$ L P $ John Ellipsoids for Log-Concave Functions

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A minimal ellipsoid will be called an Sp solution for K. Where p 1 hE p V p(K, E)= dV K , p> 0, n 1 h h ZS − K i is the normalized Lp mixed volume of K and E. More details about the solution of the two problems Sp, Sp and related inequalities see [46]. The Orlicz extension of the John ellipsoid is done by Zou and Xiong [59]. Recently, the study of the geometry of log-concave functions in the field of convex geometry has emerged, with a quite natural idea is to replace the volume of a convex body by the integral of a log-concave function. To establish functional version of the problems from the convex geometric analysis of convex body has attracted a lot of authors interest (see, e.g., [1–4,7,16–20,22,24,35,41,52]). Also an extension of the John ellipsoids to the case of log-concave functions has attracted a lot of authors interest, for example, in [5], the authors extend the notion of John’s ellipsoid to the setting of integrable log-concave functions and obtain integral ratio of a log-concave function and establish the reverse functional affine isoperimetric inequality. The extension of the LYZ ellipsoid to the log-concave functions is done by Fang and Zhou [27]. The Löwner ellipsoid function for log-concave function is invested by Li, Schütt and Werner [40]. Extensive research has been devoted to extend the concepts and inequalities from convex bodies to the setting of log-concave functions (see, e.g., [25, 29]). In fact, it was observed that the Prékopa-Leindler inequality is the functional analog of the Brunn-Minkowski inequality (see e.g., [15,30,49]) for convex bodies. Much progress has been made see [6, 8, 10, 23]. Let f be a log-concave functions of Rn such that n u f : R R, f = e− , → where u : Rn R + is a convex function. We always consider in this paper that a log-concave→ ∪{ function∞} f is integrable and such that f is nondegenerate, i.e., the interior of the support of f is non-empty, int(suppf) = . This implies that 6 ∅ 0 < Rn fdx < . u∞ v LetR f = e− , g = e− be log-concave functions, for any real α, β > 0, the Asplund sum and scalar multiplication of two log-concave functions are defined as, w α f β g := e− , where w∗ = αu∗ + βv∗. (1.4) · ⊕ · Lp JOHNELLIPSOIDSFOR LOGCONCAVEFUNCTIONS 3

Here w∗ denotes as usual the Fenchel conjugate of the convex function ω. Cor- respond to the volume V (K) of a convex body K in Rn, the total mass J(f) of a log-concave function f in Rn is firstly considered in [24]. The functional counterpart of Minkowski’s first inequality and related isoperimetric inequalities are established. The Lp extension of the Asplund sum and scalar multiplication of two log-concave functions are discussed in [26], the functional Lp Minkowski’s first inequality and the functional Lp Minkowski problem also been discussed. Our main goals in this paper are to discuss the functional Lp John ellipsoid, based on the Lp Asplund sum and Lp scalar mutiplication of two log-concave functions. Owing to the functional Lp Minkowski’s first variation of f and g, we focus on the following: Problem S . Given a log-concave function f , find a Gaussian function p ∈ A0 γφ which solves the following constrained maximization problem: J(γ ) max φ subject to δJ f,γ 1. (1.5) c p φ ≤ n 2 n φx k k Where c = (2π) 2 and φ GL(n), γ = e− 2 is the Gaussian function. n ∈ φ δJp f,γφ is the normalized first variation of the total mass J(f) with respect to theLp Asplund sum. In section 3, we prove that there exists a unique Gaussian function which solves the Problem Sp. The unique Gaussian function which solves the problem Sp is called the Lp John ellipsoid for the log-concave function f, and denoted by Epf. Moreover, we characterize a Gaussian function which is the solution of the problem Sp. In section 4, we focus on the continuity of the Lp John ellipsoid, we prove that the Lp John ellipsoid Epf is continuous with respect to f and p. By the Lp Minkowski’s first inequality for log-concave function, we prove that the total mass of the Lp John ellipsoid Epf is no more than the total mass of f. In the end of this section, we show the similar Ball’s volume ration inequality is also holds for log-concave function.

2. Preliminaries 2.1. Convex bodies. In this paper, we work in n-dimensional Euclidean space, Rn, endowed with the usual scalar product x, y and norm x . Let Bn = x n h i n 1 k k n { ∈ R : x 1 denote the standard unit ball and S − = x R : x = 1 denotek thek ≤ unit} sphere in Rn. Let n denote the class of convex{ ∈ bodiesk k in Rn}, n K and o be the subclass of convex bodies K whose relative interior int(K) is nonempty.K For i n, let i be the i-dimensional Hausdorff measure, we indicate by V (K)= n(K≤) the n-dimensionalH volume. Let h ( ):HRn R be the support function of K; i.e., for x Rn, K · → ∈ h (x) = max x, y : y K . K h i ∈ Let nK (x) be the unit outer normal at x ∂K, then hK(nK (x)) = nK(x), x . It is shown that the sublinear support function∈ characterizes a convexh body and,i conversely, every sublinear function on Rn is the support function of a nonempty 4 F. CHEN, J. FANG, M. LUO, C. YANG compact convex set. By the definition of the support function, if φ GL(n), then the support function of the image φK := φy : y K is given by∈ { ∈ } t hφK(x)= hK(φ x), where φt denotes the transpose of φ. Let K n be a convex body that contains ∈Ko the origin in its interior, the polar body K◦ is defined by n K◦ = y R : y, x 1, for all x K . ∈ h i ≤ ∈ t Obviously, for φ GL(n), then (φK)◦ = φ− K◦. The gauge function K is defined by ∈ k·k

x K = min a 0 : x αK = max x, y = hK◦ (x). k k ≥ ∈ y K◦h i ∈ It is clear that x = 1 whenever x ∂K. k kK ∈ Recall that the Lp (p 1) Minkowski combination of convex bodies K and L is deﬁned as ≥ p p p hK+pǫ L(x) = hK(x) + ǫhL(x) . (2.1) ·

