# On Mixed Quermassintegral for Log-Concave Functions

Authors: Fangwei Chen; Jianbo Fang; Miao Luo; Congli Yang
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

In this paper, the functional Quermassintegral of log-concave functions inℝn is discussed. We obtain the integral expression of the ith functional mixed Quermassintegral, which is similar to the integral expression of the ith mixed Quermassintegral of convex bodies.


---

## Body

## 1. Introduction

LetKn be the set of convex bodies (compact convex subsets with nonempty interiors) in ℝn, the fundamental Brunn-Minkowski inequality for convex bodies states that for K,L∈Kn, the volume of the bodies and of their Minkowski sum K+L=x+y:x∈K,y∈L is given by
(1)VK+L1/n≥VK1/n+VL1/n,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:
(2)V1K,L≔1nlimt→0+VK+tL−VKt≥VKn−1/nVL1/n,for K,L∈Kn. In particular, when choosing L to be a unit ball, up to a factor, V1K,L 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 V1K,L admits a simple integral representation (see [1, 2]):
(3)V1K,L=1n∫Sn−1hLdSK,where hL is the support function of L and SK is the area measure of K.The QuermassintegralsWiKi=0,1,⋯,n of K, which are defined by letting W0K=VnK, the volume of K; WnK=ωn, the volume of the unit ball B2n in ℝn and for general i=1,2,⋯,n−1,
(4)Wn−iK=ωnωi∫Gi,nvoliKξidμξi,where Gi,n is the Grassmannian manifold of i-dimensional linear subspaces of ℝn, dμξi is the normalized Haar measure on Gi,n, Kξi 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−iK (i=0,1,⋯,n−1) on Sn−1, the unit sphere in ℝn, for K,L∈Kn, the following representation holds
(5)WiK,L=1n−ilimt→0+WiK+tL−WiKε=1n∫Sn−1hLudSn−1−iK,u.The quantityWiK,L is called the ith mixed Quermassintegral of K and L.In the 1960s, the Minkowski addition was extended to theLpp≥1 Minkowski sum hK+pt·Lp=hKp+thLp. The extension of the mixed Quermassintegral to the Lp mixed Quermassintegral due to Lutwak [1], the Lp mixed Quermassintegral inequalities, and the Lp Minkowski problem are established. (See [2, 5–13] for more about the Lp Minkowski theory.) The Lp mixed Quermassintegrals are defined by
(6)Wp,iK,L≔pn−ilimt→0+WiK+pt·L−WiLt,for i=0,1,⋯,n−1. In particular, for p=1 in (6), it is WiK,L, and Wp,0K,L is denoted by VpK,L, which is called the Lp mixed volume of K and L. Similarly, the Lp mixed Quermassintegral has the following integral representation (see [1]):
(7)Wp,iK,L=1n∫Sn−1hLpudSp,iK,u.The measureSp,iK,· is absolutely continuous with respect to SiK,· and has Radon-Nikodym derivative dSp,iK,·/dSiK,·=hK·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 inKn. The classical Prékopa-Leindler inequality (see [14–18]) 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 [21–24]). The mean width for log-concave function is introduced by Klartag and Milman and Rotem [25–27]. 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 [31–37] for more about the pertinent results.)Letf=e−u, g=e−v be log-concave functions, α,β>0, the “sum” and “scalar multiplication” of log-concave functions are defined as
(8)α·f⊕β·g≔e−w,w∗=αu∗+βv∗,wherew∗ denotes as usual the Fenchel conjugate of the convex function ω. The total mass integral Jf of f is defined by Jf=∫ℝnfxdx. In paper [38] of Colesanti and Fragalà, the quantity δJf,g, which is called as the first variation of J at f along g, δJf,g=limt→0+Jf⊕t·g−Jf/t, is discussed. It has been shown that δJf,g is finite and has the following integral expression:
(9)δJf,g=∫ℝnv∗dμf,where μf is the measure of f on ℝn.Inspired by the paper [38] of Colesanti and Fragalà, in this paper, we define the ith functional Quermassintegrals Wif as the i-dimensional average total mass of f:
(10)Wif≔ωnωn−i∫Gn−i,nJn−ifdμξn−i,i=0,1,⋯,n−1,whereJif denotes the i-dimensional total mass of f defined in Section 4, Gi,n is the Grassmannian manifold of ℝn, and dμξn−i is the normalized measure on Gi,n. Moreover, we define the first variation of Wi at f along g, which is
(11)Wif,g=limt→0+Wif⊕t·g−Wift.It is a natural extension of the Quermassintegral of convex bodies inℝn; we call it the ith functional mixed Quermassintegral. In fact, if one takes f=χK, and domf=K∈ℝn, then Wif turns out to be WiK, and WiχK,χL equals to WiK,L. The main result in this paper is to show that the ith functional mixed Quermassintegral has the following integral expressions.Theorem 1.
Letf,g∈A′, be integrable functions, μif be the i-dimensional measure of f, and Wif,g be the ith functional mixed Quermassintegral of f and g. Then,
(12)Wif,g=1n−i∫ℝnhgξn−idμn−if,i=0,1,⋯,n−1,where hg∈n−i is the support function of g∈n−i.The paper is organized as follows: In Section2, 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 Wif,g similar as WiK,L. 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 Wif,g.


