On Mixed Quermassintegral for Log-Concave Functions

where G i,n is the Grassmannian manifold of i-dimensional linear subspaces of Rn, dμðξiÞ is the normalized Haar measure onG i,n, Kjξi denotes the orthogonal projection of K onto the i-dimensional subspaces ξi, and voli is the i-dimensional volume on space ξi. In the 1930s, Aleksandrov and Fenchel and Jessen (see [3, 4]) proved that for a convex body K in Rn, there exists a regular Borel measure Sn−1−iðKÞ (i = 0, 1,⋯, n − 1) on Sn−1, the unit sphere in Rn, for K , L ∈K, the following representation holds


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 = fx + y : x ∈ K, y ∈ Lg is given by with equality if and only if K and L are homothetic; namely, they agree up to a translation and a dilation. Another geometric quantity related to the convex bodies K and L is the mixed volume. The most important result concerning the mixed volume is Minkwoski's first inequality: for K, L ∈ K n . In particular, when choosing L to be a unit ball, up to a factor, V 1 ðK, LÞ is exactly the perimeter of K, and inequality (2) turns out to be the isoperimetric inequality in the class of convex bodies. The mixed volume V 1 ðK, LÞ admits a simple integral representation (see [1,2]): where h L is the support function of L and S K is the area measure of K. The Quermassintegrals W i ðKÞði = 0, 1,⋯,nÞ of K, which are defined by letting W 0 ðKÞ = V n ðKÞ, the volume of K; W n ðKÞ = ω n , the volume of the unit ball B n 2 in ℝ n and for general i = 1, 2, ⋯, n − 1, where G i,n is the Grassmannian manifold of i-dimensional linear subspaces of ℝ n , dμðξ i Þ is the normalized Haar measure on G i,n , Kj 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 1930s, Aleksandrov and Fenchel and Jessen (see [3,4]) proved that for a convex body K in ℝ n , there exists a regular Borel measure S n−1−i ðKÞ (i = 0, 1, ⋯, n − 1) on S n−1 , the unit sphere in ℝ n , for K, L ∈ K n , the following representation holds The quantity W i ðK, LÞ is called the ith mixed Quermassintegral of K and L.
In the 1960s, the Minkowski addition was extended to the L p ðp ≥ 1Þ Minkowski sum h p K+ p t·L = h p K + th p L : The extension of the mixed Quermassintegral to the L p mixed Quermassintegral due to Lutwak [1], the L p mixed Quermassintegral inequalities, and the L p Minkowski problem are established. (See [2,[5][6][7][8][9][10][11][12][13] for more about the L p Minkowski theory.) The L p mixed Quermassintegrals are defined by for i = 0, 1, ⋯, n − 1. In particular, for p = 1 in (6), it is W i ðK, LÞ, and W p,0 ðK, LÞ is denoted by V p ðK, LÞ, which is called the L p mixed volume of K and L. Similarly, the L p mixed Quermassintegral has the following integral representation (see [1]): The measure S p,i ðK, ·Þ is absolutely continuous with respect to S i ðK, ·Þ and has Radon-Nikodym derivative dS p,i ðK, ·Þ/dS i ðK, ·Þ = h K ð·Þ 1−p : In particular, p = 1 in (7) yields the representation (5).
Most recently, the interest in the log-concave functions has been considerably increasing, motivated by the analogy properties between the log-concave functions and the volume convex bodies in K n . The classical Prékopa-Leindler inequality (see [14][15][16][17][18]) firstly shows the connections of the volume of convex bodies and log-concave functions. The Blaschke-Santaló inequality for even log-concave functions is established in [19,20] by Ball (for the general case, see [21][22][23][24]). The mean width for log-concave function is introduced by Klartag and Milman and Rotem [25][26][27]. The affine isoperimetric inequality for log-concave functions is proved by Avidan et al. [28]. The John ellipsoid for log-concave functions has been establish by Alonso-Gutiérrez et al. [29]; the LYZ ellipsoid for log-concave functions is established by Fang and Zhou [30]. (See [31][32][33][34][35][36][37] 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 where w * denotes as usual the Fenchel conjugate of the convex function ω. The total mass integral Jð f Þ of f is defined by Jðf Þ = Ð ℝ n f ðxÞdx: In paper [38] of Colesanti and Fragalà, the quantity δJð f , gÞ, which is called as the first variation of J at f along g, δJð f , gÞ = lim t→0 + ðJð f ⊕ t · gÞ − Jð f ÞÞ/t, is discussed. It has been shown that δJð f , gÞ is finite and has the following integral expression: where μð f Þ is the measure of f on ℝ n . Inspired by the paper [38] of Colesanti and Fragalà, in this paper, we define the ith functional Quermassintegrals W i ðf Þ as the i-dimensional average total mass of f : where J i ðf Þ denotes the i-dimensional total mass of f defined in Section 4, G i,n is the Grassmannian manifold of ℝ n , and dμðξ n−i Þ is the normalized measure on G i,n . Moreover, we define the first variation of W i at f along g, which is It is a natural extension of the Quermassintegral of convex bodies in ℝ n ; we call it the ith functional mixed Quermassintegral. In fact, if one takes f = χ K , and dom ð f Þ = K ∈ ℝ n , then W i ðf Þ turns out to be W i ðKÞ, and W i ðχ K , χ L Þ equals to W i ðK, LÞ. The main result in this paper is to show that the ith functional mixed Quermassintegral has the following integral expressions.
Theorem 1. Let f , g ∈ A ′, be integrable functions, μ i ðf Þ be the i-dimensional measure of f , and W i ðf , gÞ be the ith functional mixed Quermassintegral of f and g. Then, where h gj ∈ n−i is the support function of gj ∈ n−i .
The paper is organized as follows: In Section 2, we introduce some notations about the log-concave functions. In Section 3, the projection of a log-concave function onto subspace is discussed. In Section 4, we focus on how we can represent the ith functional mixed Quermassintegral W i ð f , gÞ similar as W i ðK, LÞ. Owing to the Blaschke-Petkantschin formula and the similar definition of the support function of f , we obtain the integral representation of the ith functional mixed Quermassintegral W i ð f , gÞ.