One of the most important inequality related to the Lp Brunn-Minkowski combination of convex bodies K and L is p p p V (K + L) n V (K) n + V (L) n , p ≥ with equality if and only if K and L are dilation of each other. The Lp surface area measure of K is defined by 1 p dS (K, )= h − dS(K, ), (2.2) p · K · where dS(K, ) is the classical surface area measure, which is given by · V (K + ǫL) V (K) lim − = hQ(u)dS(K,u). ǫ 0+ ǫ Z n 1 → S − n p n It is easy to say that for λ> 0, Sp(λK, )= λ − Sp(K, ). If K o , then K has n· 1 · ∈K a curvature function, then fp(K, ): S − R, the Lp-curvature function of K, is defined by · → 1 p f (K, )= h − f(K, ), p · K · n 1 n where f(K, ) is the curvature function, f(K, ): S − R defined as the Radon-Nikodym· derivative · → dS(K, ) f(K, )= · , · dS n 1 and dS is the standard Lebesgue measure on S − . For quick reference about the definition and notations in convex geometry, good references are Gardner [31], Gruber [33], Schneider [53]. Lp JOHNELLIPSOIDSFOR LOGCONCAVEFUNCTIONS 5

2.2. Functional setting. In the following, we discuss in the functional setting in Rn. Let u : Rn R + be a convex function, that is u (1 t)x + ty → ∪{ ∞} − ≤ (1 t)u(x)+ tu(y) for t (0, 1). We set dom(u) = x Rn: u(x) R ,by the− convexity of u, dom(u∈) is a convex set in Rn. We{ say∈ that u is∈ proper} if dom(u) = , and u is of class 2 if it is twice differentiable on int dom(u) , with 6 ∅ C+ a positive definite Hessian matrix. Recall that the Fenchel conjuga te of uis the convex function defined by

u∗(y) = sup x, y u(x) . (2.3) x Rn h i − ∈ n It is obvious that u(x)+ u∗(y) x, y for x, y R , there is an equality if and only if x dom(u) and y is in the≥h subdifferentiali ∈ of u at x, that means ∈ u∗( u(x)) + u(x)= x, u(x) . (2.4) ∇ h ∇ i The convex function u : Rn R + is lower semi-continuous, if the subset x Rn : u(x) > t is an open→ set∪{ for∞} any t ( , + ]. Moreover, if u is a { ∈ } ∈ −∞ ∞ lower semi-continuous convex function, then also u∗ is a lower semi-continuous convex function, and u∗∗ = u. The infimal convolution of functions u and v from Rn to R + is defined by ∪{ ∞} uv(x) = inf u(x y)+ v(y) . (2.5) y Rn − ∈ The right scalar multiplication by a nonnegative real number α: αu x , ifα> 0; uα (x) := α (2.6) I 0, ifα = 0. { } The following results below gather some elementary properties of u, the Fenchel conjugate and the infimal convolution, which can be found in [24, 50]. Lemma 2.1. Let u : Rn R + , then there exist constants a and b, with a> 0, such that, for x →Rn ∪{ ∞} ∀ ∈ u(x) a x + b. (2.7) ≥ k k n Moreover u∗ is proper, and satisfies u∗(y) > , y R . −∞ ∀ ∈ Proposition 2.2. Let u : Rn R + be a convex function. Then: → ∪{ ∞} (1) uv ∗ = u∗ + v∗; x (2) (uα)∗(x)= αu∗( α ), α> 0; (3) dom(uv)= dom(u)+ dom(v); (4) it holds u∗(0) = inf(u), in particular if u is proper, then u∗(y) > ; − −∞ inf(u) > implies u∗ is proper. −∞ A function f : Rn R + is called log-concave if for x, y Rn and 0

1 t t f (1 t)x + ty f − (x)f (y). − ≥ 6 F. CHEN, J. FANG, M. LUO, C. YANG

If f is a strictly positive log-concave function on Rn, then there exist a convex n u function u : R R + such that f = e− . Following the notations in paper [24], let → ∪{ ∞} = u : Rn Rn + proper, convex, lim u(x)=+ L → ∪{ ∞}| x + ∞} | |→ ∞ n u = f : R R f = e− ,u . A → | ∈L Let f be a log-concave, according to a series of papers by Artstein-Avidan ∈A u and Milman [9], Rotem [51], the support function of f = e− is deﬁned as,

h (x)=( log f(x))∗ = u∗(x). (2.8) f − Here the u∗ is the Fenchel conjugate of u. The definition of hf is a proper generalization of the support function hK , in fact, one can easily checks hχK = hK. Obviously, the support function hf shares the most of the important properties of hK . u u The polar function of f = e− is defined by f ◦ = e− ∗ . Specifically, x,y e−h i f ◦(y) = inf , x Rn ∈ n f(x) o it follows that, f ◦ is also a log-concave function. Let φ GL(n), we always write f φ(x) = φf(x) = f(φx). The following ∈ ◦ proposition shows that hf is GL(n) covariant which is proved in [27]. Proposition 2.3. Let f . For φ GL(n) and x Rn, then ∈A ∈ ∈ t hφf (x)= hf (φ− x). Moreover, for the polar function of f, t (φf)◦ = φ− f ◦. The class of log-concave functions can be endowed with an algebraic structure which extends in a natural way as theA usual of the Minkowski’s structure on n. For example, the Asplund sum of two log-concave functions is correspondedK to the classical Minkowski sum of two convex bodies. See [24] for more about the Asplund sum and the related inequalities of the total mass of the log-concave function in which correspond to the convex bodies in n. In very recently, the A K Lp Asplund sum of log-concave functions are studied by author Fang, Xing and Ye [26]. Let u = e− : u A0 { ∈L0}⊂A with = u : u 0, (u∗)∗ = u and u(o)=0 . L0 ∈L ≥ Clearly, if u , thenu has its minimum attained at the origin o. ∈L0 u v Definition 2.1 ([26]). Let f = e− , g = e− 0, and α, β 0. The Lp (p 1) Asplund sum and multiplication of f and g is∈A defined as ≥ ≥

