JOHN's ELLIPSOID and the INTEGRAL RATIO of a LOG-CONCAVE FUNCTION 1. Introduction and Notation Asymptotic Geometric Analysis I

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JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION

DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

Abstract. We extend the notion of John’s ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.

1. Introduction and notation Asymptotic geometric analysis is a rather new branch in mathematics, which comes from the interaction of convex geometry and local theory of Banach spaces. From its beginning, the research interests in this area have been focused in un- derstanding the geometric properties of the unit balls of high-dimensional Banach spaces and their behavior as the dimension grows to infinity. The unit ball of a finite dimensional Banach space is a centrally symmetric convex body and some of these geometric properties include the study of sections and projections of convex bodies, which are also convex bodies. However, when the distribution of mass in a convex body is studied, a convex body K is regarded as a probability space with the uniform probability on K and then the projections of the measure on linear subspaces are not the uniform probability on a convex body anymore and the class of convex bodies is left. Nevertheless, as a consequence of Brunn-Minkowski’s inequality, we remain in the class of log-concave probabilities, which are the probability measures with a log-concave density with respect to the Lebesgue measure. It is natural then, to work in the more general setting of log-concave functions rather than in the setting of convex bodies and a big part of the research in the area has gone in the direction of extending results from convex bodies to log-concave functions (see, for instance, [AKM], [FM], [AKSW], [KM], [C], [CF]), while many of the open problems in the field are nowadays stated in terms of log-concave functions rather than in terms of convex bodies.

Date: April 24, 2017. The first named author is partially supported by Spanish grants MTM2013-42105-P, MTM2016-77710-P projects and by IUMA. The second named author is partially supported by FundaciónSéneca,Science and Technology Agency of the Regiónde Murcia, through the Pro- grama de FormaciónPostdoctoral de Personal Investigador, project reference 19769/PD/15, and the Programme in Support of Excellence Groups of the Regiónde Murcia, Spain, project reference 19901/GERM/15. The third named author is supported by CAPES and IMPA. The second, third, and fourth authors are partially supported by MINECO project reference MTM2015-63699- P, Spain. 1 2DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

In [J] John proved that, among all the ellipsoids contained in a convex body K, there exists a unique ellipsoid E(K) with maximum volume. This ellipsoid is called the John’s ellipsoid of K. Furthermore, he characterized the cases in which n the John’s ellipsoid of K is the Euclidean ball B2 . This characterization, together with Brascamp-Lieb inequality [BL], led to many important results in the theory of convex bodies, showing that, among centrally symmetric convex bodies, the cube is an extremal convex body for many geometric parameters like the Banach-Mazur distance to the Euclidean ball, the volume ratio, the mean width, or the mean width of the polar body, see [B], [SS], [Ba]. The non-symmetric version of these problems has also been studied, see for instance [S], [Le], [Pa], [JN], [Sch1]. A function f : Rn → R is said to be log-concave if it is of the form f(x) = e−v(x), with v : Rn → (−∞, +∞] a convex function. Note that log-concave functions are continuous on their support and, since convex functions are diﬀerentiable almost everywhere, then so are log-concave functions. In this paper we will extend John’s theorem to the context of log-concave functions. We will consider ellipsoidal functions (we will sometimes simply call them ellipsoids), which will be functions of the form a E (x) = aχE (x), with a a positive constant and χE the characteristic function of an ellipsoid E, i.e., n n an aﬃne image of the Euclidean ball (E = c + TB2 with c ∈ R and T ∈ GL(n), the set of linear matrices with non-zero determinant). The determinant of a matrix T will be denoted by |T |. The volume of a convex body K will also be denoted by |K|. Given a log-concave function f : Rn → R, we will say that an ellipsoid Ea is contained in f if for every x ∈ Rn, Ea(x) ≤ f(x). Notice that if Ea ≤ f, then necessarily 0 < a ≤ kfk∞ and that for any t ∈ (0, 1] Etkfk∞ ≤ f if and only if the ellipsoid E is contained in the convex body n Kt(f) = {x ∈ R : f(x) ≥ tkfk∞}.