## 2. Preliminaries

Letu:Ω→−∞,+∞ be a convex function; that is, u1−tx+ty≤1−tux+tuy for t∈0,1, where Ω=x∈ℝn:ux∈ℝ is the domain of u. By the convexity of u, Ω is a convex set in ℝn. We say that u is proper if Ω≠∅, and u is of class C+2 if it is twice differentiable on intΩ, with a positive definite Hessian matrix. In the following, we define the subclass of u:
(13)L=u:Ω→−∞,+∞:uisconvex,lowsemicontinuous,lim∥x∥→+∞ux=+∞.Recall that the Fenchel conjugate ofu is the convex function defined by
(14)u∗y=supx∈ℝnx,y−ux.It is obvious thatux+u∗y≥x,y 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
(15)u∗∇ux+ux=x,∇ux.Moreover, ifu is a lower semicontinuous convex function, then also u∗ is a lower semicontinuous convex function, and u∗∗=u.The infimal convolution ofu and v from Ω to −∞,+∞ is defined by
(16)u□vx=infy∈Ωux−y+vy.The right scalar multiplication by a nonnegative real numberα is
(17)uαx≔αuxα,ifα>0,I0,ifα=0.The following proposition below gathers some elementary properties of the Fenchel conjugate and the infimal convolution ofu and v, which can be found in [38, 39].Proposition 2.
Letu,v:Ω→−∞,+∞ be convex functions. Then,
(18)u□v∗=u∗+v∗(1)
uα∗=αu∗,α>0(2)
domu□v=domu+domv(3)
it holdsu∗0=−infu; in particular, if u is proper, then u∗y>−∞; infu>−∞ implies u∗ is properThe following proposition about the Fenchel and Legendre conjugates is obtained in [39].Proposition 3 (see [39]).
Letu:Ω→−∞,+∞ be a closed convex function, and set C≔intΩ, C∗≔intdomu∗. Then, C,u is a convex function of Legendre type if and only if C∗,u∗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∗.
A functionf:ℝn→−∞,+∞ is called log-concave if for all x,y∈ℝn and 0<t<1, we have f1−tx+ty≥f1−txfty. If f is a strictly positive log-concave function on ℝn, then there exists a convex function u:Ω→−∞,+∞ such that f=e−u. The log-concave function is closely related to the convex geometry of ℝn. An example of a log-concave function is the characteristic function χK of a convex body K in ℝn, which is defined by
(19)χKx=e−IKx=1,ifx∈K,0,ifx∉K,where IK is a lower semicontinuous convex function, and the indicator function of K is
(20)IKx=0,ifx∈K,∞,ifx∉K.
In the later sections, we also usef to denote f being extended to ℝn:
(21)f¯=f,x∈Ω,0,x∈RnΩ.
