John's ellipsoid and the integral ratio of a log-concave function

arXiv:1511.01266v1 [math.FA] 4 Nov 2015 JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION DAVID ALONSO-GUTI´ ERREZ, BERNARDO GONZ´ ALEZ MERINO, C. HUGO JIM´ ENEZ, 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 understanding 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 Kis regarded as a probability space with the uniform probability on Kand 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: August 24, 2018. The first named author is partially supported by Spanish grants MTM2013-42105-P, P11B2014- 35 projects and by IUMA. The second named author is partially supported by MINECO-FEDER project reference MTM2012-34037, Spain, and by Consejer´ıa de Industria, Turismo, Empresa e Innovaci´on de la CARM through Fundaci´on S´eneca, Agencia de Ciencia y Tecnolog´ıa de la Regi´on de Murcia, Programa de Formaci´on Postdoctoral de Personal Investigador. The third named author is supported by CAPES and IMPA. The forth named author were partially supported by the Spanish Ministry of Science and Innovation and FEDER Funds of European Union grant MTM2012-30748. 1 2 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, 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 the John’s ellipsoid of Kis the Euclidean ball Bn 2. 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→Ris 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 Ea(x) = aχE(x), with aa positive constant and χEthe characteristic function of an ellipsoid E,i.e., an affine image of the Euclidean ball (E=c+T Bn 2with c∈Rnand T∈GL(n), the set of linear matrices with non-zero determinant). The determinant of a matrix Twill be denoted by |T|. The volume of a convex body Kwill also be denoted by |K|. The trace of Twill be denoted by tr(T). Given a log-concave function f:Rn→R, we will say that an ellipsoid Eais contained in fif 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 Eis contained in the convex body Kt(f) = {x∈Rn:f(x)≥tkfk∞}. If f=χK(x) is the characteristic function of a convex body K, then an ellipsoid E is contained in Kif and only if Et≤ffor any t∈(0,1]. In Section 2 we will show the following: Theorem 1.1. Let f:Rn→Rbe an integrable log-concave function. There exists a unique ellipsoid E(f) = Et0kfk∞for some t0∈(0,1], such that • E(f)≤f •ZRn E(f)(x)dx = max ZRn Ea(x)dx :Ea≤f. 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 fwill allow us to define the integral ratio of f: Definition 1.1. Let f:Rn→Rbe an integrable log-concave function and E(f)its John’s ellipsoid. We define the integral ratio of f: I.rat(f) = RRnf(x)dx RRnE(f)(x)dx1 n . 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=χKis the characteristic function of a convex body then I.rat(f) = v.rat(K), the volume ratio of K(Recall that v.rat(K) = |K| |E(K)|1 n, 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 functions, finding the functions that maximize it. Namely, denoting by ∆nand Bn ∞ the regular simplex centered at the origin and the unit cube in Rn, and by k · kK the gauge function associated to a convex body Kcontaining the origin, which is defined as kxkKinf{λ > 0 : x∈λK}, we will prove the following Theorem 1.2. Let f:Rn→Rbe an integrable log-concave