# Lp-Curvature Measures and Lp,q-Mixed Volumes

Authors: Tongyi Ma
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Motivated by Lutwak et al.’sLp-dual curvature measures, we introduce the concept of Lp-curvature measures. This new Lp-curvature measure is an extension of the classical surface area measure, Lp-surface area measure, and curvature measure. In this paper, we first prove some properties of the Lp-curvature measure. Next, using the Lp-curvature measure, we define the Lp,q-mixed volume which includes Lp-mixed volume as the special cases. Further, the Minkowski-type inequality related Lp,q-mixed volume and the uniqueness of the solution for the Lp,q- Minkowski problem are obtained. Finally, we propose several problems that need to be studied further.


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## Body

## 1. Introduction

Surface area measure and integral curvature measure are two important measures in classical Brunn-Minkowski theory. Minkowski problem describing surface area measure and Aleksandrov problem describing integral curvature are two famous problems. As a generalization,Lp-surface area measure and Lp-integral curvature are defined in [1, 2], respectively. At the same time, the hyperbolic measure as the curvature measure of dual Fiedler is constructed in [3]. Lutwak et al. introduce Lp-dual curvature measure in [4], which is a generalization of the dual curvature, Lp-surface area measure and Lp-integral curvature. Lp-dual mixed volume (also known as p,q-dual mixed volume) is defined by [4] and Minkowski inequality is established. Furthermore, they study the Lp-dual Minkowski problem of Lp-dual curvature measure by reference to [5].Inspired by Lutwak et al.’sLp-dual curvature measure, a new concept of Lp-curvature measure is introduced in this paper. It includes classical surface area measure, Lp-surface area measure and curvature measure. In this paper, we first prove some properties of Lp-curvature measure. Next, based on Lp-curvature measure, we define Lp,q-mixed volume, which includes Lp-mixed volume as a special case. Furthermore, the Minkowski inequality for Lp,q-mixed volume and the uniqueness of the solution for Lp,q-Minkowski problem are obtained. Finally, some problems which need further study are put forward.LetKn represent the set of convex bodies in n-dimensional Euclidean (compact convex subsets with nonempty embedding) space ℝn, for convex bodies containing the origin inside in ℝn, we write Kon. Set B said centered on the origin of the unit sphere, B surface written as Sn−1, in ℝn. VK represents the n dimensional volume of the body K and writes VB=ωn.ForK∈Kn, its support function, hK=hK,·:ℝn⟶−∞,+∞, is defined by (see [6])
(1)hK,x=maxx·y:y∈K,x∈ℝn,where x·y denotes the standard inner product of x and y.ForK,L∈Kn and s,t≥0 (not both zero), the Minkowski combination, sK+tL∈Kn, of K and L is defined by the following:
(2)hsK+tL,·=shK,·+thL,·,i.e.,sK+tL=sx+ty:x∈K,y∈L.The surface area measureSK,· of K∈Kn can be defined by the following:
(3)ddtVK+tLt=0+=∫Sn−1hLudSK,u,for any L∈Kn. From Equation (3), the Minkowski’s first mixed volume of K and L is given as follows:
(4)V1K,L=1n∫Sn−1hLudSK,u.The mixed volumeV1K,L generalizes the concepts of volume, surface area, and mean width.We say thatK∈Kn has a positive continuous curvature function fK,·=fK·:Sn−1⟶ℝ, if for all L∈Kn,
(5)V1K,L=1n∫Sn−1hLufKudSu,where S is spherical Lebesgue measure. Clearly, Equations (4) and (5) imply the following:
(6)fK,u=dSK,udS.Letp≥1. Using the Lp-Minkowski conbinations (see Equation (60)), Lutwak [2] defined the Lp-surface area measure SpK,· of a convex body K∈Kon, namely, for each L∈Kon,
(7)ddtVK+pt·Lt=0+=1p∫Sn−1hLupdSpK,u.ForK,L∈Kon, the Lp-mixed volume VpK,L is given by the following (see [4]):
(8)VpK,L=1n∫Sn−1hLupdSpK,u.We say thatK∈Kon has a positive continuous Lp-curvature function fpK,·:Sn−1⟶ℝ, if the integral representation
(9)VpK,Q=1n∫Sn−1hQupfpK,udSu,for all Q∈Kon. For K∈Kon with a positive continuous curvature functions, it follows from Equation (8) and Equation (9) that
(10)fpK,u=dSpK,udS.TheLp-Minkowski inequality of the Lp-mixed volume is (see [2, 7]) that for p≥1,
(11)VpK,Ln≥VKn−pVLp,with equality for p>1 if and only if K and L are dilates, for p=1 and if and only if K and L are homothetic.According to Equation (10), the curvature function of Lp is the Radon-Nikodym derivative of Lp-surface area measure with respect to the spherical Lebesgue measure. The integral of Lp-curvature function (raised to an appropriate power) over the unit sphere is the Lp-affine surface area, which is an important research point of affine geometry and valuation theory, see, e.g., [8–24]. The Lp-Minkowski problem (see [2]) is a necessary and sufficient condition to find a given measure such that it is only the Lp-surface area measure of a convex body. Solving the Lp-Minkowski problem requires solving a degenerate singular Monge-Ampère-type equation on the unit sphere. The Lp-Minkowski problem has been solved for p≥1, see [2, 25, 26], but critical cases for p<1 remain open, see, e.g., [25, 27–31]. For its applications, see [5, 7, 27, 32–35].A star bodyQ⊂ℝn is a compact star-shaped set about the origin whose radial function ρQ:Sn−1⟶0,∞ is defined by the following:
(12)ρQu=maxλ≥0:λu∈Q,for u∈Sn−1. If ρK,· is positive and continuous, K will be called a star body. Denote the set of star bodies in ℝn by Son. Obviously, Kon⊂Son.The dual Brunn-Minkowski theory is the theory of dual mixed volumes of star bodies. Forq∈ℝ, the q-th dual mixed volume, V~qK,Q, of K,Q∈Son is defined by the following:
(13)V~qK,Q=1n∫Sn−1ρKquρQn−qudu,where the integration is with respect to spherical Lebesgue measure. For q≠0, the q-th dual volume V~qK of K∈Son is defined by V~qK=V~qK,B. The q-th dual volume is important in geometric tomography, one of the reasons that is that for integers q=1,2,⋯,n−1 and each K∈Son,
(14)V~qK=cn,q∫Gn,qvolqK∩ξdξ,where volq denotes volume in ℝq, Gn,qq=1,2,⋯,n−1 denote the Grassmann manifold of q-dimensional subspaces of ℝn, the integration is with respect to the rotation invariant probability measure on Gn,q and constant cn,q is trivially determined by taking K to be B.For the realq≠0, the q-th dual curvature C~qK,· of K∈Kon is a Borel measure on the unit sphere, which can be defined in [3] by using the variational formula:
(15)ddtV~qK+tLt=0+=q∫Sn−1hLvhK−1vdC~qK,v,for every L∈Kon. Similar to the critical role as Lp-surface area measures playing in the Lp Brunn-Minkowski theory, dual curvature measures is a central concept within the dual Brunn-Minkowski theory.The singularity caseq=0 of dual volume leads to dual entropy of star body. For K∈Son, the dual entropy E~K can be defined as follows:
(16)E~K=1n∫Sn−1logρKudu.TheLp-integral curvature, JpK,·, of K∈Kon (see [1]) can be defined by a variational formula:
(17)ddtE~K+pt·Lt=0+=1np∫Sn−1hLpvdJpK∗,v,for all L∈Kon, where K∗ is the polar body of K is given by K∗=x∈ℝn:x·y≤1forally∈K.In [4], Lutwak et al. introduced Lp-dual curvature measures, which are a generalization of dual curvatures, Lp-surface area measure and Lp-integral curvatures. For p,q∈ℝ, K∈Kon and Q∈Son, the Lp-dual curvature measure, C~p,q, is the Borel measure on Sn−1 defined by the following:
(18)∫Sn−1gvdC~p,qK,Q,v=1n∫Sn−1gαKuhKαKu−pρKuqρQun−qdu,for each continuous g:Sn−1⟶ℝ, where αK is the radial Gauss map (see Section 2 for details).Lp-dual mixed volume (also known as p,q-dual mixed volume) is defined by Lutwak et al. [4] using the Lp-dual curvature:Forp,q∈ℝ, K,L∈Kon, and Q∈Son, the Lp-dual mixed volume V~p,qK,L,Q is defined by the following:
(19)V~p,qK,L,Q=∫Sn−1hLpvdC~p,qK,Q,v.By Equation (18), the Lp-dual mixed volume has the following integral formula:
(20)V~p,qK,L,Q=1n∫Sn−1hLαKuphKαKu−pρKuqρQun−qdu.Specifically,C~p,nK,B=1/nSpK,·, namely,
(21)dSpK,u=nC~p,nK,B=hKαKu−pρKundu,and for K,L∈Kon,
(22)VpK,L=1n∫Sn−1hLhKpαKuρKnudu.For thep,q-dual mixed volumes, the related Minkowski inequality is given in [4]. Suppose 1≤q/n≤p, if K,L∈Kon and Q∈Son, then
(23)V~p,qK,L,Qn≥VKq−pVLpVQn−q,with equality when q>n if and only if K,L and Q are dilates; while when q=n and p>1, with equality if and only if K and L are dilates; while when q=n and p=1, with equality if and only if K and L are homothetic.In [4], the authors studied the Lp-dual Minkowski problems for Lp-dual curvature measures. The results of Lp-dual Minkowski problem caught many attentions, for example, see [3, 27, 36–42]. In addition, based on the p,q-dual mixed volumes, Ma et al. studied p,q-John ellipsoids in [43], which contain the classical John ellipsoid and the Lp-John ellipsoids. They also solved two involving optimization problem about the p,q-dual mixed volumes for all 0<p≤q. A different extension of the Lp-John ellipsoid was considered by Li et al. in [44].In this paper, motivated by Lutwak et al.’s works in [4], we introduce the following Lp-curvature measures which is a new curvature measure.Definition 1.
Forp,q∈ℝ and K,Q∈Kon, we define the Lp-curvature measure Cp,qK,Q,· by the following:
(24)∫Sn−1gvdCp,qK,Q,v=1n∫Sn−1gαKuhQq−pαKuhK−qαKuρKnudu,for each continuous g:Sn−1⟶ℝ.According to Definition1, the Lp-curvature measure Cp,qK,Q,· has the following integral expression.Property 2.
Supposep,q∈ℝ. If K,Q∈Kon, then
(25)Cp,qK,Q,η=1n∫αK∗ηhQq−pαKuhK−qαKuρKnudu,
for each Borel setη⊆Sn−1. Here,
(26)αK∗η=xx=x¯∈Sn−1wherex∈HKvforsomev∈η,and HKv is the supporting hyperplane to K with outer normal vector v∈ℝn\0.Property 3.
Supposep,q∈ℝ. If K,Q∈Kon, then for each Borel set η⊆Sn−1,
(27)Cp,qK,Q,η=1n∫x∈xKηx·νKx1−qνKxQ∗q−pdHn−1x.Among them,Hn−1· represents the n−1-dimensional Hausdorff measure, and νKx represents the regular radial vector of x∈∂K, as well as xKη represents the reverse spherical image of η⊂Sn−1.TheLp-curvature measures unify the surface area measures, Lp-surface area measures and curvature measures, as well as other measures. In particular, for p,q∈ℝ and K, Q∈Kon the Lp-surface area measures and the q-th curvature measures (see Section 3 for its definition) are special cases of the Lp-curvature measures:
(28)Cq,qK,Q,·=1nSqK,·,(29)Cp,qK,K,·=1nSpK,·,(30)Cp,qK,B,·=1nSqK,·,(31)Cp,0K,K,·=1nSpK,·,(32)C0,qK,B,·=1nSqK,·.According to theLp-curvature measures, we now define the notion of the Lp,q-mixed volumes which unifies Lp-mixed volumes and dual-mixed volumes.Definition 4.
Forp,q∈ℝ and K,L,Q∈Kon, the Lp,q-mixed volume, Vp,qK,L,Q, of K and L (with respect to Q) is defined by the following:
(33)Vp,qK,L,Q=∫Sn−1hLupdCp,qK,Q,u.The following variational formula is an extension of Equations (3) and (7).Theorem 5.
