof of Zhang’s reverse Petty projection inequality, Gardner and Zhang [GZ] extended (1) to the larger range of values p > −1 (see [GZ, Theorem 5.1]). The first goal in this paper is to also extend (2) to the larger range of values p > −1. Theorem 1.1. Let f ∈ F(Rn ) and let h : L → [0, ∞) be a concave function, where L = {(x, t) ∈ Rn+1 : f (x) ≥ e−t f ∞ }. Then, the function  p1   1 p −t h (x, t)e dtdx p→ Γ(1 + p) L e−t dtdx L is decreasing in p ∈ (−1, ∞). For any K ∈ Kn , its polar projection body Π∗ (K) is the unit ball of the norm given by xΠ∗ (K) := |x||Px⊥ K|, x ∈ Rn , where Px⊥ K is the orthogonal projection of K onto the hyperplane orthogonal to x, | · | denotes (besides the Lebesgue measure in the suitable space) the Euclidean norm and  · K denotes the Minkowski functional of K, defined for every x ∈ Rn , as xK := inf{λ > 0 | x ∈ λK} ∈ [0, ∞]. It is a norm if and only if K is centrally symmetric. The expression |K|n−1 |Π∗ (K)| is affine invariant and its extremal convex bodies are well known: Petty’s projection inequality [P] states that the (affine class of the) n-dimensional Euclidean ball, B2n , is the only maximizer and Zhang’s inequality [Z1] proves that the (affine class of the) n-dimensional simplex Δn , is the only minimizer. That is, for any convex body K ⊆ Rn , 2n |B2n |n n−1 ∗ n−1 ∗ n n−1 ∗ n n = |Δ | |Π (Δ )| ≤ |K| |Π (K)| ≤ |B | |Π (B )| = . (3) n n 2 2 nn |B2n−1 |n In recent years, many relevant geometric inequalities have been extended to the general context of log-concave functions (see for instance [AKM], [KM], [C], or [HJM] and the references therein). Let us recall that Kn and K0n naturally embed into F(Rn ), via the natural injections K → χK

and

K → e−·K ,

where χK is the characteristic function of K. These and other basic facts on convex bodies and log-concave functions used in the paper can be found in [BGVV] and [AGM].

BERWALD’S INEQUALITY AND ITS APPLICATIONS

3

For any f ∈ F(Rn ), the polar projection body of f , denoted as Π∗ (f ), is the unit ball of the norm given by  ∞  ∞ −t |Px⊥ Kt (f )| e dt = 2f ∞ xΠ∗ (Kt (f )) e−t dt, xΠ∗ (f ) := 2|x|f ∞ 0

0

where Kt (f ) := {x ∈ Rn : f (x) ≥ e−t f ∞ }, t > 0 (see [AGJV]). In [ABG], an extension of Zhang’s inequality (i.e., the left hand side inequality in (3)) was proved in the settings of log-concave functions. Theorem 1.2. Let f ∈ F(Rn ). Then,   (4) min {f (y), f (x)} dydx ≤ 2n n!f n+1 |Π∗ (f )| . 1 Rn

Rn

Moreover, if f ∞ = f (0) then equality holds if and only if some n-dimensional simplex Δn containing the origin.