function. Then, I.rat(f)≤I.rat(gc), where gc(x) = e−kxk∆n−cfor any c∈∆n. Furthermore, there is equality if and only if f kfk∞=gc◦Tfor some affine map Tand some c∈∆n. If we assume fto be even, then I.rat(f)≤I.rat(g), where g(x) = e−kxkBn ∞. with equality if and only if f kfk∞=g◦Tfor some linear map T∈GL(n). In order to do so we will prove a characterization of the situation in which the John’s ellipsoid of an integrable log-concave function is E(f) = (Bn 2)t0kfk∞. In such case we will say that a log-concave function is in John’s position. The isoperimetric inequality states that for any convex body Kthe quantity |∂K| |K|n−1 n is minimized when Kis a Euclidean ball. This inequality cannot be reversed in general. However, in [B], it was shown that for any symmetric convex body K, there exists an affine image T K such that the quotient |∂T K| |T K|n−1 n is bounded above by the corresponding quantity for the cube Bn ∞. If we do not impose symmetry then the regular simplex is the maximizer. This linear image is the one such that T K is in John’s position, i.e., the maximum volume ellipsoid contained in Kis 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 |K|n−1 n|Π∗(K)|1 n≤ |Bn 2|n−1 n|Π∗(Bn 2)|1 n, 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⊥Kthe projection of Konto the hyperplane orthogonal to x. This inequality is known as Petty’s projection inequality and there is equality in it if and only if Kis an ellipsoid. Furthermore, following the idea in the proof of the reverse isoperimetric inequality, a stability version of it was given in [A], showing that for any convex body K (1) |K|n−1 n|Π∗(K)|1 n≥1 v.rat(K)|Bn 2|n−1 n|Π∗(Bn 2)|1 n. 4 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA The isoperimetric inequality and Petty’s projection inequality have their functional extensions. Namely, Sobolev’s inequality, which states that for any function fin the Sobolev space W1,1(Rn) = f∈L1(Rn) : ∂f ∂xi ∈L1(Rn)∀i we have k|∇f|k1≥n|Bn 2|1 nkfkn n−1, and the affine Sobolev’s inequality, proved in [Z], which states that (2) kfkn n−1|Π∗(f)|1 n≤|Bn 2| 2|Bn−1 2|, where Π∗(f) is the unit ball of the norm kxkΠ∗(f)=ZRn |h∇f(y), xi|dy. We would like to recall here the fact that W1,1(Rn) is the closure of C1 00, the space of C1functions 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 1.3. Let f∈W1,1(Rn)be a log-concave function. Then kfkn n−1|Π∗(f)|1 n |Bn 2| 2|Bn−1 2|≥1 eRRnf(x) log(f(x) kfk∞)1 ndx RRnf(x)dx kfk1 n ∞RRnf(x)dx RRnf n n−1(x)dx n−1 n I.rat(f) . Remark. By (2) the left-hand side term is bounded above by 1. This lower bound is affine invariant, and if f=χKis the characteristic function of a convex body, then we recover inequality (1). Remark. Let us note that if RRnf(x)dx = 1 the previous inequality turns into e−1 nRf(x) log f(x)dx ≤I.rat(f)|Π∗(f)|1 n |Bn 2| 2|Bn−1 2|, which along with the affine Sobolev inequality (2) provides us with a bound for the power entropy of fof the following form H(f) := e−2 nRRnf(x) log f(x)dx ≤ I.rat(f) kfkn n−1!2 . For other recently studied connections between Information theory and convex geometry we refer to [BM1], [BM2] and references therein. JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 5 Let us introduce some more notation: If Kis a convex body, r(K) will