If f = χK (x) is the characteristic function of a convex body K, then an ellipsoid E is contained in K if and only if Et ≤ f for any t ∈ (0, 1]. In Section 2 we will show the following:

Theorem 1.1. Let f : Rn → R be an integrable log-concave function. There exists t0kfk∞ −n a unique ellipsoid E(f) = E for some t0 ∈ [e , 1], such that •E(f) ≤ f Z Z • E(f)(x)dx = max Ea(x)dx : Ea ≤ f . n n R R We will call this ellipsoid the John’s ellipsoid of f. The existence and uniqueness of the John’s ellipsoid of an integrable log-concave function f will allow us to deﬁne the integral ratio of f:

Deﬁnition 1.1. Let f : Rn → R be an integrable log-concave function and E(f) its John’s ellipsoid. We deﬁne the integral ratio of f:

1 R n n f(x)dx I.rat(f) = R R . n E(f)(x)dx R JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 3

Remark. This quantity is affine invariant, i.e., I.rat(f ◦ T ) = I.rat(f) for any affine map T . When f = χK is the characteristic function of a convex body then 1 |K| n I.rat(f) = v.rat(K), the volume ratio of K (Recall that v.rat(K) = |E(K)| , where E(K) is the John’s ellipsoid of K). In Section 3 we will give an upper bound for the integral ratio of log-concave n functions, finding the functions that maximize it. Namely, denoting by ∆n and B∞ n the regular simplex centered at the origin and the unit cube in R , and by k · kK the gauge function associated to a convex body K containing the origin, which is defined as kxkK = inf{λ > 0 : x ∈ λK}, we will prove the following

Theorem 1.2. Let f : Rn → R be an integrable log-concave function. Then,

I.rat(f) ≤ I.rat(gc),

−kxk∆ −c n where gc(x) = e n for any c ∈ ∆ . Furthermore, there is equality if and only f n if = gc ◦ T for some aﬃne map T and some c ∈ ∆ . If we assume f to be kfk∞ even, then I.rat(f) ≤ I.rat(g), −kxk n f where g(x) = e B∞ , with equality if and only if = g ◦ T for some linear kfk∞ map T ∈ GL(n). The value of the integral ratio of these functions will be computed and we obtain the following

Corollary 1.3. Let f : Rn → R be an integrable log-concave function. Then, e 1 n √ I.rat(f) ≤ (n!) n v.rat(∆ ) ∼ c n. n If we assume f to be even, then e 1 n √ I.rat(f) ≤ (n!) n v.rat(B ) ∼ c n. n ∞ The isoperimetric inequality states that for any convex body K the quantity |∂K| n−1 is minimized when K is a Euclidean ball. This inequality cannot be reversed |K| n in general. However, in [B], it was shown that for any symmetric convex body K, |∂T K| there exists an affine image TK such that the quotient n−1 is bounded above |TK| n n by the corresponding quantity for the cube B∞. If we do not impose symmetry then the regular simplex is the maximizer. This linear image is the one such that TK is in John’s position, i.e., the maximum volume ellipsoid contained in K is the Euclidean ball. The quantity studied in the isoperimetric inequality is not affine invariant but in [P], a stronger affine version of the isoperimetric inequality was established. Namely, it was shown that for any convex body K

n−1 1 n−1 1 n ∗ n n n ∗ n n |K| |Π (K)| ≤ |B2 | |Π (B2 )| , where Π∗(K), which is called the polar projection body of K, is the unit ball of the norm kxkΠ∗(K) = |x||Px⊥ K|, being Px⊥ K the projection of K onto the hyperplane orthogonal to x. This inequality is known as Petty’s projection inequality and there is equality in it if and only if K is an ellipsoid. Furthermore, following the idea in 4DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA the proof of the reverse isoperimetric inequality, a stability version of it was given in [A], showing that for any convex body K

n−1 ∗ 1 1 n n−1 ∗ n 1 (1) |K| n |Π (K)| n ≥ |B | n |Π (B )| n . v.rat(K) 2 2 The isoperimetric inequality and Petty’s projection inequality have their functional extensions. Namely, Sobolev’s inequality, which states that for any function f in the Sobolev space 1,1 n 1 n ∂f 1 n W (R ) = f ∈ L (R ): ∈ L (R ) ∀i ∂xi we have n 1 k|∇f|k ≥ n|B | n kfk n , 1 2 n−1 and the aﬃne Sobolev’s inequality, proved in [Z], which states that n ∗ 1 |B2 | (2) kfk n |Π (f)| n ≤ , n−1 n−1 2|B2 | where Π∗(f) is the unit ball of the norm Z kxkΠ∗(f) = |h∇f(y), xi|dy. n R 1,1 n 1 We would like to recall here the fact that W (R ) is the closure of C00, the space of C1 functions with compact support, [M]. These inequalities are actually equivalent to their geometric counterparts. In Section 4 we will follow the same ideas to obtain functional versions of the reverse isoperimetric inequality and a stability version of the aﬃne Sobolev inequality. We will prove the following extension of (1), which is a reverse form of (2) in the class of log-concave functions.