f (x) f ∞

= e−xΔn for

Observe that when f = e−·K for some convex body K ∈ K0n , then (4) recovers Zhang’s inequality. Our second goal here is to provide a new proof of the functional version of Zhang’s inequality (4) by using the extension of Berwald’s inequality given by Theorem 1.1, in a similar way as Gardner and Zhang [GZ] proved the geometrical version of Zhang’s inequality via their extension of Berwald’s inequality (1) to p > −1. A common feature in both proofs, the one given in [ABG] and the one in this paper, is the crucial role played by the functional form of the covariogram function gf associated to the function f ∈ F(Rn ). See [ABG] and its definition below. Recall that in the geometric setting the covariogram function of a convex body K is given by gK (x) = |K ∩ (x + K)|. Apart from this fact, the two proofs completely differ. We introduce further notation: S n−1 denotes the Euclidean unit sphere in Rn . If the origin is in the interior of a convex body K, the function ρK : S n−1 → [0, +∞) given by ρK (u) = sup{λ ≥ 0 | λu ∈ K} is the radial function of K. It extends to Rn \ {0} via tρK (tu) = ρK (u), for any t > 0, u ∈ S n−1 . Finally, for any function f ∈ F(Rn ) let gf be the covariogram functional of f , defined by  ∞ gf (x) := e−t |Kt (f ) ∩ (x + Kt (f ))|dt 0

(cf. [ABG]). The paper is organized as follows: Section 2 contains the aforementioned extension, Theorem 1.1, of the functional Berwald inequality to the larger range of values of p > −1. In Section 3 we recall the celebrated family (with parameter p > 0) of convex bodies associated to any log-concave function introduced by Ball in [B, pg. 74]. We also recall the properties of the covariogram functional of a log-concave function, proven in [ABG]. Another main ingredient in the proof in [GZ] is an expression that connects the covariogram function of a convex body K and Ball’s convex bodies. Such a connection can be extended to the functional form of the covariogram gf of a log-concave function and moreover, the polar projection body of f will appear as a limiting case of this new expression when the value of the parameter p tends to −1.

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´ AND B. GONZALEZ ´ D. ALONSO, J. BERNUES,

2. An extension of Berwald’s inequality In this section we will prove the aforementioned extension of Berwald’s inequality, see Theorem 1.1 above. We first state a 1-dimensional lemma that can be seen as a degenerate version of Theorem 1.1. Lemma 2.1. Let γ : [0, ∞) → [0, ∞) be a non-decreasing concave function and define   p1  ∞ 1 Φγ (p) = γ(r)p e−r dr , p > −1. Γ(1 + p) 0 Then Φγ (p) is decreasing in p in (−1, ∞). Furthermore, if there exist −1 < p1 < p2 such that Φγ (p1 ) = Φγ (p2 ), then γ is a linear function and Φγ is constant on (−1, ∞). Remark 1. As usual, we define Φγ (0) = limp→0 Φγ (p) which by straightforward computations (using L’Hˆ opital’s rule, interchanging the integral and the derivative |x=0 = −A, where A ≈ 0.577 is operations, and taking into account that ∂Γ(1+x) ∂x ∞ A the Euler-Mascheroni constant) yields Φγ (0) = e exp 0 log γ(r)e−r dr . Proof of Lemma 2.1. Fix 0 = p1 > −1 and write γ(r) = Φγ (p1 ) · r, r ≥ 0. For any p > −1,   p1  ∞ 1 p p −r Φγ (p) = Φγ (p1 ) r e dr = Φγ (p1 ). Γ(1 + p) 0 Therefore  ∞ 1 p1 p1 (5) 0 = Φγ (p1 ) − Φγ (p1 ) = (γ(r)p1 − γ(r)p1 )e−r dr, Γ(1 + p1 ) 0 or equivalently,  1 (γ(− log t)p1 − γ(− log t)p1 )dt = 0. 0

We first consider the case −1 < p1 < p2 < 0. Since the function γ is non-negative and concave and (5) holds, if γ is not identically equal to γ, i.e., γ is not linear, there exists a unique r0 ∈ (0, ∞) such that γ(r) > γ(r) if r ∈ (0, r0 ) and γ(r) < γ(r) if r ∈ (r0 , ∞). Denoting t0 = e−r0 , we have that γ(− log t) < γ(− log t) if t ∈ (0, t0 ) and γ(− log t) > γ(− log t) if t ∈ (t0 , 1). Now,  ∞ (γ(r)p2 − γ(r)p2 )e−r dr Γ(1 + p2 )(Φpγ2 (p2 ) − Φpγ2 (p2 )) = 0  1 = (γ(− log t)p2 − γ(− log t)p2 )dt 0 1 = (γ(− log t)p1 − γ(− log t)p1 )ψ(t)dt, 0

where

γ(− log t)p2 − γ(− log t)p2 . γ(− log t)p1 − γ(− log t)p1 p2 w(x) − w(y) is strictly decreasing Since w(x) = x p1 is strictly concave in (0, ∞), x−y p1 in x and y and, since γ(− log t) is non-decreasing and γ(− log t)p1 is strictly ψ(t) =