denote its inner radius, i.e., the radius of the largest centered Euclidean ball contained in it. For a set A⊆Rn, the positive hull of Ais the convex cone posA=(n X i=1 λixi:λi>0, xi∈A, n ∈N). Given a convex set E⊆Rnand x∈∂E, the boundary of E, the normal cone of E at xis defined as N(E, x) = {u∈Rn:hz−x, ui ≤ 0,∀z∈E}. The support cone of Eat xis the cone S(E, x) = cl [ λ>0 λ(E−x). The following polarity relation holds: N(E, x)∗=S(E, x), where the polarity relation is the polarity of convex cones C∗={y∈Rn:hy, xi ≤ 0,∀x∈C}. If His an affine subspace through x, then the normal cone to E∩Hat x, relative to the subspace His NH(E∩H, x) = {u∈H0:hz−x, ui ≤ 0,∀z∈E∩H}, where H0is the linear subspace parallel to H. The similar duality holds NH(E∩H, x)∗H0=S(E∩H, x), . where the duality is taken with respect to the linear subspace H0. It happens that (3) NH(E∩H, x) = PH0N(E, x), where PH0denotes the orthogonal projection onto the linear subspace H0. We refer the reader to [Sch] for these and other known facts on convex cones. For any function f:Rn→Rand any ε > 0, we will denote fεthe function given by fε(x) = fx εε . If fand gare two log-concave functions, then their Asplund product is the logconcave function f ⋆ g(z) = max z=x+yf(x)g(y) = max y∈Rnf(z−y)g(y). 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 max ZRn Ea(x)dx :Ea≤f= max t∈(0,1] φf(t), 6 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA where φf(t) = tkfk∞|Et(f)|. Thus, in order to prove Theorem 1.1 we need to prove that the function φf(t) attains a unique maximum in the interval (0,1] at some point t0. Then the ellipsoid E(f) will be the function E(f)(x) = t0kfk∞χEt0(f)(x) = (Et0(f))t0kfk∞(x), where Et0(f) is the John’s ellipsoid of the convex body Kt0(f). If f=χKwith K a convex body then the John’s ellipsoid of fwill be the characteristic function of the John’s ellipsoid of KE(K)1=χE(K). We will prove that φfattains a unique maximum in the interval (0,1]. First we prove the following: Lemma 2.1. Let fbe a log-concave function and let φf: (0,1] →Rdefined as before. For any t0, t1∈(0,1] and any λ∈[0,1] φf(t1−λ 0tλ 1)≥φf(t0)1−λφf(t1)λ. Proof. Since fis log-concave {x∈Rn:f(x)≥t1−λ 0tλ 1kfk∞} ⊇ (1 −λ){x∈Rn:f(x)≥t0kfk∞} +λ{x∈Rn:f(x)≥t1kfk∞}. Thus, if Eti(f) = ai+TiBn 2with Tia symmetric positive definite matrix, i= 0,1, then {x∈Rn:f(x)≥t1−λ 0tλ 1kfk∞} ⊇ (1 −λ)Et0(f) + λEt1(f) = (1 −λ)a0+λa1+ (1 −λ)Tt0Bn 2+λTt1Bn 2 ⊇(1 −λ)a0+λa1+ ((1 −λ)Tt0+λTt1)Bn 2.(4) Taking volumes, since by Minkowski’s determinant inequality, for any two symmetric positive definite matrices A, B we have that |A+B|1 n≥ |A|1 n+|B|1 nwith equality if and only if B=sA for some s > 0, we obtain |Et1−λ 0tλ 1(f)|1 n≥ |(1 −λ)T0+λT1|1 n|Bn 2|1 n ≥((1 −λ)|T0|1 n+λ|T1|1 n)|Bn 2|1 n = (1 −λ)|Et0(f)|1 n+λ|Et1(f)|1 n ≥ |Et0(f)|1−λ n|Et1(f)|λ n,(5) where the last inequality is the arithmetic-geometric mean inequality. Consequently, |Et1−λ 0tλ 1(f)| ≥ |Et0(f)|1−λ|Et1(f)|λ and multiplying by t1−λ 0tλ 1kfk∞ φf(t1−λ 0tλ 1)≥φf(t0)1−λφf(t1)λ.  