Theorem 1.4. Let f ∈ W 1,1(Rn) be a log-concave function. Then ∗ 1 kfk n |Π (f)| n 1 n−1 ≥ . n R f(x) n−1 |B | n f(x) log dx 2 R ( kfk∞ ) 1 R n n−1 R f(x)dx 2|B | n n f(x)dx n n 2 e R kfk∞ R n I.rat(f) R n−1 n f (x)dx R Remark. By (2) the left-hand side term is bounded above by 1. This lower bound is aﬃne invariant, and if f = χK is the characteristic function of a convex body, then we recover inequality (1). Besides, since log f(x) is a concave function, if f kfk∞ is centered we have by Jensen’s inequality that R f(x)  n xf(x)dx  R R n f(x) log dx f R R kfk∞ n f(x)dx f(0) R ≤ log  R  = log n f(x)dx kfk∞ kfk∞ R and so ∗ 1 kfk n |Π (f)| n 1 n−1 ≥ . |Bn| n−1 2 R n 2|Bn−1| 1 n f(x)dx 2 f(0) n R n I.rat(f) R n−1 n f (x)dx R JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 5

R Remark. Let us note that if n f(x)dx = 1 the inequality in Theorem 1.4 turns R into

∗ 1 −1 R f(x) log f(x) I.rat(f)|Π (f)| n e n dx ≤ , n |B2 | n−1 2|B2 | which along with the aﬃne Sobolev inequality (2) provides us with a bound for the power entropy of f of the following form

!2 −2 R I.rat(f) n f(x) log f(x)dx H(f) := e n R ≤ . kfk n n−1 For other recently studied connections between Information theory and convex geometry we refer to [BM1], [BM2] and references therein. Let us introduce some more notation: n For any function f : R → R and any ε > 0, we will denote fε the function given by xε f (x) = f . ε ε If f and g are two log-concave functions, then their Asplund product is the log- concave function f ? g(z) = max f(x)g(y) = max f(z − y)g(y). n z=x+y y∈R

2. John’s ellipsoid of a log-concave function In this section we show the existence and uniqueness of the John’s ellipsoid of an integrable log-concave function and show that the integral ratio of a function is an aﬃne invariant. For any ellipsoid Ea, its integral is a|E|. Since for any t ∈ (0, 1] the convex body Kt(f) has a unique maximum volume ellipsoid Et(f) = E(Kt(f)), then Z a a max E (x)dx : E ≤ f = max φf (t), n t∈(0,1] R where

φf (t) = tkfk∞|Et(f)|.

Thus, in order to prove Theorem 1.1 we need to prove that the function φf (t) −n attains a unique maximum in the interval (0, 1] at some point t0 ≥ e . Then the ellipsoid E(f) will be the function E(f)(x) = t kfk χ (x) = (E (f))t0kfk∞ (x), 0 ∞ Et0 (f) t0 where Et0 (f) is the John’s ellipsoid of the convex body Kt0 (f). If f = χK with K a convex body then the John’s ellipsoid of f will be the characteristic function of 1 the John’s ellipsoid of K E(K) = χE(K). We will prove that φf attains a unique maximum in the interval (0, 1]. First we prove the following:

n Lemma 2.1. Let L1,L2 ⊆ R be two convex bodies. Then, for any λ ∈ [0, 1] 1 1 1 |E ((1 − λ)L1 + λL2) | n ≥ (1 − λ)|E(L1)| n + λ|E(L2)| n . 6DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

n Proof. Let E(Li) = ai +TiB2 with Ti a symmetric positive deﬁnite matrix, i = 1, 2. Then

