The functional inequality for the mixed quermassintegral

In this paper, the functional Quermassintegrals of a log-concave function in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document} are discussed. The functional inequality for the ith mixed Quermassintegral is established. Moreover, as a special case, a weaker log-Quermassintegral inequality in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{n}$\end{document} is obtained.


Introduction
Let K n be the set of convex bodies (compact convex subsets with nonempty interiors) in R n , the fundamental Brunn-Minkowski inequality for convex bodies states that, for K, L ∈ K n , the volume of the bodies and of their Minkowski sum K + L = {x + y : x ∈ K, and y ∈ L} are given by V (K + L) with equality if and only if K and L are homothetic, namely agreeing up to a translation and a dilation. The Brunn-Minkowski inequality exposes the crucial logarithmic concavity of the volume in K n , because it has an equivalent formulation as for t ∈ (0, 1). See for example [18,19,29] for more about the Brunn-Minkowski inequality. Another important geometric inequality related to the convex bodies K and L is the mixed Quermassintegral inequality, W n (K) = ω n , the volume of the unit ball B n 2 in R n and, for general i = 1, 2, . . . , n -1, where the G i,n is the Grassmannian manifold of i-dimensional linear subspaces of R n , dμ(ξ i ) is the normalized Haar measure on G i,n , K| ξ i denotes the orthogonal projection of K onto the i-dimensional subspaces ξ i , and vol i is the i-dimensional volume on space ξ i . In the 1960s, the Minkowski addition was extended to the L p (p ≥ 1) Minkowski sum by defining h p K+ p t·L = h p K + th p L . The extension of the mixed Quermassintegrals to the L p mixed Quermassintegrals is due to Lutwak [24]. The inequalities for the L p mixed Quermassintegrals are established by Lutwak. Let K, L ∈ K n with origin in their interiors, 0 ≤ i < ni and p > 1, then with equality if and only if K and L are dilates. Here the L p mixed Quermassintegrals are defined by for i = 0, 1, . . . , n -1. In particular, for p = 1 in (0.4), it becomes W i (K, L), and W p,0 (K, L) is denoted by V p (K, L), which is called the L p mixed volume of K and L. Motivated by the analogy properties between the log-concave functions and the volume of convex bodies in K n , the interest in studying the log-concave functions has been considerably increasing. For example, the functional Blaschke-Santaló inequality for even logconcave function is discussed by Ball in [6,7]; for the general case see [8,17,21,28]. The mean width for a log-concave function is introduced by Klartag, Milman and Rotem (see [22,26,27]). The affine isoperimetric inequality for the log-concave function is proved by Artstein-Avidan, Klartag, Schütt and Werner [5]. The John ellipsoid for log-concave function has been establish by Gutiérrez, Merino Jiménez and Villa [2], the LYZ ellipsoid for log-concave function is established by Fang and Zhou [16]. See [1,4,9,[12][13][14]23] for more about the pertinent results.
Let f = e -u , g = e -v be log-concave functions, α, β > 0, the "sum" and "scalar multiplication" of log-concave functions are defined as here w * denotes as usual the Fenchel conjugate. The total mass integral J(f ) of f is defined as J(f ) = R n f (x) dx. In [15], the quantity δJ(f , g), which is called the first variation of J at f along g, is defined by Colesanti and Fragalà, The authors show that the functional form of Minkowski's first inequality is where Ent(f ) is the entropy of f defined by Ent(f ) = R n f log f dx -J(f ) log J(f ). We have inequality in (0.5) if and only if there exist x 0 ∈ R n such that g(x) = f (xx 0 ). Inspired by Ref. [15] of Colesanti and Fragalà, in this paper, we define the ith functional Quermassintegrals W i (f ) as the i-dimensional average total mass of f , where J i (f ) denotes the i-dimensional total mass of f defined in (3.1), G i,n is the Grassmannian manifold of R n and dμ(ξ n-i ) is the normalized measure on G i,n . The first variation of W i at f along g is defined by (see Definition 3.3) W i (f , g) is a natural extension of the mixed Quermassintegrals of convex bodies in R n , we call it the ith functional mixed Quermassintegral. In fact, if one takes f = χ K , and dom(f ) = K ∈ R n , then W i (χ K ) turns out to be W i (K), and W i (χ K , χ L ) equals W i (K, L). In this paper, our main result is to show the inequality for the ith functional mixed Quermassintegrals. Let A denote the integrable functions in R n .
Theorem 0.1 Let f and g are integrable functions on A , then with equality if and only if there exists x 0 ∈ R n such that g( The paper is organized as follows, in Sect. 1, we introduce some notations about the log-concave function. In Sect. 2, the projection of log-concave function is discussed. In Sect. 3, we turn our attention to the functional inequalities involving W i (f , g), we prove the ith functional mixed Quermassintegral inequality. Specially, the weaker log-Quermassintegral inequality for convex bodies is obtained as a corollary.