BERWALD’S INEQUALITY AND ITS APPLICATIONS

5

increasing in t, ψ(t) is strictly decreasing. Now, by the mean value theorem, there exist c1 ∈ (0, t0 ) and c2 ∈ (t0 , 1) such that  1 (γ(− log t)p1 − γ(− log t)p1 )ψ(t)dt  0 t0  1 p1 p1 = (γ(− log t) − γ(− log t) )ψ(t)dt + (γ(− log t)p1 − γ(− log t)p1 )ψ(t)dt 0 t0  t0  1 = ψ(c1 ) (γ(− log t)p1 − γ(− log t)p1 )dt + ψ(c2 ) (γ(− log t)p1 − γ(− log t)p1 )dt 0 t0  t0 p1 p1 = (ψ(c1 ) − ψ(c2 )) (γ(− log t) − γ(− log t) )dt > 0, 0

since ψ is strictly decreasing, γ(− log t) < γ(− log t) for t ∈ (0, t0 ) and p1 < 0. Therefore, if γ is not linear, Φγ (p2 ) < Φγ (p2 ) = Φγ (p1 ) = Φγ (p1 ). The case 0 < p1 < p2 follows analogously with straightforward changes (in this case, if γ is not linear w is strictly convex and ψ is strictly decreasing). The continuity of Φγ at 0 then implies that Φγ (p) is decreasing in p > −1. If Φγ (p1 ) = Φγ (p2 ) for some −1 < p1 < p2 , since Φγ (p) would not be strictly decreasing in [p1 , p2 ], then γ would be linear, thus concluding the case of equality.  Our next result is the aforementioned extension of [AAGJV, Lemma 3.3] to p ∈ (−1, ∞). Proof of Theorem 1.1. Consider the probability measure on Rn+1 given by e−t χL (x, t) dtdx. dμ(x, t) := −t e dtdx L Denote Cs (h) = {(x, t) ∈ L : h(x, t) ≥ s} and define the function Ih : [0, ∞) → [0, ∞) as  1 e−t dtdx = μ(Cs (h)). Ih (s) := −t e dtdx Cs (h) L Ih is non-increasing, Ih (0) = μ(L) = 1 and since h is concave, Ih is log-concave (see [AAGJV, Lemma 3.2]). if and only if Observe that (x, t) ∈ L if and only if x ∈ Kt (f ), which happens  ρKt (f ) (x) ≥ 1, and that, by Fubini’s theorem, define h1 : L → [0, ∞) as

∞

e−t dtdx =

L



h1 (x, t) := sup s ∈ [0, ∞) : Ih (s) >

0

1 ρnKt (f ) (x)

e−t |Kt (f )|dt. Now

.

h1 has two important properties: - h and h1 are equally distributed with respect to μ, that is Ih1 ≡ Ih . In order to prove this, notice that for every s ≥ 0, and every (x, t) ∈ L, we have that h1 (x, t) > s if and only if ρnKt (f ) (x) > Ih1(s) and so by Fubini’s theorem,   ∞ dt Ih1 (s) = = Ih (s). dμ(x, t) = e−t |Kt (f )|Ih (s) −t e dtdx Cs (h1 ) 0 L

´ AND B. GONZALEZ ´ D. ALONSO, J. BERNUES,

6

- h1 (rρKt (f ) (u)u, t) does not depend on t and u since for any r, t > 0, u ∈ S n−1 , h1 (rρKt (f ) (u)u, t) = sup {s ∈ [0, ∞) : Ih (s) > rn } := γ(r). Therefore, for any p > 0,  hp (x, t)dμ(x, t) =





 ∞ 1 χ{hp (x,t)≥r} drdμ(x, t) = Ih (r p )dr n 0 0 R∞  1 p Ih1 (r p )dr = h1 (x, t)dμ(x, t).