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] →Rdefined as before. Then φfis continuous in (0,1] and lim t→0+φf(t) = 0. Consequently φfattains its maximum value at some t0∈(0,1]. Furthermore, such t0is unique. JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 7 Proof. In order to show the continuity of φfin (0,1] from the right it is enough to show that for any t0∈(0,1) , if dt0(ε) is the smallest number such that Kt0+ε(f)⊆Kt0(f)⊆dt0(ε)Kt0+ε(f), then limε→0+dt0(ε) = 1, since then |Et0+ε(f)| ≤ |Et0(f)| ≤ dt0(ε)n|Et0+ε(f)| and consequently lim ε→0+|Et0+ε(f)| ≤ |Et0(f)| and lim ε→0+|Et0+ε(f)| ≥ lim ε→0+ 1 dt0(ε)n|Et0(f)|=|Et0(f)|. Let us see then that limε→0+dt0(ε) = 1. Notice that for every x∈Kt0(f)\limε→0+Kt0+ε(f) we have that f(x) = t0kfk∞. Assume that limε→0+dt0(ε) is not 1. Then there exists a segment [x0, x1] and a point c∈(x0, x1) such that [x0, c] is contained in Kt0(f)\limε→0+Kt0+ε(f) and x1∈Kt1(f) for some t1> t0. Then, since fis log-concave f(c)> f(x0) = t0, which contradicts the fact that [x0, c] is contained in Kt0(f)\limε→0+Kt0+ε(f). A similar argument proves that φfis continuous from the left. Let us now prove that limt→0+φf(t) = 0. Let ε > 0. Since fis integrable, we can find R(ε) big enough such that ZRn\R(ε)Bn 2 f(x)dx < ε 2. Now, for any t < ε 2kfk∞|R(ε)Bn 2|we have that tkfk∞|Kt(f)|=tkfk∞|Kt(f)∩R(ε)Bn 2|+tkfk∞|Kt(f)\R(ε)Bn 2| < tkfk∞|R(ε)Bn 2|+ZKt(f)\R(ε)Bn 2 f(x)dx <ε 2+ZRn\R(ε)Bn 2 f(x)dx <ε 2+ε 2=ε. Then, 0≤lim t→0+tkfk∞|Et(f)| ≤ lim t→0+tkfk∞|Kt(f)|= 0 and so lim t→0+φf(t) = 0. Consequently φfattains its maximum for some t0∈(0,1]. Let us prove that such t0is unique. Assume that there exist two different t1< t2at which φfattains its maximum. Then, by Lemma 2.1 for any λ∈[0,1] φf(t1−λ 1tλ 2) = φf(t1)1−λφf(t2)λ. Thus, for any λ∈[0,1] |Et1−λ 1tλ 2(f)|1 n=|Et1(f)|1−λ n|Et2(f)|λ n and all the inequalities in (5) are equalities. This implies that Tt2is a multiple of Tt1and so the ellipsoids Et1and Et2are homothetic. Besides, since there is equality 8 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA in the arithmetic-geometric mean inequality both ellipsoids have the same volume and, Et2is a translate of Et1. Thus, for any λ∈[0,1] we have |Et1−λ 1tλ 2(f)|=|Et1(f)|. But then φf(t2)> φf(t1), which contradicts the assumption of the maximum being attained at two different points.  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 fas I.rat(f) = RRnf(x)dx RRnE(f)(x)dx1 n =RRnf(x)dx maxt∈(0,1] φf(t)1 n . 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→Rbe an integrable log-concave function and let Tbe an affine map. Then for any t∈(0,1] Et(f◦T−1) = TEt(f). As a consequence φf◦T−1(t) = |T|φf(t), the maximum of φf◦T−1and φfis attained for the same t0, and E(f◦T−1) = E(f)◦T−1. Proof. Notice that Kt(f◦T−1) = {x∈Rn:f(T−1x)≥tkfk∞}=T{x∈Rn:f(x)≥tkfk∞}=T Kt(f). Consequently Et(f◦T−1) = TEt(f).  3. John’s position of a log-concave function and maximal integral ratio A log-concave function will be said to be in John’s position if E(f) = (Bn 2)t0kfk∞ for some t0∈(0,1]. By Lemma 2.3, for any log-concave integrable function there exists an affine map Tsuch that f◦Tis in John’s position. In this section we will give a characterization for a function to be in John’s position. As a consequence 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. We will follow the ideas in [GS] and prove the following Theorem 3.1. Let f:Rn→Rbe an even integrable log-concave function and t0∈(0,1]. Assume that (Bn 2)t0kfk∞≤f. Then the following are equivalent: 1. E(f) = (Bn 2)t0kfk∞. 