[(u pα)p(v pβ)] α f β g = e− · · , (2.9) ·p ⊕p ·p where 1 p p p ∗ (u α) (v β)= α(u∗) + β(v∗) . ·p p ·p h i Lp JOHNELLIPSOIDSFOR LOGCONCAVEFUNCTIONS 7

The Lp Asplund sum is an extension of the Asplund sum on 0. Specially, when p = 1, it reduces to the Asplund sum of two functions on A, that is A0 x y α y β α f β g (x) = sup f − g . (2.10) · ⊕ · y Rn α β ∈ 1 x β p Moreover, when α = 0 and β > 0, we have (α p f p β p g)(x)= g 1 ; when · ⊕ · β p 1 x α p α > 0 and β = 0, then (α p f β p g)(x)= f 1 ; ﬁnally, when α = β = 0, · ⊕ · α p we set α p f p β p g = I 0 . We say that the Lp Asplund sum for log-concave · ⊕ · { } n functions is closely related to the Lp Minkowski sum for convex bodies in R . For examples, K, L n, let ∈K

IK (x) 1, if x K; χK (x)= e− = ∈ (2.11) 0, if x / K, ∈ where IK is the indicator function of K, and it is a lower semi-continuous convex function, 0, if x K; IK (x)= ∈ (2.12) , if x / K. ∞ ∈ The characteristic function χK is log-concave functions with u = IK belongs to , u∗ = h belongs to if 0 int(K), for p [1, + ), the function L K L ∈ ∈ ∞ 1 p p p ∗ (IK ) p α p (IL) p β = α(IK∗ ) + β(IL∗ ) · · h i 1 p p p ∗ = αhK + βhL h i = Iα pK+pβ pL. · · [IK pαpIL pβ] Then α χK β χL = e− · · = χα pK+pβ pL. · ⊕ · · · The following Proposition assert that the Lp Asplund sum of log-concave functions is closed in . A0 Proposition 2.4 ([26]). Let f and g belong both to the same class 0, and α, β 0. Then f α β g belongs to . A ≥ ·p ⊕p ·p A0 The total mass function of f is deﬁned as

J(f)= f(x)dx. (2.13) ZRn

Clearly, when f = χK, J(f)= V (K). Similar to the integral expression of mixed u v volume V (K, L), for f = e− and g = e− in 0, the quantity δJ(f,g), which is called as the ﬁrst variation of J at f along g isA deﬁned by (see [24]) J(f t g) J(f) δJ(f,g) = lim ⊕ · − . t 0+ t → It has been shown that δJ(f,g) has the following integral expression,

δJ(f,g)= hgdµ(f, x), (2.14) ZRn 8 F. CHEN, J. FANG, M. LUO, C. YANG where µ(f, x) is the surface area measure of f on Rn and is given by u(f, x)=( u(x)) f( n), (2.15) ∇ ♯ H here u is the gradient of u in Rn, that means, for any Borel function g , ∇ ∈A u(x) g(x)dµ(f, x)= g( u(x))e− dx. (2.16) ZRn ZRn ∇ Specially, if take f = g in (2.14), then

δJ(f, f)= nJ(f)+ f log fdx = J(nf + f log f). (2.17) ZRn

In the following sections we write J(nf + f log f) in terms of J(f ⋄) for simplicity. The L surface area measure of f, denoted by µ (f, ) is given in [26]. p p · u Deﬁnition 2.2 ([26]). Let f = e− be a log-concave function, the L ∈ A0 p surface area measure of f, denoted as µp(f, ), is the Boreal measure on Ω such that ·

1 p g(y)dµp(f,y)= g( u(x))(hf ( u(x))) − f(x)dx, (2.18) ZΩ Z x dom(u): u(x) Ω ∇ ∇ { ∈ ∇ ∈ } holds for every Borel function g such that g L1 µ (f, ) . ∈ p · Similarly, the first variation of the total mass at f along g with respect to the Lp Asplund sum is defined as, Definition 2.3 ([26]). Let f, g . For p 1, the first variation of the total ∈A0 ≥ mass of f along g with respect to the Lp Asplund sum is defined by

J(f p t p g) J(f) δJp(f,g) = lim ⊕ · − , (2.19) t 0+ t → whenever the limit exists.

The following integral expression of δJp(f,g) is Lp extension of (2.14) which is IK (x) IL(x) established in [26]. Specially, if take f = e− and g = e− , where IK and IL are the indicator function of K and L. So J(f) = V (K) and J(g) = V (L), then we have δJ(f,g)= Vp(K, L). u v Theorem 2.5 ([26]). Let f = e− 0 and g = e− 0. For p 1, assume that g is an admissible p-perturbation∈A for f. In addition,∈A suppose that≥ there exists a constant k > 0 such that 2 p n(p 1) 2 det (h ) k(h ) − det( h ), ∇ f ≤ f ∇ f holds for all x Rn o . Then ∈ \{ } 1 p δJp(f,g)= (hg) dµp(f, x). (2.20) p ZRn Note that the support function of the log-concave function is nondecreasing, it’s easy to get that if g g , then 1 ≤ 2 δJ (f,g ) δJ (f,g ). p 1 ≤ p 2 Lp JOHNELLIPSOIDSFOR LOGCONCAVEFUNCTIONS 9

u v In the following, we normalize the δJp(f,g). For f = e− , g = e− 0, and 1 p< , we deﬁne ∈A ≤ ∞ 1 p 1 p δJp(f,g) p 1 hg p δJp(f,g)= · = hf dµ(f, x) , (2.21) J(f ⋄) hJ(f ⋄) ZRn hf i Note that hf dµ(f,x) is a probability measure on Rn. For p = deﬁne J(f ⋄) ∞ h (x) δJ (f,g) = max g : x Rn . (2.22) ∞ nhf (x) ∈ o Unless hg(x) is a constant on Rn, by the Jensen’s inequality, it follows that hf (x) δJp(f,g) < δJq(f,g), for 1 p