Preliminaries
Let u : Ω → ð−∞,+∞ be a convex function; that is, uðð1 − tÞx + tyÞ ≤ ð1 − tÞuðxÞ + tuðyÞ for t ∈ ð0, 1Þ, where Ω = fx ∈ ℝ n : uðxÞ ∈ ℝg is the domain of u. By the convexity of u, Ω is a convex set in ℝ n . We say that u is proper if Ω ≠ ∅, and u is of class C 2 + if it is twice differentiable on int ðΩÞ, with a positive definite Hessian matrix. In the following, we define the subclass of u: Journal of Function Spaces Recall that the Fenchel conjugate of u is the convex function defined by It is obvious that uðxÞ + u * ðyÞ ≥ hx, yi for all x, y ∈ Ω, and there is an equality if and only if x ∈ Ω and y is in the subdifferential of u at x, which means Moreover, if u is a lower semicontinuous convex function, then also u * is a lower semicontinuous convex function, and u * * = u.
The infimal convolution of u and v from Ω to ð−∞, + ∞ is defined by The right scalar multiplication by a nonnegative real number α is The following proposition below gathers some elementary properties of the Fenchel conjugate and the infimal convolution of u and v, which can be found in [38,39].
If f is a strictly positive log-concave function on ℝ n , then there exists a convex function u : Ω → ð−∞,+∞ 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 semicontinuous convex function, and the indicator function of K is In the later sections, we also use f to denote f being extended to ℝ n : Let A = f f : ℝ n → ð0,+∞: f = e −u , u ∈ Lg be the subclass of f in ℝ n . The addition and multiplication by nonnegative scalars in A are defined by the following (see [38]).
Lemma 5 (see [38]). Let u ∈ L, then there exist constants a and b, with a > 0, such that, for x ∈ Ω, Moreover, u * is proper and satisfies u * ðyÞ > −∞, ∀y ∈ Ω. Lemma 5 grants that L is closed under the operations of infimal convolution and right scalar multiplication defined in (16) and (17) which are closed.

Journal of Function Spaces
Proposition 6 (see [38]). Let u and v belong both to the same class L, and α, β ≥ 0. Then, uα□vβ belongs to the same class as u and v.
Let f ∈ A, according to papers of [26,40], the support function of f = e −u is defined as where 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 . Obviously, the support function h f share the most of the important properties of support functions h K . Specifically, it is easy to check that the function h : A → L has the following properties [27]: The following proposition shows that h f is GLðnÞ covariant.
Proposition 7 (see [30]). Let f ∈ A, A ∈ GLðnÞ and x ∈ ℝ n . Then, Let u, v ∈ L, denote by u t = u□vtðt > 0Þ, and f t = e −u t . The following lemmas describe the monotonicity and convergence of u t and f t , respectively. Lemma 8 (see [38]). Let f = e −u , g = g −v ∈ A. For t > 0, set u t = u□ðvtÞ and f t = e −u t . Assume that vð0Þ = 0, then for every fixed x ∈ ℝ n , u t ðxÞ and f t ðxÞ are, respectively, pointwise decreasing and increasing with respect to t; in particular, it holds Lemma 9 (see [38]). Let u and v belong both to the same class L and, for any t > 0, set u t ≔ u□ðvtÞ. Assume that vð0Þ = 0, then (1) ∀x ∈ Ω, lim t→0 + u t ðxÞ = uðxÞ (2) ∀E ⊂ ⊂Ω, lim t→0 + ∇u t ðxÞ = ∇u uniformly on E Lemma 10 (see [38]). Let u and v belong both to the same class L and for any t > 0, let u t ≔ u□ðvtÞ. Then, ∀x ∈ int ðΩ t Þ, and ∀t > 0, where ψ ≔ v * .