2. There exist • {uij} ⊆ ∂Kt0∩Sn−1,1≤i≤m, 1≤j≤m′(i) • {λi}m i=1,{µij}m′(i) j=1 , with λi, µij >0and • {αi}m i=1, with αi∈ht0d dt+r(Kt)t=t0, t0d dt−r(Kt)t=t0i JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 9 with 1≤m≤1 + n(n+1) 2and n≤m′(i)≤n(n+1) 2for any 1≤i≤m, such that m X i=1 λiαi=−1, In= m X i=1 m′(i) X j=1 λiµijuij ⊗uij, and for any t∈(0,1], and any 1≤i≤mif T Bn 2⊆Kt(f) −αilog t t0+ m′(i) X j=1 µij hT uij, uiji ≤ m′(i) X j=1 µij. Proof. Any ellipsoid E ⊆ Rnis E=T Bn 2with Ta symmetric positive definite matrix. For any symmetric positive definite matrix T, we will call xT∈Rn2the vector xT= (t11,...,t1n, t21,...,t2n, tn1,...,tnn)t. Notice that the set Cspd ={xT:Tsymmetric positive definite}is a convex cone contained in a linear subspace L⊆Rn2of dimension n(n+1) 2. We will consider the following two sets in R×L: E={(s, xT)∈[0,+∞]×L:T∈Cspd,(T Bn 2)e−skfk∞≤f} and C1={(s, xT)∈[0,+∞]×L:T∈Cspd, e−s|T| ≥ t0}. First of all, notice that both of them are convex. In order to show that Eis convex set let (si, xTi)∈E,i= 1,2. From the definition of E, this means that Ke−si⊇TiBn 2. By (4) we get that Ke−(1−λ)s1−λs2⊇((1 −λ)T1+λT2)Bn 2 from which ((1 −λ)s1+λs2, x(1−λ)T1+λT2)∈E. In order to see that C1is convex let (si, xTi)∈C1,i= 1,2. Then e−si|Ti| ≥ t0, i= 1,2. Minkowski’s determinant inequality and the arithmetic-geometric mean imply that e−(1−λ)s1+λs2 n|(1 −λ)T1+λT2|1 n≥e−(1−λ)s1+λs2 n((1 −λ)|T1|1 n+λ|T2|1 n) ≥e−(1−λ)s1+λs2 n|T1|(1−λ) n|T2|λ n =e−s1|T1|1−λ ne−s2|T2|λ n ≥t1 n 0, from which we conclude ((1 −λ)s1+λs2,(1 −λ)T1+λT2)∈C1. Second, notice that if s6= 0 and e−s|T|=t0, then the point (s, xT) belongs to the boundary of C1, which is smooth around it. Then there exists a unique supporting hyperplane of C1at (s, xT). Since the function g(s) = es nis convex, its graph is above its tangent line at the point (−log t0, t−1 n 0). Thus, for any (s, xT)∈C1, h(−1, xIn),(s, xT)i=−s+ tr(T)≥ −s+n|T|1 n 16 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA =v.rat(K)n t0Z1 0 1−log t t01 n!n dt. Changing variables t=t0e−swe have I.rat(fK,t0)n=v.rat(K)Z+∞ log t01 + 1 nsn e−sds, which is clearly decreasing in t0∈[e−n,1].  Now, we have the following, which in particular, since I.rat(fBn ∞,t0) and I.rat(f∆n,t0) decrease in t0, implies Theorem 1.2. Theorem 3.5. Let t0∈(0,1] and let f:Rn→Rbe an integrable log-concave such that maxt∈(0,1] φf(t) = φf(t0), i.e., its John’s ellipsoid is E(f) = Et0(f)t0kfk∞. Then we have that I.rat(f)≤I.rat(f∆n,t0) with equality if and only if f kfk∞=f∆n−c,t0◦Tfor some affine map Tand some c∈∆n. If fis even I.rat(f)≤I.rat(fBn ∞,t0) with equality if and only if f kfk∞=fBn ∞,t0◦Tfor some T∈GL(n). Proof. Let f:Rn→Rbe such that maxt∈(0,1] φf(t) = φf(t0). Then I.rat(f)n=1 t0kfk∞|Ef(t0)|ZRn f(x)dx =1 t0|Ef(t0)|Z1 0 |Kt(f)|dt =1 t0|Ef(t0)|Z1 0 v.rat(Kt)n|Ef(t)|dt ≤v.rat(∆n)n t0|Ef(t0)|Z1 0 |Ef(t)|dt ≤v.rat(∆n)n t0Z1 0 1−log t t01 n!n dt =I.rat(f∆n,t0)n. Besides, if there is equality, all the inequalities are equalities and so v.rat(Kt) = v.rat(∆n), which implies that Kt=Tt∆n, for some affine map Ttand |Ef(t)|= 1−log t t01 nn |Ef(t0)|, which by Corollary 3.3 implies that the John’s ellipsoid of every level set Ef(t) = ct+1−log t t01 nEf(t0) and so Tt=ct+ 1−log t t01 