u v v0 Lemma 2.7. Let f = e− , g = e− , g = e− , then 0 ∈A0 hg hg0 δJ (f,g) δJ (f,g ) k − k∞ , (2.24) p − p 0 ≤ min h : x Rn {| f | ∈ } for all p [1, ], where denotes the norms. ∈ ∞ k·k∞ ∞ Proof. First suppose that p < , by (2.21) and the triangle inequality for Lp norms, we have ∞ p 1 1 hg hg0 p δJp(f,g) δJp(f,g0) hf dµ(f, x) − ≤ J(f ) ZRn hf − hf h ⋄ i 1 1 1 p p hf dµ(f, x) hg hg0 ≤ J(f ) ZRn hf k − k∞ h ⋄ | | i hg hg0 k − k∞ . ≤ min h (x) : x Rn {| f | ∈ } The third inequality we use the fact that hf dµ(f,x) is a probability measure on Rn. J(f ⋄) For p , the continuous of the Lp norm with p shows that (2.24) holds for p = →as ∞ well. ∞ The following Lemma shows some Properties of δJp(f,g) and its normalizer. u v Lemma 2.8. Suppose that f = e− , g = e− 0, then 1 ∈A (1) δJp(f, f)= p J(f ⋄). (2) δJp(f, f) = 1. (3) δJp(f,λ p g)= λδJp(f,g), for λ> 0. · 1 (4) δJp(f,λ p g)= λ p δJp(f,g), for λ> 0. · 1 1 (5) δJp(φf,g)= det φ − δJp(f,φ− g), for all φ GL(n). | | 1 ∈ (6) δJ (φf,g)= δJ (f,φ− g), for all φ GL(n). p p ∈ 10 F. CHEN, J. FANG, M. LUO, C. YANG

Proof. By formula (2.20) and Definition 2.3, it immediately gives (1) and (2). 1 p In order to prove (3), by Definition 2.1, we have hλ pf = λ hf . So we have δJ (f,λ g)= λδJ (f,g). By (3), it yields (4) directly. · p ·p p By the integral formula of the first variation (2.20), and note that x(φu) = φt u, we have ∇ ∇φx 1 p δJp(φf,g)= hg(x)dµp(φf,x) p ZRn

1 p 1 p φu = hg x(φu) hφf− x(φu) e− dx p ZRn ∇ ∇ 1 p 1 p u(φx) = hφ 1g φxu hf− φxu e− dx p ZRn − ∇ ∇ 1 1 p 1 p u = det φ− hφ 1g u hf− u e− dx | |p ZRn − ∇ ∇ 1 1 = det φ − δJ (f,φ− g). | | p 1 On other hand, note that J (φf)⋄ = det φ − J(f ⋄) and together with (5), it 1 | | follows that δJp(φf,g)= δJp(f,φ − g). So we complete the proof. The following result will been used in the next section (see [52]).

Proposition 2.9 ([52]). Let D be a relatively open convex sets, and f1, f2, , be a sequence of ﬁnite convex functions on D. Suppose that the real number··· f (x), f (x), , is bounded for each x D. It is then possible to selected a 1 2 ··· ∈ subsequence of f1, f2, , which converges uniformly on closed bounded subset of D to some ﬁnite convex··· function f.

3. Lp John ellipsoid for log-concave functions

x 2 k k Let γ = e− 2 be the standard Gaussian function. In the following, we set

φx 2 k k γφ(x)= e− 2 , where φ GL(n). It is worth noting that the Gaussian function γφ plays an important∈ role in the study of the extremal problem of log-concave functions as the ellipsoids do for the study of the extremal problems of convex bodies. In fact, it is the unique function of which is self-dual, that is A x 2 k k o f = e− 2 f = f. ⇐⇒ Now, Let us consider the following optimization problem. The L optimization problem S (p 1) for log-concave functions f: Given p p ≥ a log-concave function f 0, ﬁnd a Gaussian function γφ which solves the following constrained maximization∈ A problem: J(γ ) max φ subject to δJ f,γ 1. (3.1) c p φ ≤ n Lp JOHN ELLIPSOIDS FOR LOG CONCAVE FUNCTIONS 11

The dual L optimization problem S (p 1) for log-concave functions f: p p ≥ Given a log-concave function f 0, ﬁnd a Gaussian function γφ which solves the following constrained minimization∈ A problem: J(γ ) min δJ f,γ subject to φ 1. (3.2) p φ c ≥ n Lemma 3.1. These above optimization problems for log-cocave functional are equivalent to: (1) The problem Sp is equivalent to:

J(γφ) max subject to δJp(f,γφ)=1. cn (2) The dual problem Sp is equivalent to:

J(γφ) min δJp(f,γφ) subject to =1. cn

φx 2 k k J(γφ) Proof. (1) Assume that γφ = e− 2 is the solution of the problem Sp, if cn obtains the maximum, but δJ (f,γ ) = 1, assume that δJ (f,γ ) < 1, then let p φ 6 p φ 1 γ = p γφ. φ p δJp(f,γφ) · 1 Then hγ = hγφ . Moreover, we have φ δJp(f,γφ) 1 1 p p δJp(f, γ )= h dµp(f, x) φ γφ hJ(f ⋄) ZRn i 1 1 1 p p = hγφ dµp(f, x) δJp(f,γφ)hJ(f ⋄) ZRn i =1.