Projection of Functions onto Linear Subspace
Let G i,n ð0 ≤ i ≤ nÞ be the Grassmannian manifold of i -dimensional linear subspace of ℝ n . The elements of G i,n will usually be denoted by ξ i , and ξ ⊥ i stands for the orthogonal complement of ξ i which is a ðn − iÞ-dimensional subspace of ℝ n . Let ξ i ∈ G i,n and f : ℝ n → ℝ. The projection of f onto ξ i is defined by (see [25,41]) where ξ ⊥ i is the orthogonal complement of ξ i in ℝ n and Ωj ξ i is the projection of Ω onto ξ i . By the definition of the logconcave function f = e −u , for every x ∈ Ωj ξ i , one can rewrite (29) as Regarding 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.
Proposition 11. Let f , g ∈ A, ξ i ∈ G i,n , and α, β > 0. Then, Proof. Let f , g ∈ A, let x 1 , x 2 , x ∈ ξ i such that x = αx 1 + βx 2 , then we have Taking the supremum of the second right-hand inequality over all ≥ sup Journal of Function Spaces Since f , g ≥ 0, the inequality max f f · gg ≤ max f f g · max fgg holds. So, we complete the proof.

Proposition 12.
Let ξ i ∈ G i,n , f and g are functions on ℝ n , such that f ðxÞ ≤ gðxÞ holds. Then, holds for any x ∈ ξ i .
By the definition of the projection, we complete the proof.
For the convergence of f , we have the following.
Hence, each f f n j ξ i g has a convergent subsequence; we denote it also by f f n j ξ i g, converging to some f ′ 0 j ξ i . Then, for x ∈ ξ i , we have By the arbitrary of ε, we have f ′ 0 j ξ i = f 0 j ξ i , so we complete the proof. Combining with Proposition 13 and Lemma 9, it is easy to obtain the following proposition.

Proposition 14.
Let u and v belong both to the same class L and Ω ∈ ℝ n be the domain of u, for any t > 0, set u t = u□ðvtÞ. Assume that vð0Þ = 0 and ξ i ∈ G i,n , then Now, let us introduce some facts about the functions u t = u□ðvtÞ with respect to the parameter t. Lemma 15. Let ξ i ∈ G i,n , u and v belong both to the same class L, u t ≔ u□ðvtÞ and Ω t be the domain of u t ( t > 0). Then, for where Indeed, by the definition of Fenchel conjugate and the definition of projection u, it is easy to see that ðu ξ i Þ * = u * j ξ i and ðu□utÞj ξ i = u j ξ i □utj ξ i hold. Proposition 6 and the property of the projection grant the differentiability. Set φ ≔ u * j ξ i and ψ ≔ v * j ξ i , and φ t = φ + tψ, then φ t belongs to the class C 2 + on ξ i . Then, locally defines a map y = yðx, tÞ which is of class C 1 . By Proposition 3, we have ∇ðu t j ξ i Þ is the inverse map of ∇φ t , that is, ∇φ t ð∇ðu t j ξ i ðxÞÞ = x, which means that for every x ∈ int ðD t Þ and every t > 0, t → ∇ðu t j ξ i Þ is differentiable. Using equation (15) again, we have Moreover, note that φ t = φ + tψ, we have Differential the above formal we obtain, d/dtðu t j ξ i ÞðxÞ = −ψð∇ðu t j ξ i ÞðxÞÞ: Then, we complete the proof of the result.
Definition 16. Let f ∈ A ′ , ξ i ∈ G i,n ði = 1, 2,⋯,n − 1Þ, and x ∈ Ωj ξ i . The ith total mass of f is defined as where f j ξ i is the projection of f onto ξ i defined by (29) and dx is the i-dimensional volume element in ξ i .