LetA=f:ℝn→0,+∞:f=e−u,u∈L be the subclass of f in ℝn. The addition and multiplication by nonnegative scalars in A are defined by the following (see [38]).Definition 4.
Letf=e−u, g=e−v∈A, and α,β≥0. The sum and multiplication of f and gare defined as
(22)α·f⊕β·g=e−uα□vβ.
That means,(23)α·f⊕β·gx=supy∈ℝnfx−yααgyββ.
In particular, whenα=0 and β>0, we have α·f⊕β·gx=gx/ββ; when α>0 and β=0, then α·f⊕β·gx=fx/αα; finally, when α=β=0, we have α·f⊕β·g=I0.The following lemma is obtained in [38].Lemma 5 (see [38]).
Letu∈L, then there exist constants a and b, with a>0, such that, for x∈Ω,
(24)ux≥a∥x∥+b.
Moreover,u∗ is proper and satisfies u∗y>−∞, ∀y∈Ω.
Lemma5 grants that L is closed under the operations of infimal convolution and right scalar multiplication defined in (16) and (17) which are closed.Proposition 6 (see [38]).
Letu 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.
Letf∈A, according to papers of [26, 40], the support function of f=e−u is defined as
(25)hfx=−logfx∗=u∗x,where u∗ is the Legendre transform of u. The definition of hf is a proper generalization of the support function hK. In fact, one can easily check hχK=hK. Obviously, the support function hf share the most of the important properties of support functions hK. Specifically, it is easy to check that the function h:A→L has the following properties [27]:
(1)
h is a bijective map from A→L(2)
h is order preserving: f≤g if and only if hf≤hg(3)
h is additive: for every f,g∈A, we have hf⊕g=hf+hgThe following proposition shows thathf is GLn covariant.Proposition 7 (see [30]).
Letf∈A, A∈GLn and x∈ℝn. Then,
(26)hf∘Ax=hfA−tx.Letu,v∈L, denote by ut=u□vtt>0, and ft=e−ut. The following lemmas describe the monotonicity and convergence of ut and ft, respectively.Lemma 8 (see [38]).
Letf=e−u, g=g−v∈A. For t>0, set ut=u□vt and ft=e−ut. Assume that v0=0, then for every fixed x∈ℝn, utx and ftx are, respectively, pointwise decreasing and increasing with respect to t; in particular, it holds
(27)u1x≤utx≤ux,fx≤ftx≤f1x∀x∈ℝn,∀t∈0,1.Lemma 9 (see [38]).
Letu and v belong both to the same class L and, for any t>0, set ut≔u□vt. Assume that v0=0, then
(1)
∀x∈Ω, limt→0+utx=ux(2)
∀E⊂⊂Ω, limt→0+∇utx=∇u uniformly on ELemma 10 (see [38]).
Letu and v belong both to the same class L and for any t>0, let ut≔u□vt. Then, ∀x∈intΩt, and ∀t>0,
(28)ddtutx=−ψ∇utx,where ψ≔v∗.