nTfor every t∈(0,1]. Thus, we have that Kt=ct+ 1−log t t01 n!T∆n. JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 17 By Lemma 2.3 we can assume without loss of generality that Kt0=n∆n. In such case ct0= 0. Then, calling t=e−sand t0=e−s0we have that Ke−s=ce−s+1 + s n−s0 nKe−s0. By log-concavity, we have that for every s∈[0, s0] Ke−s⊇s s0 Ke−s0+1−s s0K1 =1−s s0c1+1 + s n−s0 nKe−s0 and then ce−s=1−s s0c1. If s≥s0we have that Ke−s0⊇s0 sKe−s+1−s0 sK1 =s0 sce−s+1−s0 sc1+Ke−s0 and also in this case ce−s=1−s s0c1. Thus, for any s≥0 Ke−s=1−s s0c1+1 + s n−s0 nKe−s0. Consequently, f kfk∞=e−v(·)◦Twith Tan affine map and v(x) = inf {s:x∈Ke−s} = inf s:x∈1−s s0c1+1 + s n−s0 nKe−s0 = inf s:x∈n s0 c1+ (s+n−s0)Ke−s0 n−1 s0 c1 = inf s:x−n s0 c1∈+(s+n−s0)Ke−s0 n−1 s0 c1 = max       x−n s0 c1   1 nKe−s0−1 s0c1−(n−s0),0   = max       x+n log t0 c1   1 nKt0+1 log t0c1−(n+ log t0),0   = max       x+n log t0 c1   ∆n+1 log t0c1−(n+ log t0),0   . Notice that for vto be well defined necessarily c=1 −log t0c1∈∆nand then there exists c∈∆nsuch that f kfk∞ =e−max{k·−nck(∆n−c)−(n+log t0),0}◦T or, equivalently, f kfk∞ =e−max{k·k(∆n−c)−(n+log t0),0}◦T 18 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA The same proof works in the even case. In the even case we know that ct= 0 for any tand then Tt=1−log t t01 nT. Thus, we can assume without loss of generality that Kt0=nBn ∞and then it implies that f kfk∞=fBn ∞,t0◦T. 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. Lemma 3.6. Let α≥1and f(x) = e−kxkα K. Then I.rat(f) = eαΓ1 + n αα n n!1 α v.rat(K)∼v.rat(K). Proof. On one hand ZRn e−kxkα Kdx =ZRnZ+∞ kxkα K e−tdtdx =Z+∞ 0Zt1 αK e−tdxdt =|K|Z+∞ 0 tn αe−tdt =|K|Γ1 + n α. On the other hand, for any t∈(0,1] Kt= (−log t)1 αK and then, Et(f) = (−log t)1 αE(K), where E(K) is the John ellipsoid of K. Thus, φf(t) = t(−log t)n α|E(K)|. Let us find maxt∈(0,1] t(−log t)n α|E(K)|= maxs∈[0,+∞)e−ssn α|E(K)|. Taking derivatives we obtain that this maximum is attained at s=n αand so max t∈(0,1] t(−log t)n α|E(K)|=n αn αe−n α|E(K)|. Consequently I.rat(f) = eαΓ1 + n αα n n!1 α v.rat(K).  4. Reverse Sobolev-type inequalities In this section we will prove Theorem 1.3. 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∈Rnthen it defines a norm kxk= 2|x|Zx⊥ max s∈Rfy+sx |x|dy. Besides, if f∈W1,1(Rn)this norm equals kxk=ZRn |h∇f(y), xi|dy. JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 19 The unit ball of this norm is the polar projection body of f, which will be denoted by Π∗(f). Proof. Notice that kxk= 2|x|Zx⊥ max s∈Rfy+sx |x|dy = 2|x|Z+∞ 0y∈x⊥: max s∈Rfy+sx |x|≥t dt = 2|x|kfk∞Z1 0 |Px⊥Kt|dt = 2kfk∞Z1 0 kxkΠ∗(Kt)dt and it is clear that it is a norm. If f∈W1,1(Rn), for almost every tthe boundary of Ktis {x∈Rn:f(x) = tkfk∞}and we have kxkΠ∗(f)= 2|x|kfk∞Z1 0 |Px⊥Kt|dt =|x|Zkfk∞ 0Z{f(x)=t}ν(y),x |x| dHn−1(y)dt where ν(y) is the outer normal unit vector to {x∈Rn:f(x)≥t}and dHn−1is the Haussdorff measure on the boundary of it. Since ν(y) = ∇f(y) |∇f(y)|almost everywhere the above expression is Zkfk∞ 0Z{f(x)=t}∇f(y) |∇f(y)|, x dHn−1(y)dt which, by the co-area formula, equals ZRn |h∇f(y), xi|dy.  