L

=

∞

L

0

By Fubini’s theorem and integrating in polar coordinates,    ∞ hp1 (x, t)e−t dxdt = e−t hp1 (x, t)dxdt L K (f ) 0  ρK (f ) (u)  ∞t  t n = n|B2 | e−t hp1 (ru, t)rn−1 drdσ(u)dt n−1 S 0 1 0 ∞ n −t = n|B2 | e γ p (r)ρnKt (f ) (u)rn−1 drdσ(u)dt n−1 S 0 0  ∞  1 = n e−t |Kt (f )| γ p (r)rn−1 drdt and so, since

∞ 0

0

−t

e |Kt (f )|dt = 

0

L

−t

e dtdx, 

p

1

h (x, t)dμ(x, t) = n L

γ p (r)rn−1 dr.

0

If p < 0 the same equality holds. Indeed, we have  ∞  ∞  1 p 1 dμ(x, t)dr = h (x, t)dμ(x, t) = χ (1 − Ih (r p ))dr p} {h(x,t)≤r L 0  0 ∞ Rn  1 p p = (1 − Ih1 (r ))dr = h1 (x, t)dμ(x, t) L

0

and we proceed as before. If p = 0 the equality is obviously true. Notice that since Ih is log-concave the function γ is non-increasing and for every r1 , r2 ∈ [0, 1], γ(r11−λ r2λ ) ≥ (1 − λ)γ(r1 ) + λγ(r2 ). If we denote γ1 (r) = γ(e−r/n ) the previous statement means that γ1 is nondecreasing and concave in [0, ∞) and we have  1  ∞  p p n−1 h (x, t)dμ(x, t) = n γ (r)r dr = γ1p (r)e−r dr. L

0

0

We can apply now Lemma 2.1 to the function γ1 and conclude that  Γ(1 + p) is non-decreasing in (−1, ∞).

1 L

 e−t dtdx

p

−t

 p1

h (x, t)e dtdx L



BERWALD’S INEQUALITY AND ITS APPLICATIONS

7

3. Proof of functional Zhang’s inequality In this section we will give the proof of the functional version of Zhang’s inequality (4). For any g ∈ F(Rn ) such that g(0) > 0 and p > 0, we will consider the following important family of convex bodies, which was introduced by K. Ball in [B, pg. 74]. We denote    ∞ g(0)  p (g) := x ∈ Rn : K . g(rx)rp−1 dr ≥ p 0  p (g) is given by It follows from the definition that the radial function of K  ∞ 1 prp−1 g(ru)dr. ρpK (g) (u) = p g(0) 0 Remark 2. It is well known (cf. [BGVV, Proposition 2.5.7]) that, for any g ∈ F(Rn ) such that g∞ = g(0) and 0 < p ≤ q, 1

Γ(1 + p) p    1 Kq (g) ⊆ Kp (g) ⊆ Kq (g). Γ(1 + q) q We will make use of the following well known relation (cf. [B]) between the  n (g) and the integral of g. Lebesgue measure of K Lemma 3.1 ([B]). Let g ∈ F(Rn ) be such that g(0) > 0. Then  1  g(x)dx. |Kn (g)| = g(0) Rn For any f ∈ F(Rn ), we collect below the properties of its covariogram functional gf , whose proof can be found in [ABG, Lemma 2.1]. Lemma 3.2. Let f ∈ F(Rn ). Then the function gf : Rn → R defined by  ∞ gf (x) = e−t |Kt (f ) ∩ (x + Kt (f ))|dt 0