If realsp,q≠0 and K,L,Q∈Kon, then the Lp,q-mixed volume Vp,qK,L,Q via the variational formula of K and L (with respect to Q) by the following:
(34)Vp,qK,L,Q=pqlimt⟶0VqK,Q+pt·L−VqK,Qt.Using Equation (24), the Lp,q-mixed volume can be written by the following integral formula:
(35)Vp,qK,L,Q=1n∫Sn−1hLαKuphQαKuq−phKαKu−qρKundu.It will be shown that theLp-mixed volume (Equation (8)) is the special case of the Lp,q-mixed volumes of convex bodies, i.e.,
(36)Vp,qK,L,K=VpK,L,Vp,pK,L,Q=VpK,L,Vp,qK,L,L=VqK,L,Vp,pK,L,B=VpK,L,V0,qK,L,Q=VqK,Q.The Minkowski-type inequality forLp,q-mixed volume is as follows:Theorem 6.
LetK,L,Q∈Kon and q≥1,p<0. Then,
(37)Vp,qK,L,Qn≥VKn−qVLpVQq−p,
with equality if and only ifK,L,Q are dilates when q>1 and K,Q are homothetic when q=1.ForQ,K∈Kon, we say that the convex body Q with respect to K has a positive continuous p,q-curvature function fp,qK,Q,·:Sn−1⟶ℝ, if
(38)Vp,qK,L,Q=1n∫Sn−1hLupfp,qK,Q,udSu,for all L∈Kon. From Equations (33) and (38), we get that for K∈Kon with a positive continuous curvature functions and a fixed Q∈Kon,
(39)1nfp,qK,Q,u=dCp,qK,Q,udS.Forq∈ℝ and t∈0,∞, the normalized power function tq¯ can be defined by the following:
(40)tq¯=1qtq,ifq≠0,logt,ifq=0.Forq∈ℝ and K,Q∈Kon, the normalized Lp-mixed volume Vq¯K,Q is defined by the following:
(41)Vq¯K,Q=1n∫Sn−1hQhKq¯αKuρKnudu.Note that forq≠0, we have qVq¯K,Q=VqK,Q, while for p=0 the normalized Lp-mixed volume Vp¯K,Q is not just VpK,Q multiplied by a constant but it can be considered from the mixed entropy (see Section 2 for details).Another aim of this paper is to show that forp,q∈ℝ and K,Q∈Kon, there exists a variational formula that defines the Lp-curvature measure Cp,qK,Q,· by the following:
(42)ddtVq¯K,Q+pt·Lt=0+=∫Sn−1hLvp¯dCp,qK,Q,v,for every L∈Kon. This plays a key role to solve the associated Minkowski-type problems using a variational method.Associated withLp-curvature measures, p,q-Minkowski problem related to Lp-curvature measure asks: For a given Borel measure ϕ on a sphere, what are the necessary and sufficient conditions for the existence of a K convex body whose Lp-curvature measure is ϕ? The uniqueness of the problem is to ask to what extent is a convex body uniquely determined by its Lp-curvature measure?The newp,q-Minkowski problem is equivalent to a degenerate singular Monge-Ampère equation on Sn−1: For fixed p,q∈ℝ,
(43)h1−pν∘∇¯h+hℓQ∗q−pdet∇¯2h+hI=f,where f:Sn−1⟶0,∞ is the given “data” function, h:Sn−1⟶0,∞ is the unknown function, and ℓ:Sn−1⟶Sn−1 is the identity map. Here, ∇¯h and ∇¯2h denote the gradient vector and the Hessian matrix of h, respectively, with respect to an orthonormal frame on Sn−1, and I is the identity matrix. If we assume that the range of the gradient function ∇¯h is D, then ν:D⟶Sn−1 is also an unknown function related to h.Finally, we propose some problems that need further study, i.e.,Lp,q-affine surface area problem, Lp,q-geominimal surface area problem and Lp,q-John ellipsoid problem.