## 3. Projection of Functions onto Linear Subspace

LetGi,n0≤i≤n be the Grassmannian manifold of i-dimensional linear subspace of ℝn. The elements of Gi,n will usually be denoted by ξi, and ξi⊥ stands for the orthogonal complement of ξi which is a n−i-dimensional subspace of ℝn. Let ξi∈Gi,n and f:ℝn→ℝ. The projection of f onto ξi is defined by (see [25, 41])
(29)fξix≔maxfy:y∈x+ξi⊥,∀x∈Ωξi,where ξi⊥ is the orthogonal complement of ξi in ℝn and Ωξi is the projection of Ω onto ξi. By the definition of the log-concave function f=e−u, for every x∈Ωξi, one can rewrite (29) as
(30)fξix=expmax−uy:y∈x+ξi⊥=e−uξix.Regarding the “sum” and “multiplication” off, we say that the projection keeps the structure on ℝn. In other words, we have the following proposition.Proposition 11.
Letf,g∈A, ξi∈Gi,n, and α,β>0. Then,
(31)α·f⊕β·gξi=α·fξi⊕β·gξi.Proof.
Letf,g∈A, let x1,x2,x∈ξi such that x=αx1+βx2, then we have
(32)α·f⊕β·gξix≥α·f⊕β·gαx1+βx2+ξi⊥≥fx1+ξi⊥αgx2+ξi⊥β.
Taking the supremum of the second right-hand inequality over allξi⊥, we obtain α·f⊕β·gξi≥α·fξi⊕β·gξi. On the other hand, for x∈ξi,x1,x2∈ξi such that x1+x2=x, then
(33)α·f∣ξi⊕β·gξix=supx1+x2=xmaxfαx1α+ξi⊥maxgβx2β+ξi⊥≥supx1+x2=xmaxfαx1α+ξi⊥gβx2β+ξi⊥=maxsupx1+x2=xfαx1α+ξi⊥gβx2β+ξi⊥=α·f⊕β·gξix.
Sincef,g≥0, the inequality maxf·g≤maxf·maxg holds. So, we complete the proof.Proposition 12.
Letξi∈Gi,n, f and g are functions on ℝn, such that fx≤gx holds. Then,
(34)fξi≤gξiholds for any x∈ξi.Proof.
Fory∈x+ξi⊥, since fy≤gy, then fy≤maxgy:y∈x+ξi⊥. So, maxfy:y∈x+Li⊥≤maxgy:y∈x+ξi⊥. By the definition of the projection, we complete the proof.For the convergence off, we have the following.Proposition 13.
Letfi be functions such that limn→∞fn=f0, ξi∈Gi,n, then limn→∞fnξi=f0ξi.Proof.
Sincelimn→∞fn=f0, it means that ∀ε>0, there exist N0, ∀n>N0, such that f0−ε≤fn≤f0+ε. By the monotonicity of the projection, we have f0ξi−ε≤fnξi≤f0ξi+ε. Hence, each fnξi has a convergent subsequence; we denote it also by fnξi, converging to some f′0ξi. Then, for x∈ξi, we have
(35)f0ξix−ε≤f′0ξix=limn→∞fnξix≤f0ξix+ε.
By the arbitrary ofε, we have f′0ξi=f0ξi, so we complete the proof.
Combining with Proposition13 and Lemma 9, it is easy to obtain the following proposition.Proposition 14.
Letu and v belong both to the same class L and Ω∈ℝn be the domain of u, for any t>0, set ut=u□vt. Assume that v0=0 and ξi∈Gi,n, then
(1)
∀x∈Ωξi, limt→0+utξix=uξix(36)∀x∈intΩξi,limt→0+∇utξi=∇uξi
Now, let us introduce some facts about the functionsut=u□vt with respect to the parameter t.Lemma 15.
Letξi∈Gi,n, u and v belong both to the same class L, ut≔u□vt and Ωt be the domain of ut(t>0). Then, for x∈Ωtξi,
(37)ddtutξix=−ψ∇utξix,where ψ≔v∗ξi.Proof.
SetDt≔Ωtξi⊂ξi, for fixed x∈intDt, the map t→∇utξix is differentiable on 0,+∞. Indeed, by the definition of Fenchel conjugate and the definition of projection u, it is easy to see that uξi∗=u∗ξi and u□utξi=uξi□utξi hold. Proposition 6 and the property of the projection grant the differentiability. Set φ≔u∗ξi and ψ≔v∗ξi, and φt=φ+tψ, then φt belongs to the class C+2 on ξi. Then, ∇2φt=∇2φ+t∇2ψ is nonsingular on ξi. So, the equation
(38)∇φy+t∇ψy−x=0locally defines a map y=yx,t which is of class C1. By Proposition 3, we have ∇utξi is the inverse map of ∇φt, that is, ∇φt∇utξix=x, which means that for every x∈intDt and every t>0, t→∇utξi is differentiable. Using equation (15) again, we have
(39)utξix=x,∇utξix−φt∇utξix,∀x∈intDt.
Moreover, note thatφt=φ+tψ, we have
(40)utξix=x,∇utξix−φ∇utξix−tψ∇utξix=utξi∇utξix−tψ∇utξix.
Differential the above formal we obtain,d/dtutξix=−ψ∇utξix. Then, we complete the proof of the result.