verifies that



 gf (x) =

min Rn

f (y) f (y − x) , f ∞ f ∞

 dy

 ∞ e−t |Kt (f )|dt = is even, log-concave, 0 ∈ int(supp gf ) with gf ∞ = gf (0) = 0       f (y) f (x) f (x) dx > 0, and gf (x)dx = min , dydx. f ∞ f ∞ Rn f ∞ Rn Rn Rn In the particular case of gf as in Lemma 3.2, we can provide an alternative  p (gf ) in terms of its radial function that will allow us to obtain the definition for K polar projection body of f as a limiting case of this expression when p tends to −1. Lemma 3.3. Let f ∈ F(Rn ) and let gf : Rn → R be the function  ∞ gf (x) = e−t |Kt (f ) ∩ (x + Kt (f ))|dt. 0

Then, for any u ∈ S n−1 and p > 0,  ∞  1 −t ρpK (g ) (u) = e |Kt (f ) ∩ (y + u)|p+1 dydt, f (x) p f (p + 1) Rn f ∞ dx 0 Pu⊥ Kt (f )

´ AND B. GONZALEZ ´ D. ALONSO, J. BERNUES,

8

where u denotes the 1-dimensional subspace spanned by u. Remark 3. Notice that the right hand side in the equality above is defined for p > −1 and that, since (p + 1)Γ(1 + p) = Γ(2 + p), if p → −1+ then  ∞  uΠ∗ (f ) 1 −t e |Kt (f )∩(y + u)|p+1 dt → . f (x) 2f 1 (p + 1)Γ(1 + p) Rn f ∞ dx 0 Pu⊥ Kt (f ) Proof of Lemma 3.3. By Lemma 3.2, gf (0) > 0 and ρpK = = = = = =

(u) =  ∞ p rp−1 gf (ru)dr gf (0) 0  ∞ ∞ 1 p−1 pr e−t |Kt (f ) ∩ (ru + Kt (f ))|dtdr gf (0) 0 0  ∞  ρK (f )−K (f ) (u) t t 1 −t e prp−1 |Kt (f ) ∩ (ru + Kt (f ))|drdt gf (0) 0 0  ∞  ρK (f )−K (f ) (u)  t t 1 e−t prp−1 max{|Kt (f ) ∩ (y + u)| − r, 0}dydrdt gf (0) 0 Pu⊥ Kt (f ) 0  ∞   |Kt (f )∩(y+ u )| 1 −t e prp−1 (|Kt (f ) ∩ (y + u)| − r) drdydt gf (0) 0 Pu⊥ Kt (f ) 0  ∞  1 e−t |Kt (f ) ∩ (y + u)|p+1 dydt. (p + 1)gf (0) 0 Pu⊥ Kt (f )

p (gf )

 Proof of inequality (4). Let u ∈ S n−1 and consider a function h : L → [0, ∞) given by h(x, t) = |Kt (f ) ∩ {(x, t) + λu : λ ≥ 0}|, where L is the epigraph of − log ff∞ . Since L is convex, h is concave. For any p > −1 we have,  ∞   ∞ 1 −t p+1 e |Kt (f ) ∩ (y + u)| dxdt = e−t h(x, t)p dxdt. (p + 1) 0 Pu⊥ Kt (f ) Kt (f ) 0 Therefore, by Theorem 1.1, for every −1 < p < 0,  ∞  1 e−t |Kt (f ) ∩ (y + u)|p+1 dtdx ≤ (p + 1)Γ(1 + p) Rn ff∞ 0 Pu⊥ Kt (f )  ≤

1 (n + 1)n! Rn

 f f ∞

∞

e 0

−t

 np

 |Kt (f ) ∩ (y + u)|

n+1

Pu⊥ Kt (f )

dtdx

=

ρK n (gf ) (u)p p

n! n

.