(1 − λ)L1 + λL2 ⊇ (1 − λ)E(L1) + λE(L2) n n = (1 − λ)a1 + λa2 + (1 − λ)T1B2 + λT2B2 n ⊇ (1 − λ)a1 + λa2 + ((1 − λ)T1 + λT2)B2 . Since by Minkowski’s determinant inequality, for any two symmetric positive deﬁ- 1 1 1 nite matrices A, B we have that |A + B| n ≥ |A| n + |B| n with equality if and only if B = sA for some s > 0, we obtain

1 1 1 n n n n |E ((1 − λ)L1 + λL2) | ≥ |(1 − λ)T1 + λT2| |B2 | 1 1 1 n n n n ≥ ((1 − λ)|T1| + λ|T2| )|B2 | 1 1 = (1 − λ)|E(L1)| n + λ|E(L2)| n . (3) Now, Theorem 1.1 will be a consequence of the following

Lemma 2.2. Let f : Rn → [0, +∞) be an integrable log-concave function and let φf : (0, 1] → R deﬁned as before. Then φf is continuous in (0, 1),

lim φf (t) = 0, t→0+

and φf attains its maximum value at some t0 ∈ (0, 1]. Furthermore, such t0 is unique.

Proof. Since f is log-concave, for any u1, u2 ∈ [0, ∞) and any λ ∈ [0, 1],

n −((1−λ)u1+λu2) n −u1 {x ∈ R : f(x) ≥ e kfk∞} ⊇ (1 − λ){x ∈ R : f(x) ≥ e kfk∞} n −u2 + λ{x ∈ R : f(x) ≥ e kfk∞} and then, by Lemma 2.1

0 ≤ lim tkfk∞|Et(f)| ≤ lim tkfk∞|Kt(f)| = 0 t→0+ t→0+ and so

lim φf (t) = 0, t→0+ or, equivalently, − u lim e n g(u) = 0. u→∞ − u Besides, since g is concave there exists limu→0+ e n g(u) ∈ R. Consequently, − u e n g(u) attains its maximum for some u0 ∈ [0, ∞) and so φf attains its maximum −u0 for some t0 = e ∈ (0, 1]. Let us prove that such u0 (and thus t0) is unique. Assume that there exist − u two diﬀerent u1 < u2 at which e n g(u) attains its maximum. Then, the function u h(u) = − n + log g(u), which is concave since g is concave, attains its maximum at u1 and u2. Thus, for every λ ∈ [0, 1]

h((1 − λ)u1 + λu2) = (1 − λ)h(u1) + λh(u2) and so 1−λ λ g((1 − λ)u1 + λu2) = g (u1)g (u2). Consequently, all the inequalities in (4) are equalities and, since there is equality in the arithmetic-geometric mean inequality, g(u1) = g(u2). But then h(u1) > h(u2), which contradicts the assumption of the maximum being attained at two diﬀerent points. −n It is left to prove that t0 ≥ e . This will be done as an observation in the proof of Lemma 3.1. Now that we have established the existence and uniqueness of the John’s ellipsoid of an integrable log-concave function f, we can deﬁne the integral ratio of f as

1 1 R n R n n f(x)dx n f(x)dx I.rat(f) = R R = R . n E(f)(x)dx maxt∈(0,1] φf (t) R The integral ratio of a function is an aﬃne invariant, i.e., for any aﬃne map T we have that I.rat(f ◦T ) = I.rat(f). This is a consequence of the following lemma.

Lemma 2.3. Let f : Rn → R be an integrable log-concave function and let T be an aﬃne map. Then for any t ∈ (0, 1]

−1 Et(f ◦ T ) = T Et(f). As a consequence

φf◦T −1 (t) = |T |φf (t), the maximum of φf◦T −1 and φf is attained for the same t0, and E(f ◦ T −1) = E(f) ◦ T −1. 8DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

Proof. Notice that −1 n −1 n Kt(f◦T ) = {x ∈ R : f(T x) ≥ tkfk∞} = T {x ∈ R : f(x) ≥ tkfk∞} = TKt(f). Consequently −1 Et(f ◦ T ) = T Et(f).

n t0kfk∞ A log-concave function will be said to be in John’s position if E(f) = (B2 ) for some t0 ∈ (0, 1]. As a consequence of the previous lemma, for any log-concave integrable function there exists an aﬃne map T such that f ◦T is in John’s position.