We will use the following lemma to prove Theorem 1.3. Lemma 4.2. Let f:Rn→Rbe a log-concave function and g(x) = (Bn 2)a(x). Then lim ε→0+f ⋆ gε(z) = f(z) and lim ε→0+ f ⋆ gε(z)−f(z) ε=|∇f(z)|+f(z) log aalmost everywhere. Proof. By definition of the Asplund product, since fis continuous, lim ε→0+f ⋆ gε(z) = lim ε→0+sup z=x+y f(x)aεχBn 2y ε= lim ε→0+sup y∈Bn 2 f(z−εy)aε=f(z). Besides, if fis differentiable in z, lim ε→0+ f ⋆ gε(z)−f(z) ε= lim ε→0+sup y∈Bn 2 f(z−εy)aε−f(z)aε+f(z)aε−f(z) ε 20 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA = lim ε→0+sup y∈Bn 2 f(z−εy)aε−f(z)aε ε+f(z) lim ε→0+ aε−1 ε. Since lim ε→0+sup y∈Bn 2 f(z−εy)−f(z) ε=|∇f(z)|, 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→Rbe an integrable log-concave function. Then lim ε→0+RRnf ⋆ fε(x)dx −RRnf(x)dx ε=nZRn f(x)dx +ZRn f(x) log f(x)dx Proof. First of all, notice that if f(x) = e−u(x)with ua convex function, then f ⋆ fε(z) = e−(1+ε)u(z 1+ε), since, as uis convex, its epigraph epi uis a convex set and then inf z=x+yu(x) + εu y ε= inf z=x+εy u(x) + εu(y) = inf{µ: (z, µ)∈(1 + ε)epi u} = (1 + ε)uz 1 + ε. Then, RRnf ⋆ fε(x)dx −RRnf(x)dx ε=1 ε(1 + ε)nZRn e−(1+ε)u(x)dx −ZRn e−u(x)dx =(1 + ε)n−1 εZRn e−(1+ε)u(x)dx +ZRn e−u(x)e−εu(x)−1 εdx. Now, taking limit when εtends to 0 we obtain the result. The monotone convergence theorem and possibly a translation of the function uallows us to interchange limits.  Now we are able to prove Theorem 1.3: Proof. Since all the quantities in the statement of the theorem are affine invariant, i.e., they take the same value for fand for f◦T, we can assume that fis in John’s position. That is, E(f) = (Bn 2)t0kfk∞. On the one hand, by Jensen’s inequality |Π∗(f)|1 n=|Bn 2|1 n ZSn−1ZRn |h∇f(z), θi|dz−n dσ(θ)!1 n ≥ |Bn 2|1 nZSn−1ZRn |h∇f(z), θi|dzdσ(θ)−1 =|Bn 2|1 n2 n |Bn−1 2| |Bn 2|ZRn |∇f(z)|dz−1 . JOHN’S ELLIPSOID AND THE INTEGRAL RATIO OF A LOG-CONCAVE FUNCTION 21 On the other hand, let g(x) = E(f)(x). By Lemma 4.2, we have that |∇f(z)|+f(z) log(t0kfk∞) = lim ε→0+ f ⋆ gε(z)−f(z) ε ≤lim ε→0+ f ⋆ fε(z)−f(z) ε By Lemma 4.3, integrating in z∈Rnwe have that ZRn |∇f(z)|dz +ZRn f(z)dz log(t0kfk∞)≤nZRn f(z)dz +ZRn f(z) log f(z)dz. Then ZRn |∇f(z)|dz ≤nZRn f(z)dz +ZRn f(z) log f(z) t0kfk∞ dz. Consequently, kfkn n−1|Π∗(f)|1 n |Bn 2| 2|Bn−1 2|is bounded below by   (t0kfk∞)1 nI.rat(f)   RRnf(x)dx RRnfn n−1(x)dx!n−1 n +RRnf(x) log f(x) t0kfk∞1 ndx kfkn n−1kfk1 n 1      −1 . Since t0≥e−n, we can write t0=e−s0nfor some s0∈[0,1] and then t1 n 0   RRnf(x)dx RRnfn n−1(x)dx!n−1 n +RRnf(x) log f(x) t0kfk∞1 ndx kfkn n−1kfk1 n 1    =e−s0  (1 + s0) RRnf(x)dx RRnfn n−1(x)dx!n−1 n +RRnf(x) log f(x) kfk∞1 ndx kfkn n−1kfk1 n 1   . Since the maximum of g(s) = e−s[(1 + s)A+B] with A≥0, B≤0 and s≥0 is attained when s=−B Awe have that kfkn n−1|Π∗(f)|1 n |Bn 2| 2|Bn−1 2|is bounded below by  eRRnf(x) log(f(x) kfk∞)1 ndx RRnf(x)dx kfk1 n ∞ RRnf(x)dx RRnfn n−1(x)dx!n−1 n I.rat(f)  −1 .  References [A] Alonso-Guti´ errez D. On a reverse Petty projection inequality for projections of convex bodies. Advances in geometry 14, (2014) pp. 215–223. [ABBW] Alonso-Guti´ errez D. bastero J. Bernu´ es J., Wolff P. On the isotropy constant of projections of polytopes. J. Functional Analysis 258, (2010) pp. 1452–1465. [AKSW] Arstein-Avidan S. Klartag B., Sch¨ utt C. Werner E. Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality. J. 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