Taking limit as p → −1 and by Lemma 3.3 we obtain 1

ρK n (gf ) (u) ≤ 2(n!) n f 1 ρΠ∗ (f ) (u), that is,  n (gf ) ⊆ 2(n!) n1 f 1 Π∗ (f ) . K Taking Lebesgue measure and using Lemmas 3.2 and 3.1 we obtain inequality (4). 

BERWALD’S INEQUALITY AND ITS APPLICATIONS

9

References ´rrez D., Artstein-Avidan S., Gonza ´ lez Merino B., Jime ´nez C.H., [AAGJV] Alonso-Gutie Villa R. Rogers-Shephard and local Loomis-Whitney type inequalities, Math. Annalen 374 (2019), pp. 1719–1771. ´rrez D., Gonza ´ lez Merino B., Jime ´nez, R. Villa C. H. John’s ellipsoid [AGJV] Alonso-Gutie and the integral ratio of a log-concave function, J. Geom. Anal. 28 (2) (2018), pp. 1182– 1201. ´rrez D., Bernue ´s J., Gonza ´ lez Merino B. Zhang’s inequality for log[ABG] Alonso-Gutie concave functions, Geometric Aspects of Functional Analysis - Israel Seminar (GAFA) 20172019”, Lecture Notes in Mathematics 2256. [AGM] Artstein-Avidan S., Giannopoulos A., Milman V. D. Asymptotic geometric analysis, Part I (Vol. 202). American Mathematical Soc, 2015. [AKM] Artstein-Avidan S., Klartag M., and Milman V.D The Santal´ o point of a function and a functional form of Santal´ o inequality, Mathematika 51 (2004),pp. 33–48. [B] Ball K. Logarithmically concave functions and sections of convex sets in Rn , Studia Math. 88 (1), (1988). pp. 69–84. [Ber] Berwald L. Verallgemeinerung eines Mittelwetsatzes von J. Favard, f¨ ur positive konkave Funktionen, Acta Math. 79 (1947), pp. 17–37. [BGVV] Brazitikos S., Giannopoulos A., Valettas P., Vritsiou B.-H. Geometry of isotropic convex bodies. Mathematical Surveys and Monographs, 196. American Mathematical Society, Providence, RI, (2014). [C] Colesanti, A. Log-Concave Functions, Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, 161. Springer, New York, 2017, pp. 487-524. [GZ] Gardner R.J., Zhang G. Affine inequalities and radial mean bodies, Amer. J. Math. 120, no.3 (1998), pp. 505–528. ´nez C. H., Montenegro M. Sharp affine Sobolev type inequalities via [HJM] Haddad J., Jime the Lp Busemann-Petty centroid inequality., J. Funct. Anal. 271, no.2 (2016), pp. 454–473. [KM] Klartag B., Milman V. Geometry of Log-concave Functions and Measures, Geom. Dedicata. 112 (1), (2005). pp. 169–182. [P] Petty C.M. Isoperimetric problems. Proceedings of the Conference on Convexity and Combinatorial Geometry, pp. 26–41. University of Oklahoma, Norman (1971). [Z1] Zhang G. Restricted chord projection and affine inequalities, Geom. Dedicata. 39, (1991), n.2, pp.213–222. ´ ´ lisis Matema ´ tico, Departamento de Matema ´ ticas, Facultad de Ciencias, Area de Ana Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza (Spain), IUMA Email address, (David Alonso-Guti´ errez): [email protected] ´ ´ lisis matema ´ tico, Departamento de matema ´ ticas, Facultad de Ciencias, Area de ana Universidad de Zaragoza, Pedro cerbuna 12, 50009 Zaragoza (Spain), IUMA Email address, (Julio Bernu´es): [email protected] ´ ctica de las Ciencias Matema ´ ticas y Sociales, Facultad de Departamento de Dida ´ n, Universidad de Murcia, Murcia, Spain Educacio Email address, Bernardo Gonz´ alez Merino: [email protected]