3. Maximal value of the integral ratio of log-concave functions

In this section we will obtain an estimate for the function φf (t) that will allow us to give an upper bound for the integral ratio of any integrable log-concave function. In order to do that we start proving the following

Lemma 3.1. Let f be an integrable log-concave function such that maxt∈(0,1] φf (t) = t0kfk∞ φf (t0), i.e., its John’s ellipsoid is E(f) = Et0 (f) . Then for every t ∈ (0, 1] 1 t n |Et(f)| ≤ 1 − log |Et0 (f)|. n t0 n Besides, if there is equality for every t ∈ (0, 1], then for some ct ∈ R 1 t Et(f) = ct + 1 − log Et0 (f). n t0

1 u0−u n − n Since the function h(u) = |Ee−u0 (f)| e is convex in R and the function 1 g(u) = |Ee−u (f)| n is concave in [0, ∞), as we have seen in Lemma 2.2 the graph of g is under the tangent at u0 to the graph of h. Thus, for every u ∈ [0, ∞) u − u g(u) ≤ g(u ) 1 + 0 . 0 n −n Observe that since g(0) ≥ 0, it must be u0 ≤ n and thus t0 ≥ e . Setting u = − log t we obtain that for every t ∈ (0, 1] 1 t n |Et(f)| ≤ 1 − log |Et0 (f)|. n t0 If there is equality for every t ∈ (0, 1], then for every u ∈ [0, ∞) u − u g(u) = g(u ) 1 + 0 0 n and the function g is an aﬃne function. Thus, for every u1, u2 ∈ [0, ∞) and any λ ∈ [0, 1] all the inequalities in (4) are equalities and, by the equality cases in JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 9

Minkowski’s determinant inequality, Ee−u1 (f) and Ee−u2 (f) are homothetic for every u1, u2 ∈ [0, ∞) and so, for every t ∈ (0, 1] 1 t Et(f) = ct + 1 − log Et0 (f) n t0 n for some ct ∈ R . The maximizers of the integral ratio will be log-concave functions like the ones deﬁned in the following lemma. Let us study some of their properties

−n n Lemma 3.2. For any t0 ≥ e and convex body K ⊆ R with 0 ∈ K, let

− max{kxkK −(n+log t0),0} fK,t0 (x) = e . Then 1 t • Kt(fK,t ) = 1 − log Kt (fK,t ) 0 n t0 0 0 1 t •Et(fK,t ) = 1 − log Et (fK,t ) 0 n t0 0 0 • max φ (t) = φ (t ) t∈(0,1] fK,t0 fK,t0 0 1 v.rat(K) Z 1 1 t n n • I.rat(fK,t0 ) = 1 1 − log dt n 0 n t0 t0 −n • I.rat(fK,t0 ) is decreasing in t0 in the interval [e , 1].

Proof. Notice that kfK,t0 k∞ = 1. Then, by deﬁnition of Kt(fK,t0 ) n Kt(fK,t0 ) = {x ∈ R : max{kxkK − (n + log t0), 0} ≤ − log t} t 1 t = n − log K = n 1 − log K t0 n t0 Consequently, for any t ∈ (0, 1] 1 t Kt(fK,t0 ) = 1 − log Kt0 (fK,t0 ). n t0 Then 1 t Et(fK,t0 ) = 1 − log Et0 (fK,t0 ) n t0 and t 1 t n φ (t) = 1 − log φ (t ). fK,t0 fK,t0 0 t0 n t0 Since the function g(x) = x(1 − log x) attains its maximum at x = 1, φ (t) fK,t0 attains its maximum at t = t0. Consequently Z n 1 I.rat(fK,t0 ) = fK,t0 (x)dx t |E (f )| n 0 t0 K,t0 R 1 Z 1 = |Kt(fK,t0 )|dt t0|Et0 (fK,t0 )| 0 |K (f )| Z 1 1 t n = t0 K,t0 1 − log dt t0|Et0 (fK,t0 )| 0 n t0 v.rat(K)n Z 1 1 t n = 1 − log dt. t0 0 n t0 10DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

−s Changing variables t = t0e we have Z +∞ n n 1 −s I.rat(fK,t0 ) = v.rat(K) 1 + s e ds, log t0 n −n which is clearly decreasing in t0 ∈ [e , 1].