Preliminaries
Let u : → (-∞, +∞] be a convex function, that is, The convexity of u implies that is a convex set in R n . We say that u is proper if = ∅, and u is of class C 2 + if it is twice differentiable on int( ), with a positive definite Hessian matrix. Let It is obvious that u(x) + u * (y) ≥ x, y for all x, y ∈ , and there is an equality if and only if x ∈ , and y is the subdifferential of u at x, which means Moreover, if u is a lower semi-continuous convex function, then also u * is a lower semicontinuous convex function, and u * * = u. The infimal convolution of functions u and v is defined by The right scalar multiplication by a nonnegative real number α is given by (1.4) The following propositions below gather some elementary properties of the Fenchel conjugate and the infimal convolution of u and v, which can be found in [15,25].  is. In this case (C * , u * ) is the Legendre conjugate of (C, u) (and conversely). Moreover, ∇u := C → C * is a continuous bijection, and the inverse map of ∇u is precisely ∇u * .
If f is a strictly positive log-concave function on R n , then there exists a convex function u : → (-∞, +∞] such that f = e -u . The log-concave function is closely related to the convex geometry of R n . An example of a log-concave function is the characteristic function χ K of a convex body K in R n , which is defined by where I K is a lower semi-continuous convex function, and the indicator function of K is Let us generalize f to the domain of R n by (1.8) In the later sections, we also use f to denote f having been extended to R n , let A = {f : R n → (0, +∞] : f = e -u , u ∈ L} be the subclass of f .
and α, β ≥ 0. The sum and multiplication of f and g are de- In particularly, when α = 0 and β > 0, we  Let f ∈ A, the support function of f = e -u is defined by here the u * is the Legendre transform of u. The definition of h f is a proper generalization of the support function h K , in fact, one can easily check h χ K = h K (see [3,26]). Specifically, the function h : A → L has the following properties [27]: The following proposition shows that h f is GL(n) covariant.
Let u, v ∈ L, denote u t = u vt (t > 0), and f t = e -u t . The following lemmas describe the monotonicity and convergence of u t and f t , respectively.
For t > 0, set u t = u (vt), f t = e -u t , and assume that v(0) = 0. Then, for every fixed x ∈ R n , u t (x) and f t (x) are, respectively, pointwise decreasing and increasing with respect to t; in particular (1.12) Lemma 1.7 ([15]) Let u and v belong both to the same class L and, for any t > 0, set u t := u (vt), assume that v(0) = 0. Then Then ∀x ∈ int( t ), and ∀t > 0, where ψ := v * .

Projection of functions onto linear subspace
The elements of G i,n will usually be denoted by ξ i and ξ ⊥ i stands for the orthogonal The projection of f onto ξ i is defined by (see [20,22]) Here ξ ⊥ i is the orthogonal complement of ξ i in R n , | ξ i is the projection of onto ξ i . By the definition of the log-concave function f = e -u , for every x ∈ | ξ i , one can rewrite (2.1) as As regards the "sum" and "multiplication" of f , we say that the projection keeps the structure on R n . In other words, we have the following proposition.
Then, by the definition of the projection, we have Taking the supremum of the second right hand inequality over all Since f , g ≥ 0, the inequality max{f · g} ≤ max{f } · max{g} holds. So we complete the proof.

Proposition 2.2 Let
by the definition of the projection, we complete the proof.
For the convergence of f we have the following.
Proof Since lim n→∞ f n = f 0 , it means that, for ∀ > 0, there exists N 0 , ∀n > N 0 , such that f 0 -≤ f n ≤ f 0 + . By the monotonicity of the projection, we have Hence each {f n | ξ i } has a convergent subsequence, we denote it also by {f n | ξ i }, converging to some f 0 | ξ i . Then, for x ∈ ξ i , we have By the arbitrariness of we have f 0 | ξ i = f 0 | ξ i , so we complete the proof.
Combining with Proposition 2.3 and Proposition 1.7, it is easy to obtain the following proposition.