φx 2 k k On the other hand, since γφ = e− 2 , by the Deﬁnition 2.1 of the Lp multiplication, a simple computation shows that, for λ R, ∈ 2 φ(x/λ) n p λ k k J λ p γφ = e− 2 dx = λ 2 J(γφ). (3.3) · ZRn n Note that δJp(f,γφ)− 2 > 1, then we have

1 n J p γφ = δJp(f,γφ)− 2 J(γφ) J(γφ). p δJp(f,γφ) · ≥ This means that γφ is not a solution to the problem Sp, which is contract with our assumption, so we complete the proof of the ﬁrst one. The similar with the proof of Problem Sp, so we complete the proof. Lemma 3.2. Suppose f . If γ is a Gaussian function that is an solution ∈ A0 φ for the problem Sp of f, then 2p cn n p γφ J(γφ) · 12 F. CHEN, J. FANG, M. LUO, C. YANG is an solution for problem Sp of f. Conversely, if γφ is a Gaussian function that is an solution for the Sp problem of f, then 1 p γ p φ δJp(f,γφ) · is an solution for problem Sp of f.

Tx k k Proof. Let γT = e− 2 , where T GL(n), and satisﬁes J(γT ) cn. By Lemma 1 ∈ ≥ 2.8, it obviously has δJp f, p p γT =1. Since γφ is an Sp solution for f, δJp(f,γT ) · then 1 J(γφ) J p γT . p ≥ δJp(f,γT ) · By (3.3) it shows

1 n J p γT = δJp(f,γT )− 2 J(γT ). (3.4) p δJp(f,γT ) ·

According to Lemma 3.1, we have δJp(f,γφ) = 1. Then 2 2 2p J(γT ) n cn n cn δJ (f,γ ) = δJ f, n γ . p T ≥ J(γ ) ≥ J(γ ) p J(γ ) ·p φ φ φ φ 2p cn n On the other hand, since J p γφ = cn, it implies that the ellipsoid J(γφ) · 2p cn n p γφ is a solution to Problem Sp. We complete the ﬁrst assertion’s J(γφ) · proof. Tx 2 k k Let γ = e− 2 , T GL(n), such that δJ (f,γ ) 1. Then we have T ∈ p T ≤ cn 2p J n γ c . J(γ ) ·p T ≥ n T Since γφ is solution of Sp, we have 2 2p cn n cn δJ (f,γ )= δJ f, n γ δJ (f,γ ). J(γ ) p T p J(γ ) ·p T ≥ p φ T T By Lemma 3.1 we have J(γφ)= cn. Hence the above inequality can be rewritten as J(γφ) J(γT ) n n . δJp(f,γφ) 2 ≥ δJp(f,γT ) 2 By formula (3.4) again, it means 1 1 J p γ J p γ J(γ ). p φ p T T δJp(f,γφ) · ≥ δJp(f,γT ) · ≥ On the other hand, since 1 δJp f, p γ =1. p φ δJp(f,γφ) · This completes the proof. Lp JOHN ELLIPSOIDS FOR LOG CONCAVE FUNCTIONS 13

Theorem 3.3. Suppose that f . Then there exist a solution, γ , satisﬁes ∈ A0 φ the Optimization Problem Sp.

Proof. By the definition of the optimization problem Sp, let φ GL(n), if γφ is Gaussian function subject to J(γ )= c , then we have φ SL(n∈). The existence φ n ∈ of the solution of the Problem S is equivalent to find the φ SL(n) which p 0 ∈ solves the following. Let φ SL(n), choose ǫ0 > 0 sufficiently small so that for all ǫ ( ǫ , ǫ ), I + ǫφ is invertible.∈ For ǫ ( ǫ , ǫ ) define φ SL(n) by ∈ − 0 0 ∈ − 0 0 ǫ ∈ I + ǫφ φǫ = 1 , det(I + ǫφ) n here I is the identity matrix and detφ = 1. Then find φ such that | ǫ| 0 d δJp f,γφǫ =0. (3.5) dǫ ǫ=0 It is equivalent to

t 2 p d φǫ− x ǫ=0 k k dµp(f, x)=0. ZRn dǫ 2

Since the norm of the vector and tp are continuous functions, it grants there exists a solution of the above equation. So there exists a φ SL(n) such that ∈ δJp(f,γφ).

We say that the γφ is the solution of the Lp functional optimization problem Sp, and we rewrite it as the γf . The following Corollary are obviously. Corollary 3.4. Suppose f , γ be the solution of the optimization problem ∈ A0 f Sp, then

δJp(f,γf )=1. (3.6) By Lemma 3.2 and Theorem 3.3, it guarantees that there exists a unique solution for the optimization problem Sp. Theorem 3.5. Let f . Then problem S has a unique solution. Moreover ∈ A0 p a Gaussian function γφ solves Sp if and only if it satisﬁes

n 2 p 1 δJp(f,γφ)hγo (y)= x, y hγ−o (x)dµp(f, x), (3.7) φ 4p ZRn |h i| φ for any y Rn. ∈ In order to prove Theorem 3.5, we need the following Lemma.

u Lemma 3.6. Let f = e− and φ GL(n). If the Gaussian function γ ∈ A0 ∈ φ solves the optimization functional problem Sp, then

n 2 p 1 δJp(f,γφ)hγo (y)= x, y hγ−o (x)dµp(f, x). φ 4p ZRn |h i| φ 14 F. CHEN, J. FANG, M. LUO, C. YANG