Now, we have the following, which in particular, since I.rat(f n ) and I.rat(f n ) B∞,t0 ∆ ,t0 decrease in t0, implies Theorem 1.2. n Theorem 3.3. Let t0 ∈ (0, 1] and let f : R → R be an integrable log-concave t0kfk∞ such that maxt∈(0,1] φf (t) = φf (t0), i.e., its John’s ellipsoid is E(f) = Et0 (f) . Then we have that

n I.rat(f) ≤ I.rat(f∆ ,t0 ) f with equality if and only if = f∆ −c,t ◦ T for some aﬃne map T and some kfk∞ n 0 c ∈ ∆n. If f is even I.rat(f) ≤ I.rat(f n ) B∞,t0 f with equality if and only if = fBn ,t ◦ T for some T ∈ GL(n). kfk∞ ∞ 0 n Proof. Let f : R → R be such that maxt∈(0,1] φf (t) = φf (t0). Then 1 Z I.rat(f)n = f(x)dx t kfk |E (t )| n 0 ∞ f 0 R 1 Z 1 = |Kt(f)|dt t0|Ef (t0)| 0 Z 1 1 n = v.rat(Kt) |Ef (t)|dt t0|Ef (t0)| 0 v.rat(∆n)n Z 1 ≤ |Ef (t)|dt t0|Ef (t0)| 0 v.rat(∆n)n Z 1 1 t n ≤ 1 − log dt t0 0 n t0 n n = I.rat(f∆ ,t0 ) .

Besides, if there is equality, all the inequalities are equalities and so v.rat(Kt) = n n v.rat(∆ ), which implies that Kt = Tt∆ , for some aﬃne map Tt and |Ef (t)| = n 1 t 1 − log |Ef (t0)|, which by Corollary 3.1 implies that the John’s ellip- n t0 1 t soid of every level set Ef (t) = ct + 1 − log Ef (t0) and so Tt = ct + n t0 1 − 1 log t T for every t ∈ (0, 1]. Thus, we have that n t0 1 t n Kt = ct + 1 − log T ∆ . n t0 n By Lemma 2.3 we can assume without loss of generality that Kt0 = n∆ . In −s −s0 such case ct0 = 0. Then, calling t = e and t0 = e we have that s s0 K −s = c −s + 1 + − K −s . e e n n e 0

By log-concavity, we have that for every s ∈ [0, s0] s s Ke−s ⊇ Ke−s0 + 1 − K1 s0 s0 JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 11 s s s0 = 1 − c1 + 1 + − Ke−s0 s0 n n

s and then c −s = 1 − c1. If s ≥ s0 we have that e s0

s0 s0 K −s ⊇ K −s + 1 − K e 0 s e s 1 s0 s0 = c −s + 1 − c + K −s s e s 1 e 0

s and also in this case c −s = 1 − c1. Thus, for any s ≥ 0 e s0 s s s0 Ke−s = 1 − c1 + 1 + − Ke−s0 . s0 n n Consequently, f = e−v(·) ◦ T with T an aﬃne map and kfk∞

v(x) = inf {s : x ∈ Ke−s } s s s0 = inf s : x ∈ 1 − c1 + 1 + − Ke−s0 s0 n n n Ke−s0 1 = inf s : x ∈ c1 + (s + n − s0) − c1 s0 n s0 n Ke−s0 1 = inf s : x − c1 ∈ +(s + n − s0) − c1 s0 n s0  

 n  = max x − c1 − (n − s0), 0 s 1 1 0 K −s − c1  n e 0 s0   

 n  = max x + c1 − (n + log t0), 0 log t 1 1 0 Kt + c1  n 0 log t0   

 n  = max x + c1 − (n + log t0), 0 . log t n 1 0 ∆ + c1  log t0 

1 n Notice that for v to be well deﬁned necessarily c = c1 ∈ ∆ and then there − log t0 exists c ∈ ∆n such that f = e− max{k·−nck(∆n−c)−(n+log t0),0} ◦ T kfk∞ or, equivalently, f = e− max{k·k(∆n−c)−(n+log t0),0} ◦ T kfk∞

The same proof works in the even case. In the even case we know that ct = 0 1 t for any t and then Tt = 1 − log T . Thus, we can assume without loss of n t0 n f generality that Kt = nB and then it implies that = fBn ,t ◦ T . 0 ∞ kfk∞ ∞ 0

Finally, we will compute the integral ratio of this maximizing function in the following lemma. We will do it for a whole class of functions that include the maximizing one. 12DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