## 4. Functional Quermassintegrals of Log-Concave Function

A functionf∈A is nondegenerate and integrable if and only if lim∥x∥→+∞ux/∥x∥=+∞. Let L′=u∈L:u∈C+2ℝn,lim∥x∥→+∞ux/∥x∥=+∞, and A′=f:ℝn→0,+∞:f=e−u,u∈L′. Now, we define the ith total mass of f.Definition 16.
Letf∈A′, ξi∈Gi,ni=1,2,⋯,n−1, and x∈Ωξi. The ith total mass of f is defined as
(41)Jif≔∫ξifξixdx,where fξi is the projection of f onto ξi defined by (29) and dx is the i-dimensional volume element in ξi.Remark 17.
(1)
The definition ofJif follows the i-dimensional volume of the projection a convex body. If i=0, we defined J0f≔ωn, the volume of the unit ball in ℝn, for the completeness(2)
When takingf=χK, the characteristic function of a convex body K, one has Jif=ViK, the i-dimensional volume in ξiDefinition 18.
Letf∈A′. Set ξi∈Gi,n be a linear subspace and for x∈Ωξi, the ith functional Quermassintegrals of f (or the i-dimensional mean projection mass of f) are defined as
(42)Wn−if≔ωnωi∫Gi,nJifdμξi,i=1,2,⋯,n,where Jif is the ith total mass of f defined by (41) and dμξi is the normalized Haar measure on Gi,n.Remark 19.
(1)
The definition ofWif follows the definition of the ith Quermassintegrals WiK, that is, the ith mean total mass of f on Gi,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)
Wheni equals to n in (42), we have W0f=∫ℝnfxdx=Jf, the total mass function of f defined by Colesanti and Fragalá [38]. Then, we can say that our definition of Wif is a natural extension of the total mass function of Jf(3)
From the definition of the QuermassintegralsWif, the following properties are obtained (see also [42]):(43)Positivity::0≤Wif≤+∞(i)
Monotonicity:Wif≤Wig, if f≤g(ii)
Generally speaking,Wif has no homogeneity under dilations. That is, Wiλ·f=λn−iWifλ, where λ·fx=λfx/λ,λ>0Definition 20.
Letf, g∈A′, ⊕, and · denote the operations of “sum” and “multiplication” in A′. Wif and Wig are, respectively, the ith Quermassintegrals of f and g. Whenever the following limit exists,
(44)Wif,g=1n−ilimt→0+Wif⊕t·g−Wift,we denote it by Wif,g and call it as the first variation of Wi at f along g, or the ith functional mixed Quermassintegrals of f and g.Remark 21.
Letf=χK and g=χL, with K, L∈Kn. In this case Wif⊕t·g=WiK+tL, then Wif,g=WiK,L. In general, Wif,g has no analog properties of WiK,L; for example, Wif,g is not always nonnegative and finite.The following is devoted to proving thatWif,g exists under the fairly weak hypothesis. First, we prove that the first i-dimensional total mass of f is translation invariant.Lemma 22.
Letξi∈Gi,n, f=e−u,g=e−v∈A′. Let c=infuξi≕u0, d=infvξi≔v0, and set u~ix=uξix−c, v~ix=vξix−d, φ~iy=u~i∗y, ψ~iy=v~i∗y, f~i=e−u~i, g~i=e−v~i, and f~ti=f~⊕t·g~. Then, if limt→0+Jif~t−Jif~/t=∫ξiψ~idμif~ holds, then we have limt→0+Jift−Jif/t=∫ξiψidμif.Proof.
By the construction, we haveu~i0=0,v~i0=0, and v~i≥0,φ~i≥0,ψ~i≥0. Further, ψ~iy=ψiy+d, and f~i=ecfi. So,
(45)limt→0+Jif~t−Jif~t=∫ξiψ~idμif~=ec∫ξiψidμif+dec∫ξidμif.
On the other hand, sincefi⊕t·gi=e−c+dtf~i⊕t·g~i, we have, Jif⊕t·g=e−c+dtJif~i⊕t·g~i. By derivation of both sides of the above formula, we obtain
(46)limt→0+Jif⊕t·g−Jift=−de−climt→0+Jif~i⊕tg~idx+e−climt→0+Jif~t−Jif~t=−de−cJif~i+∫ξiψidμif+d∫ξidμif=∫ξiψidμif.
So, we complete the proof.Theorem 23.
Letf,g∈A′, with −∞≤inflogg≤+∞ and Wif>0. Then, Wjf,g is differentiable at f along g, and it holds