Proof. By the SL(n) invariance of the δJp(f,g), we may assume that γφ = γ is the solution of problem S . Let φ SL(n), choose ǫ > 0 sufficiently small so p ∈ 0 that for all ǫ ( ǫ0, ǫ0), I + ǫφ is invertible. For ǫ ( ǫ0, ǫ0) define φǫ SL(n) by ∈ − ∈ − ∈ I + ǫφ φǫ = 1 , det(I + ǫφ) n here I is the identity matrix and detφ = 1. Then | ǫ| δJ f,γ δJ f,γ , p φ0 ≤ p φǫ for all ǫ ( ǫ , ǫ ). That means ∈ − 0 0 d δJp f,γφǫ =0. (3.8) dǫ ǫ=0 o t On the other hand, by Proposition 2.3, we have γ = φ− γ. Then hγo (x) = φǫ ǫ φǫ t hφǫ− γ(x)= hγ(φǫx). By the definition of δJp(f,g), we have

d p hγo (x)dµp(f, x)=0. dǫ ǫ=0 ZRn φ

It is equivalent to the following 2 p d φǫx ǫ=0 k k dµp(f, x)=0. ZRn dǫ 2 Or equivalently,

d 2p p n 2 ǫ=0 det(I + ǫφ)− x x +2ǫ x φx + ǫ φx φx dµp(f, x)=0. ZRn dǫ h · i h · i h · i Since d det(I + ǫφ)= trace(φ), by a simple computation, we have, dǫ |ǫ=0 2 p 1 n x − trace(φ)δJp(f,γ)= x, φx k k dµp(f, x). 2p ZRn h i 2 Choosing an appropriate φ for each i, j 1, 2, , n gives ∈{ ··· } n p 1 δi,jδJp(f,γ)= x, ei x, ej hγ− (x)dµp(f, x), 2p ZRn h ih i n where e1, , en is an otrhonormal basis of R , and δi,j is the Koronecker symbols. Which in··· turn gives

2 n 2 p 1 y δJp(f,γ)= x, y hγ− (x)dµp(f, x). (3.9) k k 2p ZRn |h i| That means

n 2 p 1 δJp(f,γ)hγ (y)= x, y hγ− (x)dµp(f, x). (3.10) 4p ZRn |h i| So we have

n 2 p 1 δJp(f,γφ)hγo (y)= x, y hγ−o (x)dµp(f, x). φ 4p ZRn |h i| φ We complete the proof. Lp JOHN ELLIPSOIDS FOR LOG CONCAVE FUNCTIONS 15

Now we prove Theorem 3.5.

Proof of Theorem 3.5. Lemma 3.6 grants that if γφ is and Sp solution of f, then the above formula holds. Covnersely, without loss of generality, we may prove that (3.7) holds when γ = γ, that is φ = I. Then for any φ GL(n), we shall prove that if J(γ )= c φ 1 ∈ φ1 n for some φ GL(n), 1 ∈ δJ (f,γ ) δJ (f,γ), p φ1 ≥ p with equality if and only if γφ1 = γ. Equivalently, we shall prove that if φ1 is φ x k 1 k 2 a positive deﬁnite symmetric matric with φ1 = 1, and note that γφ1 = e− then | |

1 p hγφ (x)dµp(f, x) 1, pδJp(f,γ) ZRn 1 ≥ or equivanently,

t 2 p 2 p 1 φ1− x x k 2k k k dµp(f, x) 1, pδJp(f,γ) ZRn x 2 ≥ k k

t φ− x k 1 k Rn with equality if and only x = 1, for x . t t k k ∈ Write φ1− = OTO , where T = diag(λ1, ,λn) is a diagonal matrix with eigenvalues λ , ,λ and O is an orthogonal··· matrix. To establish our inequality 1 ··· n we need to show that if φ1 is a positive deﬁnite symmetric matrix with det φ1 = 1, then

t p φ1− x x log k k k k dµp(f, x) 0. ZRn x 2 ≥ k k t On the other hand, since (φu) = φ φxu for each φ GL(n), and (φu)∗ = t ∇ ∇ ∈ φ− u∗, we have

T x 2 p x 2 p log k k2 k k dµp(Of,x) ZRn x 2 k k t t 2 p TO Oxu O Oxu 1 p t t − Ou =2p log k t ∇ k k ∇ k u∗(O− O Oxu) e− dx ZRn O Oxu 2 ∇ k ∇ k t 2 p TO u u 1 p u =2p log k ∇ k k∇ k u∗( u) − e− dx (3.11) ZRn u 2 ∇ k∇ k TOtx x 2 p =2p log k k k k dµp(f, x). ZRn x 2 k k 16 F. CHEN, J. FANG, M. LUO, C. YANG

Then we have t 2 p 2 p φ1− x x log k 2k k k dµp(f, x) ZRn x 2 k k TOtx x 2 p =2p log k k k k dµp(f, x) ZRn x 2 k k T x x 2 p =2p log k k k k dµp(Of,x) ZRn x 2 nk k 2 p 2 2 x = p log xi λi k k dµp(Of,x) (3.12) ZRn 2 Xi=1 n 2 p 2 2 x p xi log λi k k dµp(Of,x) ≥ ZRn 2 h Xi=1 i n

=2pδJp(Of,γ) log λi =0. Xi=1 x Where xi = x ei . Then we have h k k · i 1 1 p p hγφ (x)dµp(f, x) (3.13) hpδJp(f,γ) ZRn 1 i t 2 p 1 2 φ1− x x p exp log k k k k dµp(f, x) ≥ δJp(f,γ) ZRn x 2 h k k i 1. ≥ The ﬁrst inequality in (3.13) is a consequence of the Jensen’s inequality, with t φ− x k 1 k equality holds if and only if there exist a constant c> 0 such that x = c for all x Rn. k k Moreover,∈ note that from the strict concave of the log-concave function that equality in (3.12) is possible only if x1 xn = 0 which implies λ1 = = λn, n ··· 6 ··· for x R . Thus T x = λi when xi = 0. Now equality in (3.13) would force t ∈ k k 6 φ1− x t n k k = c or equivalently T − x = c for x R , so that λ = c for all i. This x k k ∈ i togetherk k with the fact λ λ = 1 shows that equality in (3.13) would imply 1 ··· n T = I and hence φ1 = I. Theorem 3.7. Let f . Then problem S has a unique solution. Moreover a ∈A0 p Gaussian function γφ solves Sp if and only if it satisﬁes

2 cn n t n 2 t p 1 δJp(f,γφ)hγ(√αφ− x)(y)= x, y hγ (√αφ− y) − dµp(f, x). J(γφ) 4p ZRn |h i| (3.14) for any y Rn. ∈ p v αv( x ) p Proof. By computation shows α p e− = e− α , then α p γφ = γ φ . So we · · √α o t obtain h p = hγ(√αφ− x). α pγ · φ Lp JOHN ELLIPSOIDS FOR LOG CONCAVE FUNCTIONS 17

Together with Lemma 3.2 and Theorem 3.5 we have 2 cn n t n 2 t p 1 δJp(f,γφ)hγ (√αφ− y)= x, y hγ (√αφ− y) − dµp(f, x). J(γφ) 4p ZRn |h i|

Now we deﬁne Lp John ellipsoid for log-concave functions.