−kxkα Lemma 3.4. Let α ≥ 1 and f(x) = e K . Then

α 1 ! α eαΓ 1 + n n I.rat(f) = α v.rat(K) ∼ v.rat(K). n

Proof. On one hand Z Z Z +∞ Z +∞ Z −kxkα −t −t e K dx = e dtdx = e dxdt 1 n n α α R R kxkK 0 t K Z +∞ n −t n = |K| t α e dt = |K|Γ 1 + . 0 α On the other hand, for any t ∈ (0, 1]

1 Kt = (− log t) α K and then, 1 Et(f) = (− log t) α E(K), where E(K) is the John ellipsoid of K. Thus,

n φf (t) = t(− log t) α |E(K)|.

n −s n Let us ﬁnd maxt∈(0,1] t(− log t) α |E(K)| = maxs∈[0,+∞) e s α |E(K)|. Taking deriva- n tives we obtain that this maximum is attained at s = α and so

n n n α − n max t(− log t) α |E(K)| = e α |E(K)|. t∈(0,1] α Consequently

α 1 ! α eαΓ 1 + n n I.rat(f) = α v.rat(K). n

4. Reverse Sobolev-type inequalities In this section we will prove Theorem 1.4. First we will deﬁne the polar projection body of a function

Proposition 4.1. Let f : Rn → [0, +∞) be a log-concave integrable function. If the following quantity is ﬁnite for every x ∈ Rn then it deﬁnes a norm Z x kxk = 2|x| max f y + s dy. x⊥ s∈R |x| Besides, if f ∈ W 1,1(Rn) this norm equals Z kxk = |h∇f(y), xi|dy. n R The unit ball of this norm is the polar projection body of f, which will be denoted by Π∗(f). JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 13

Proof. Notice that Z x kxk = 2|x| max f y + s dy x⊥ s∈R |x| Z +∞ ⊥ x = 2|x| y ∈ x : max f y + s ≥ t dt 0 s∈R |x| Z 1 = 2|x|kfk∞ |Px⊥ Kt| dt 0 Z 1 ∗ = 2kfk∞ kxkΠ (Kt)dt 0 and it is clear that it is a norm. 1,1 n n If f ∈ W (R ), for almost every t the boundary of Kt is {x ∈ R : f(x) = tkfk∞} and we have Z 1 kxkΠ∗(f) = 2|x|kfk∞ |Px⊥ Kt| dt 0 Z kfk∞ Z x = |x| ν(y), dHn−1(y)dt 0 {f(x)=t} |x| n where ν(y) is the outer normal unit vector to {x ∈ R : f(x) ≥ t} and dHn−1 is the ∇f(y) Haussdorﬀ measure on the boundary of it. Since ν(y) = |∇f(y)| almost everywhere the above expression is

Z kfk∞ Z ∇f(y) , x dHn−1(y)dt 0 {f(x)=t} |∇f(y)| which, by the co-area formula, equals Z |h∇f(y), xi|dy. n R We will use the following lemma to prove Theorem 1.4.

n n a Lemma 4.2. Let f : R → R be a log-concave function and g(x) = (B2 ) (x). Then

lim f ? gε(z) = f(z) ε→0+ and f ? g (z) − f(z) lim ε = |∇f(z)| + f(z) log a almost everywhere. ε→0+ ε Proof. By deﬁnition of the Asplund product, since f is continuous,

ε y ε lim f ? gε(z) = lim sup f(x)a χBn = lim sup f(z − εy)a = f(z). ε→0+ ε→0+ 2 ε ε→0+ n z=x+y y∈B2 Besides, if f is diﬀerentiable in z, f ? g (z) − f(z) f(z − εy)aε − f(z)aε + f(z)aε − f(z) lim ε = lim sup ε→0+ ε ε→0+ n ε y∈B2 f(z − εy)aε − f(z)aε aε − 1 = lim sup + f(z) lim . ε→0+ n ε ε→0+ ε y∈B2 14DAVID ALONSO-GUTIERREZ,´ BERNARDO GONZALEZ´ MERINO, C. HUGO JIMENEZ,´ AND RAFAEL VILLA

Since f(z − εy) − f(z) lim sup = |∇f(z)|, ε→0+ n ε y∈B2 the previous limit equals |∇f(z)| + f(z) log a. The following lemma was proved in [CF]. We reproduce it here for the sake of completeness:

Lemma 4.3. Let f : Rn → R be an integrable log-concave function. Then R R Z Z n f ? fε(x)dx − n f(x)dx lim R R = n f(x)dx + f(x) log f(x)dx ε→0+ ε n n R R Proof. First of all, notice that if f(x) = e−u(x) with u a convex function, then −(1+ε)u( z ) f ? fε(z) = e 1+ε , since, as u is convex, its epigraph epi u is a convex set and then y inf u(x) + εu = inf u(x) + εu(y) z=x+y ε z=x+εy = inf{µ :(z, µ) ∈ (1 + ε)epi u} z = (1 + ε)u . 1 + ε Then, R R Z Z n f ? fε(x)dx − n f(x)dx 1 R R = (1 + ε)n e−(1+ε)u(x)dx − e−u(x)dx ε ε n n R R (1 + ε)n − 1 Z = e−(1+ε)u(x)dx ε n R Z e−εu(x) − 1 + e−u(x) dx. n ε R Now, taking limit when ε tends to 0 we obtain the result. The monotone conver- gence theorem and possibly a translation of the function u allows us to interchange limits. Now we are able to prove Theorem 1.4: Proof. Since all the quantities in the statement of the theorem are aﬃne invariant, i.e., they take the same value for f and for f ◦ T , we can assume that f is in John’s n t0kfk∞ position. That is, E(f) = (B2 ) . On the one hand, by Jensen’s inequality 1 −n ! n 1 1 Z Z ∗ n n n |Π (f)| = |B2 | |h∇f(z), θi|dz dσ(θ) n−1 n S R −1 1 Z Z n n ≥ |B2 | |h∇f(z), θi|dzdσ(θ) n−1 n S R n−1 −1 1 2 |B | Z n n 2 = |B2 | n |∇f(z)|dz . n |B | n 2 R On the other hand, let g(x) = E(f)(x). By Lemma 4.2, we have that

f ? gε(z) − f(z) |∇f(z)| + f(z) log(t0kfk∞) = lim ε→0+ ε JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 15

f ? f (z) − f(z) ≤ lim ε ε→0+ ε

By Lemma 4.3, integrating in z ∈ Rn we have that Z Z Z Z |∇f(z)|dz + f(z)dz log(t0kfk∞) ≤ n f(z)dz + f(z) log f(z)dz. n n n n R R R R Then Z Z Z f(z) |∇f(z)|dz ≤ n f(z)dz + f(z) log dz. n n n t kfk R R R 0 ∞ ∗ 1 kfk n |Π (f)| n n−1 Consequently, n is bounded below by |B2 | n−1 2|B2 | −1   n−1 f(x)  ! n R R n f(x) log dx 1 n f(x)dx R t0kfk∞ n R (t0kfk∞) I.rat(f)  n + 1  . R n−1 n f (x)dx nkfk n kfk n R n−1 1

−n −s0n Since t0 ≥ e , we can write t0 = e for some s0 ∈ [0, 1] and then

 n−1 f(x)  ! n R 1 R n f(x) log dx n f(x)dx t0kfk∞ n R R t0  n + 1  R n−1 n f (x)dx nkfk n kfk n R n−1 1  n−1 f(x)  ! n R R n f(x) log dx n f(x)dx kfk∞ −s0 R R = e (1 + s0) n + 1  . R n−1 n f (x)dx nkfk n kfk n R n−1 1 Since the maximum of g(s) = e−s [(1 + s)A + B] with A ≥ 0, B ≤ 0 and s ≥ 0 is ∗ 1 kfk n |Π (f)| n −B n−1 attained when s = we have that n is bounded below by A |B2 | n−1 2|B2 |

n−1 −1  R f(x)  n f(x) log dx ! n R ( kfk∞ ) 1 R R n f(x)dx n n f(x)dx n R e R kfk∞ n I.rat(f) . R n−1 n f (x)dx R

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(D. Alonso) Universidad de Zaragoza (Spain) E-mail address: [email protected]

(B. Gonz´alez) University Centre of Defence at the Spanish Air Force Academy, MDE- UPCT, Murcia (Spain) E-mail address: [email protected]

(C. H. Jim´enez) Pontif´ıcia Universidade Catolica´ do Rio de Janeiro (Brazil) E-mail address: [email protected]

(R. Villa) Universidad de Sevilla (Spain) E-mail address: [email protected]