Proposition 2.4
Let u and v belong both to the same class L, ∈ R n be the domain of u, for any t > 0, set u t = u (vt). Assume that v(0) = 0 and ξ i ∈ G i,n , then Now let us introduce some facts about the functions u t = u (vt) with respect to the parameter t. Lemma 2.5 Let ξ i ∈ G i,n , u and v belong both to the same class L, u t := u (vt) (t > 0) and t be the domain of u t . Then, for Indeed, by the definition of Fenchel conjugate and the definition of projection u, it is easy to see that (u| ξ i ) * = u * | ξ i and (u ut)| ξ i = u| ξ i ut| ξ i hold. Lemma 1.4 and the property of the projection grant the differentiability. Set ϕ := u * | ξ i and ψ := v * | ξ i , and ϕ t = ϕ + tψ, then ϕ t belongs to the class C 2 + on ξ i . Then ∇ 2 ϕ t = ∇ 2 ϕ + t∇ 2 ψ is nonsingular on ξ i . So the equation locally defines a map y = y(x, t) which is of class C 1 . By Proposition 1.3, ∇(u t | ξ i ) is the inverse map of ∇ϕ t , that is, ∇ϕ t (∇(u t | ξ i (x)) = x, which means that, for every x ∈ int(D t ) and every t > 0, t → ∇(u t | ξ i ) is differentiable. Using Eq. (1.2) again, we have Moreover, note that ϕ t = ϕ + tψ and we have Taking the differential of the above formally we obtain Then we complete the proof.  1, 2, . . . , n), and x ∈ | ξ i . The ith total mass of f is defined as

Inequality for functional mixed quermassintegral
where f | ξ i is the projection of f onto ξ i defined by (2.1), dx is the i-dimensional volume element in ξ i . (2) When one takes f = χ K , the characteristic function of a convex body K , one has J i (f ) = V i (K), the i-dimensional volume in ξ i .

Definition 3.2
Let f ∈ A . Set ξ i ∈ G i,n be a linear subspace and, for any x ∈ | ξ i , the ith functional Quermassintegrals of f (or the i-dimensional mean projection mass of f ) is defined as where J i (f ) is the ith total mass of f defined by (3.1), dμ(ξ i ) is the normalized Haar measure on G i,n .

Remark 3.2 (1) The definition of the W i (f ) follows the definition of the ith Quermassintegral W i (K)
, that is, the ith mean total mass of f on G i,n . Also in the recent paper of Bobkov, Colesanti and Fragala [10], the authors give the same definition by defining the Quermassintegral of the support set for the quasi-concave functions.
(3) From the definition of the Quermassintegrals W i (f ), the following properties are obtained (see also [10]): • Generally speaking, the W i (f ) has no homogeneity under dilations. That is,