Deﬁnition 3.1. Let f 0 be log-concave function, the unique Gaussian function that solves the constrained∈A maximization problem max J(φγ) subject to δJ (f,φγ) 1, (3.15) p ≤ is called the Lp John ellipsoid of f and denoted by Epf. The unique Gaussian function that solves the constrained minimization problem J(φγ) min δJp(f,φγ) subject to =1, (3.16) cn is called the normalized Lp John ellipsoid of f and denoted by Epf.

x 2 k kK n Specially, if we take f = e− 2 for K o , since the Gaussian function γφ x 2 ∈ K k kE can be viewed as γφ = e− 2 , where E is the origin-centered ellipsoid. Then Definition 3.1 deduce the definition of Lp John ellipsoid defined in [46]. 1 Since δJp(φf,g) = δJp(f,φ− g) for φ GL(n), then we have the following result. ∈ Proposition 3.8. Let f . Then ∈A0 Ep(T f)= T (Epf). (3.17)

Proof. By the deﬁnition of problem Sp, set Epf = γφ, since δJp(f,γφ) = 1. By the Lemma 2.8, we have 1 δJp(Tf,Tγφ)= δJp(f, T − Tγφ)= δJp(f,γφ)=1. (3.18)

By the uniqueness of the problem Sp, we have Ep(T f)= T (Epf). So we complete the proof.

If we take f = γ, then we have Epγ = γ. Moreover by Proposition 3.8, we have the following.

φx 2 k k Corollary 3.9. Let γ = e− 2 , for φ GL(n), then φ ∈ Ep(γφ)= γφ.

4. Continuity

In this section, we will show that the family of Lp John ellipsoids for log- u concave functions is continuous in p [1, ). First let f = e− , note ∈ ∞ ∈ A0 that if Epf is a solution of problem Sp, then there exist a φ SL(n) such that φx 2 ∈ k k Epf = γφ = e− 2 . 18 F. CHEN, J. FANG, M. LUO, C. YANG

n Theorem 4.1. Let f and fi in 0, such that limi fi = f on R . Then { } A →∞ lim Epfi = Epf. (4.1) i →∞ To prove Theorem 4.1, we need the following Theorem. First we prove that dµ (f , x) dµ (f, x) p i → p u u Lemma 4.2. Let f = e− , f = e− i , if f f, then dµ (f , x) dµ (f, x). i ∈A0 i → p i → p Proof. We only need to prove that, for any function g L1(µ (f, )), ∈ p ·

lim g(x)dµp(fi, x)= g(x)dµp(f, x). (4.2) i →∞ ZRn ZRn n n Set a = max g(x) : x R , b = max f(x) : x R bi = max fi(x) : x Rn , c {|= max| h ∈(x) }: x Rn ,{|c = max| ∈h (x)} : x Rn{|. Note| ∈ } i {| fi | ∈ } {| f | ∈ } that f and fi 0 are integrable, and hfi hf whenever fi f, then there ∈N A ǫ → → N exist an N1 such that fi f < 2acp for i N1, and N2 such that ∈ | − | i ≥ ∈ ǫ 1 p hf hf < p 1 p 1 j for i N2. Since dµp(f, x) = h − fdx, so we can i 2ab P − cj c − − f | − | j=0 i ≥ choose i max N , N , then ≥ { 1 2} gdµ (f , x) gdµ (f, x) ghp f ghp f dx, p i p fi i f ZRn − ZRn ≤ ZRn −

ghp f f dx + gf hp hp dx. fi i fi f ≤ ZRn | || − | ZRn | || − | ǫ. ≤ So we complete the proof.

u ui v vj Theorem 4.3. Suppose f = e− , fi = e− , g = r− , gj = e− 0, where i, j N. If f f, g g, then ∈ A ∈ i → i → lim δJp(fi,gj)= δJp(f,g). (4.3) i,j →∞ Proof. Since f f, g g, by the Deﬁnition of δJ (f,g), we have i → i → p p p hgi dµp(fi, x) hgdµp(f, x) (4.4) ZRn − ZRn

p p p p hgi dµp(fi, x) hgi dµp(f, x) + hgi hg dµp(f, x). ≤ ZRn − ZRn ZRn | − |

Rn Note that hg and hgi are bounded, set ci = max gi(x) : x and c0 = Rn {| | ∈p 1 j} p 1 j max g(x) : x , then ci, c0 are bounded. Let c = j=1− ci c0− − and {| | ǫ ∈ } hg hg , then P | i − | ≤ 2c p p ǫ hgi hg dµp(f, x) . (4.5) ZRn − | ≤ 2

By Lemma 4.2, we can show that

p p ǫ hgi dµp(fi, x) hgi dµp(f, x) . (4.6) ZRn − ZRn ≤ 2

Lp JOHN ELLIPSOIDS FOR LOG CONCAVE FUNCTIONS 19

So together with formulas (4.4), (4.5) and (4.6) we can choose uniﬁed N such that

p p hgi dµp(fi, x) hgdµp(f, x) ǫ. ZRn − ZRn ≤

We complete the proof. Now we give a proof of Theorem 4.1

Proof of Theorem 4.1. By Theorem 4.3, and J(fi⋄) J(f ⋄) when i , then we have → → ∞

lim δJp(fi, Epfi) = lim min δJp(fi,γφ) i i J(γ )=c →∞ →∞ φ n = min lim δJp(fi,γφ) J(γ )=c i φ n →∞ = min δJp(f,γφ) J(γφ)=cn