(47)Wjf,g∈−k,+∞,where k=maxd,0Wif.Proof.
Letξi∈Gi,n, since uξi≔−logfξi=−logfξi and vξi≔−loggξi=−logfξi. By the definition of ft and Proposition 11, we obtain ftξi=f⊕t·gξi=fξi⊕t·gξi. Notice that vξi0=v0, set d≔v0, v~ξix≔vξix−d, g~ξix≔e−v~ξix, and f~tξi≔fξi⊕t·g~ξi. Up to a translation of coordinates, we may assume infv=v0. Lemma 8 says that for every x∈ξi,
(48)fξi≤f~tξi≤f~1ξi,∀x∈ℝn,∀t∈0,1.
Then, there existsf~ξix≔limt→0+f~tξix. Moreover, it holds f~ξix≥fξix and f~tξi is pointwise decreasing as t→0+. Lemma 5 and Proposition 6 show that fξi⊕t·g~ξi∈A′, ∀t∈0,1. Then, Jif≤Jif~t≤Jif~1, −∞≤Jif,Jif~1<∞. Hence, by monotonicity and convergence, we have limt→0+Wif~t=Wif~. In fact, by definition, we have f~tξix=e−infuξix−y+tvξiy/t,
(49)−infuξix−y+tvξiyt≤−infuξix−y−tinfvξiyt.
Note that−∞≤infvξi≤+∞, then −infuξix−y−tinfvξiy/t is a continuous function of variable t, then
(50)f~ξix≔limt→0+f~tξix=fξix.
Moreover,Wif~t is a continuous function of t∈0,1; then, limt→0+Wif~t=Wif. Since ftξi=e−dtf~ξix, we have
(51)Wift−Wift=Wife−dt−1t+e−dtWif~t−Wift.
Notice that,f~tξi≥fξi, we have the following two cases, that is, ∃t0>0:Wif~t0=Wif or Wif~t=Wif, ∀t>0.
For the first case, sinceWif~t is a monotone increasing function of t, it must hold Wif~t=Wif for every t∈0,t0. Hence, we have limt→0+Wift−Wif/t=−dWif; the statement of the theorem holds true.
In the latter case, sincef~tξi is an increasing nonnegative function, it means that logWif~t is an increasing concave function of t. Then, ∃logWif~t−logWif/t∈0,+∞. On the other hand, since
(52)log′Wif~tt=0=limt→0+logWif~t−logWifWif~t−Wif=1Wif.
Then,(53)limt→0+Wif~t−WiflogWif~t−logWif=Wif>0.
From above, we infer that∃limt→0+Wif~t−Wif/t∈0,+∞. Combining the above formulas, we obtain
(54)limt→0+Wift−Wift∈−maxd,0Wif,+∞.
So, we complete the proof.In view of the example of the mixed Quermassintegral, it is natural to ask whether, in general,Wif,g has some kind of integral representation.Definition 24.
Letξi∈Gi,n and f=e−u∈A′. Consider the gradient map ∇u:ℝn→ℝn, the Borel measure μifonξi is defined by
(55)μif≔∇uξi#∥x∥n−ifξi.
Recall that the following Blaschke-Petkantschin formula is useful.Proposition 25 (see [43]).
Letξi∈Gi,ni=1,2,⋯,n be linear subspace of ℝn and f be a nonnegative bounded Borel function on ℝn, then
(56)∫ℝnfxdx=ωnωi∫Gi,n∫ξifx∥x∥n−idxdμξi.Now, we give a proof of Theorem1.Proof of Theorem 1.
By the definition of theith Quermassintegral of f, we have
(57)Wift−Wift=ωnωn−i∫Gn−i,nJn−ift−Jn−iftdμξn−i.
Lett>0 be fixed, take C⊂⊂Ωξn−i, and by reduction 0∈intΩξn−i, we have C⊂⊂Ωξn−i, by Lemma 15, we obtain
(58)limh→0Jn−ift+hx−Jn−iftxh=∫ξn−iψ∇ut∣ξn−ixftξn−ixdx,where ψ=hgξn−i=vξn−i∗. Then, we have
(59)limh→0Wift+h−Wifth=ωnωn−i∫Gn−i,n∫ξn−iψ∇ut∣ξn−ixftξn−ix∥x∥n−i∥x∥n−idxdμξn−i,=∫ℝnψ∇ut∣ξn−ixftξn−ix∥x∥n−idx=∫ℝnψdμn−ift.
So, we haveWift+h−Wift=∫0t∫ℝnψdμn−ifsds. The continuity of ψ implies lims→0+∫ℝnψdμn−ifsds=∫ℝnψdμn−ifds. Therefore,
(60)limt→0+Wift−Wift=ddtWiftt=0+=lims→0+ddtWiftt=s=lims→0+ddt∫0t∫ℝnψdμn−ifsds=∫ℝnψdμn−if.
Sinceψ=hgξ, we have
(61)Wif,g=1n−ilimt→0+Wift−Wift=1n−i∫ℝnhgξn−idμn−if.
So, we complete the proof.Remark 26.
From the integral representation (12), the ith functional mixed Quermassintegral is linear in its second argument, with the sum in A′, forf,g,h∈A′, then we have Wif,g⊕h=Wif,g+Wif,h.



---

*Source: 8811566-2020-11-17.xml*