Definition 3.3
Let f , g ∈ A , ⊕ and · denote the operations of "sum" and "multiplication" in A , W i (f ) be the ith Quermassintegrals of f . Whenever the following limit exists: we denote it by W i (f , g), and call it the first variation of W i at f along g, or the ith functional mixed Quermassintegrals of f and g.
The following is devoted to proving that W i (f , g) exists under the fairly weak hypothesis. First, we prove that the first variation of i-dimensional total mass of f is translation invariant.
Proof By the construction, we have u i (0) = 0, v i (0) = 0, and v i ≥ 0, ϕ i ≥ 0, ψ i ≥ 0. Further, we have ψ i (y) = ψ i (y) + d, and f i = e c f i . Then we have On the other hand, since By derivation of both sides of the above formula, we obtain So we complete the proof.
Theorem 3.5 Let f , g ∈ A , with -∞ ≤ inf(log g) ≤ +∞, and W i (f ) > 0. Then W j (f , g) is differentiable at f along g, and By the definition of f t and the Proposition 2.1 we obtain Without loss of generality, we may assume inf(v) = v(0). Lemma 1.6 says that, for every x ∈ ξ i , Then there exists f | ξ i (x) := lim t→0 and f t | ξ i is pointwise decreasing as t → 0 + . By Lemma 1.1 and Proposition 1.4, one shows that Then Hence, by the monotonicity and convergence, we have lim t→0 Since f t | ξ i ≥ f | ξ i , we have the following two cases: For the first case, since W i ( f t ) is a monotone increasing function of t, W i ( f t ) = W i (f ) for every t ∈ [0, t 0 ]. Hence we have the statement of the theorem holds true.
In the latter case, since f t | ξ i is an increasing nonnegative function, log(W i ( f t )) is an increasing concave function of t. Then On the other hand, Then From the above we infer that Combining the above formulas we obtain So we complete the proof.
Now taking the limit when t → 0 + , we obtain Then we have Here we use the (f log f ) The following lemma is useful in proving Minkowski's first inequality for Quermassintegrals. Lemma 3.7 Let f , g ∈ A , and 0 < t < 1. Then Proof First by Lemma 3.4, without loss of generality, we may assume that the function v = -log g satisfies the condition v(0) = 0. For t ∈ (0, 1), letting s(t) = t 1-t , by (1.9) we obtain Concerning the first term of the right hand side (3.13), by Lemma 1.6 we know that the function f s(t) (x) converges decreasingly to some pointwise limit f (x) as t → 0 + , since s(t) → 0 + as t → 0 + . In fact, we have lim t→0 Concerning the second term, we have Then one can show the conclusion by combining with (3.14) and (3.15).
Now we give the proof of Theorem 0.1.
Proof of Theorem 0.1 Let 0 ≤ t ≤ 1, we construct a function In fact, for f , g, h ∈ A and 0 ≤ t ≤ 1, By the Prékopa-Leindler inequality, for 0 ≤ t ≤ 1, we have That means that Taking the integral of both sides of (3.16) on G i,n with measure μ(ξ i ), by the Prékopa-Leindler inequality once again, we obtain It means that (t) | t=0 ≥ (1) -(0). By Lemma 3.7, we have On the other hand, note that (1) -(0) = log(W i (g)) -log(W i (f )). Therefore, we obtain Then, combining with formula (3.10), we obtain Concerning the equality case, first, assume that g(x) = f (xx 0 ), by (3.10) and the invariance of the integral by translation of coordinates, we know that (0.6) holds with equality. On the other hand, if (0.6) holds with equality, by inspection of the above proof, one may see that the inequalities (3.16), (3.17) and (3.18) must hold as equalities. Moreover, whenever inequalities (3.16) and (3.17) hold with equality sign, then (3.18) automatic holds with equality. This entails that the Prékopa-Leindler inequality holds as an equality, therefore f and g must agree up to a translation.
The inequality (0.6) is called the functional Brunn-Minkwoski first inequality for ith mixed Quremassintegrals or functional mixed Quermassintegral inequality. In the following we will give some special case of (0.6).
In fact, we take f = χ K and g = χ L , with K, L ∈ K n . In this case χ K ⊕ t · χ L = χ K+tL , J i (χ K ) = V i (K), here V i denotes the i-dimensional volume in ξ i , W i (χ K ) = W i (K), and W i (χ K , χ L ) = W i (K, L). Moreover, by (1.6) and (1.7) we have, for any x, f (x) log f (x) = -e -I K (x) I K (x) ≡ 0.
Then (0.6) turns out to be . (3.19) We can rewrite the above formula (3.19) equivalently as the following: (3.20) By defining the i-cone volume probability measure V iK similar to the V K defined in [11] by Böröczky, where dS iK is the ith Borel measue on S n-1 . The normalized i-cone volume probability measure V iK is defined as Then the normalized i-mixed Quermassintegrals W i (K, L) can be expressed as We call (3.22) the weaker of the ith log Quermassintegral inequality. In fact, for all u ∈ S n-1 , and the equality holds if and only if h L h K = 1, that is, K = L. For i = 0 and n = 2, since dV 0K = dV K , the cone volume probability measure of K , then by (3.23) and (3.22) we obtain (3.24) So we have the following corollary.
Corollary 3.8 Let K, L ∈ K n , W i (K) denotes the ith Quermassintegral of K , V iK be the normalized i-cone volume probability measure. Then When h K = h L , equality holds.