= δJp(f, Epf). We complete the proof. Note that E f is bounded for p [0, ]. Thus in order to establish the p ∈ ∞ continuity of Epf in p [0, ], (2.24) shows that the δJp(K, ) is continuity of p [1, ]. ∈ ∞ · ∈ ∞ Lemma 4.4. If p [1, ], then 0 ∈ ∞

lim δJp(f,γφ)= δJp0 (f,γφ). (4.7) p p0 → for some φ SL(n). ∈ u Theorem 4.5. Let f = e− 0, and 1 p q < , Epf be the solution of constrained maximization problem,∈ A then ≤ ≤ ∞

J(E f) J(Eqf) J(Epf) J(E1f). (4.8) ∞ ≤ ≤ ≤

Proof. The deﬁnition of δJp(f,g), together with Jensenn’s inequality, for 1 p q < , we have ≤ ≤ ∞ p 1 1 hγ p δJp(f,γ)= hf dµ(f, x) hJ(f ⋄) ZRn hf i q 1 1 hγ q hf dµ(f, x) ≤ hJ(f ⋄) ZRn hf i = δJq(f,γ). By Deﬁnition 3.1, we have

E f = max E′f : δJ (f,Ef) 1 max E′f : δJ (f,Ef) 1 = E f. q q ≤ ≤ q ≤ p This implies J(E f) J(E f). For p , by the deﬁnition of (2.22 ), and the q ≤ p →∞ continuity of p [1, ], we have E f = limp Epf, we complete the proof. ∈ ∞ ∞ →∞ 20 F. CHEN, J. FANG, M. LUO, C. YANG

u Theorem 4.6. Let f = e− 0, such that J(f) > 0. Let 1 p < , Epf be the solution of constrained maximization∈ A problem, then ≤ ∞ J(E f) J(f). (4.9) p ≤ Proof. By the deﬁnition 3.1, we have

p p δJp(f, Epf) 1= δJp(f, Epf) = · . J(f ⋄)

The Lp Minkowski inequality for log-concave functions (see [26]) says that J(f ) J(E f) ⋄ δJ (f, f)+ J(f) log p . p ≥ p J(f) This means that J(E f) J(f) log p 0. J(f) ≤ Sine J(f) > 0, then log J(Epf) 0. That means J(f) ≤ J(E f) J(f). p ≤ The functional Blaschke-Santal´oinequality, proved for even functions in [11], and given in full generality in [6] says that for log-concave function f 0, the following inequality holds ∈A P (f) P (γ), ≤ that is J(f)J(f o) J(γ)2 = (2π)n. ≤ So combine with the Theorem 4.6, we have the following Theorem. u Theorem 4.7. Let f = e− 0, such that J(f) > 0. Let 1 p < , Epf be the solution of constrained maximization∈ A problem, then ≤ ∞ J(E f)J(E f o) c2 . (4.10) p p ≤ n n where cn = (2π) 2 . n In the following, denoting by n and B the regular simplex centered at the n △ ∞ origin and the unit cube in R . To establish the Lp Ball’s ratio inequality for log-concave function, we need the following results, more details see [5]. Theorem 4.8 ([5]). Let f , Ef be the functional John ellipsoid of f, then ∈A0 J(f) J(g ) c , (4.11) J(Ef) ≤ J(Egc)

x n c where gc(x) = e−k k△ − for any c n. Furthermore , there is equality if and f ∈ △ only if f = Tgc for some aﬃne map T and some c n. If we assume f to ∞ ∈△ be even,k thenk J(f) J(g) , (4.12) J(Ef) ≤ J(Eg) Lp JOHN ELLIPSOIDS FOR LOG CONCAVE FUNCTIONS 21

x Bn f where g(x)= e−k k ∞ , with equality if and only if f = Tg for some linear map T GL(n). k k∞ ∈ Moreover, by compute the right hand of the above formulas, it gives Lemma 4.9 ([5]). Let f , Ef be the functional John ellipsoid of f, then ∈A0 J(f) e 1 n (n!) n |△ | , (4.13) J(Ef) ≤ n E | △n| If we assume f to be even, then n e 1 B I.rat(f) (n!) n | ∞| . (4.14) ≤ n EBn | ∞ | Now we give the Ball’s volume ration inequality for log-concave function.

u Theorem 4.10. Let f = e− 0, such that J(f) > 0. Let 1 p< , Epf be the solution of constrained maximization∈A problem, then ≤ ∞

n 2 n+1 J(f) n −n (n + 1) 2 e n 1 . (4.15) − J(Epf) ≤ (n!) n ωn If f is even then n J(f) e 1 2 (n!) n . (4.16) J(Epf) ≤ n ωn n where ωn is the volume of ball B . Proof. By Theorem 4.5, and the fact E f = Ef, then we have ∞ J(f) J(f) e 1 n (n!) n |△ | . J(E f) ≤ J(Ef) ≤ n J p | △n| √n+1 On the other hand, note that the volume of n is given by n = n , and the 2 2 n! 1 △ |△ | inradius of n is given by r n = . Then by a simple computation gives △ △ √2n(n+1)

n 2 n+1 − J(f) e 1 n n n (n + 1) 2 e n (n!) |△ | = n 1 . J(Epf) ≤ n J n (n!) −n ω | △ | n If f is even, then n n J(f) e 1 B e 1 2 n n (n!) | ∞n| = (n!) . J(Epf) ≤ n JB n ωn | ∞| So we complete the proof. Acknowledgments

The authors would like to strongly thank the anonymous referee for the very valuable comments and helpful suggestions that directly lead to improve the orig- inal manuscript. 22 F. CHEN, J. FANG, M. LUO, C. YANG

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1. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou 550025, People’s Republic of China. Email address: [email protected] Email address: [email protected] 2. School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, People’s Republic of China Email address: [email protected] Email address: [email protected]