Remark 17.
(1) The definition of J i ð f Þ follows the i-dimensional volume of the projection a convex body. If i = 0, we defined J 0 ð f Þ ≔ ω n , the volume of the unit ball in ℝ n , for the completeness (2) When taking f = χ K , the characteristic function of a convex body K, one has J i ð f Þ = V i ðKÞ, the i-dimensional volume in ξ i Definition 18. Let f ∈ A ′ . Set ξ i ∈ G i,n be a linear subspace and for x ∈ Ωj ξ i , the ith functional Quermassintegrals of f (or the i-dimensional mean projection mass of f ) are defined as where J i ðf Þ is the ith total mass of f defined by (41) and dμ ðξ i Þ is the normalized Haar measure on G i,n .

Remark 19.
(1) The definition of W i ðf Þ follows the definition of the i th Quermassintegrals W i ðKÞ, that is, the ith mean total mass of f on G i,n . Also, in a recent paper [42], the authors give the same definition by defining the Quermassintegral of the support set for the quasiconcave functions (2) When i equals to n in (42), we have W 0 ð f Þ = Ð ℝ n f ðxÞdx = Jð f Þ, the total mass function of f defined by Colesanti and Fragalá [38]. Then, we can say that our definition of W i ðf Þ is a natural extension of the total mass function of Jð f Þ (3) From the definition of the Quermassintegrals W i ðf Þ, the following properties are obtained (see also [42]): where λ · f ðxÞ = λf ðx/λÞ, λ > 0 Definition 20. Let f , g ∈ A ′, ⊕ , and · denote the operations of "sum" and "multiplication" in A ′. W i ðf Þ and W i ðgÞ are, respectively, the ith Quermassintegrals of f and g. Whenever the following limit exists, we denote it by W i ðf , gÞ and call it as the first variation of W i at f along g, or the ith functional mixed Quermassintegrals of f and g.
In general, W i ð f , gÞ has no analog properties of W i ðK, LÞ; for example, W i ð f , gÞ is not always nonnegative and finite.
The following is devoted to proving that W i ð f , gÞ exists under the fairly weak hypothesis. First, we prove that the first i-dimensional total mass of f is translation invariant.
Proof. By the construction, we haveũ i ð0Þ = 0,ṽ i ð0Þ = 0, On the other hand, since f i ⊕ t · g i = e −ðc+dtÞ ðf i ⊕ t ·g i Þ, we have, J i ðf ⊕ t · gÞ = e −ðc+dtÞ J i ðf i ⊕ t ·g i Þ: By derivation of both sides of the above formula, we obtain So, we complete the proof.

6
Journal of Function Spaces By the definition of f t and Prop- Notice that vj ξ i ð0Þ = vð0Þ, set d ≔ vð0Þ,ṽj ξ i ðxÞ ≔ vj ξ i ðxÞ − d, Up to a translation of coordinates, we may assume inf ðvÞ = vð0Þ: Lemma 8 says that for every x ∈ ξ i , Then, there existsf j ξ i ðxÞ ≔ lim Hence, by monotonicity and convergence, we have lim t→0 + W i ðf t Þ = W i ðf Þ: In fact, by definition, we Note that −∞ ≤ inf ðvj ξ i Þ ≤ +∞, then −inf u j ξ i ðx − yÞ − t inf vj ξ i ðy/tÞ is a continuous function of variable Moreover, W i ðf t Þ is a continuous function of ðt ∈ ½0, 1Þ; then, lim t→0 + W i ðf t Þ = W i ð f Þ: Since f t j ξ i = e −dtf j ξ i ðxÞ, we have Notice that,f t j ξ i ≥ f j ξ i , we have the following two cases, that is, ∃t 0 > 0 : W i ðf t 0 Þ = W i ðf Þ or W i ðf t Þ = W i ð f Þ, ∀t > 0: For the first case, since W i ðf t Þ is a monotone increasing function of t, it must hold W i ðf t Þ = W i ðf Þ for every t ∈ ½0, t 0 . Hence, we have lim t→0 + ðW i ð f t Þ − W i ð f ÞÞ/t = −dW i ð f Þ; the statement of the theorem holds true.
In the latter case, sincef t j ξ i is an increasing nonnegative function, it means that log ðW i ðf t ÞÞ is an increasing concave function of t.
From above, we infer that ∃lim t→0 + ðW i ðf t Þ − W i ð f ÞÞ/t ∈ ½0,+∞: Combining the above formulas, we obtain So, we complete the proof.
In view of the example of the mixed Quermassintegral, it is natural to ask whether, in general, W i ð f , gÞ has some kind of integral representation.
Definition 24. Let ξ i ∈ G i,n and f = e −u ∈ A ′ . Consider the gradient map ∇u : ℝ n → ℝ n , the Borel measure μ i ð f Þ on ξ i is defined by Recall that the following Blaschke-Petkantschin formula is useful.