## 2. Preliminaries

### 2.1. Basics in Convex Geometry

We work in then-dimensional Euclidean space ℝn. For x,y∈ℝn, we use x·y to denote the standard inner product of x and y, and x=x·x to denote the Euclidean norm of x. For x∈ℝn\0, we will use both x¯ and x to abbreviate x/x.We denote byCSn−1 the family of continuous functions defined on Sn−1 as endowed with the topology induced by the max-norm: f∞=maxv∈Sn−1fv, for f∈CSn−1.For the support function, we know that forλ>0 and x∈ℝn,
(44)hλKx=λhKx.Generally, forϕ∈GLn, the image ϕK=ϕx:x∈K satisfies that for x∈ℝn,
(45)hϕKx=hKϕtx,where ϕt denotes the transpose of ϕ.Since the support function is positive homogeneous of degree1, we can restricted it on the unit sphere. For convex bodies K,L∈Kn, their Hausdorff metric is given by the following:
(46)δHK,L≔hK−hL∞=maxu∈Sn−1hKu−hLu.At the pointv∈Sn−1 where hK is differentiable, the gradient of hK in ℝn is as follows:
(47)∇hKv=∇¯hKv+hKvv,where ∇¯hK denotes the gradient of hK on Sn−1 with respect to the standard metric of Sn−1.For the radial function, we see that forK∈Son, ϕ∈GLn and x∈ℝn\0,
(48)ρϕKx=ρKϕ−1x.Using the radial function, the volume ofK∈Son can be expressed as follows:
(49)VK=1n∫Sn−1ρKundSu.ForK∈Kon, the polar body K∗ of K is defined by the following:
(50)K∗=x∈ℝn:x·y≤1forally∈K.From this definition, we get that foru∈Sn−1,
(51)ρKu=1hK∗u,hKu=1ρK∗u,and for K∈Kon,
(52)K∗∗=K∗∗=K.ForK∈Kon, the Minkowski function of K is defined by the following:
(53)xK=mina≥0:x∈aK.Obviously, it is a continuous function onℝn, and
(54)xK∗=ρK∗x−1=hKx.In the whole process,Ω⊂Sn−1 will represent a closed set that cannot be contained in any of the closed hemispheres of Sn−1. Wulff shape h∈Kon, a continuous function h:Ω⟶0,∞, also known as h of the Aleksandrov body, is defined by the following:
(55)h=⋂v∈Ωx∈ℝn:x·v≤hv.IfK∈Kon, then it is easily seen that
(56)hK=K.Assume that the functionρ:Ω⟶0,∞ is continuous. Since Ω⊂Sn−1 is assumed to be closed, and ρ is continuous, we have ρuu:u∈Ω is a compact set in ℝn. The convex hull ρ generated by ρ,
(57)ρ=convρuu:u∈Ω,is compact as well (see Schneider [40], Theorem 1.1.11). Since Ω is not contained in any closed hemisphere of Sn−1, we get that ρ contains the origin in its interior; namely, ρ∈Kon. Obviously, if K∈Kon,
(58)ρK=K.The following lemma will be required.Lemma 7 (see [3]).
LetΩ⊂Sn−1 be a closed set that is not contained in any closed hemisphere of Sn−1. Let h:Ω⟶0,∞ be continuous. Then, the Wulff shape h determined by h and the convex hull 1/h generated by the function 1/h are polar reciprocals of each other; namely,
(59)h∗=1h.
LetK,L∈Kon and p≥1. The Lp-Minkowski combination s·K+pt·L is the convex body whose support function is given by the following (see [2]):
(60)hs·K+pt·L,·p=shK,·p+thL,·p.From Equation (53), we can extend the Lp-Minkowski combinations to the cases of p<1.Letp≠0. For K,L∈Kon, and s,t∈ℝ such that shKp+thLp is a strictly positive function on Sn−1, Lutwak et al. [4] defined the Lp-Minkowski combination s·K+pt·L∈Kon by the following:
(61)s·K+pt·L=shKp+thLp1/p.Whenp=0, define s·K+0t·L by the following:
(62)s·K+0t·L=hKshLt.Note thats·K+0t·L is defined for all s,t∈ℝ, since hK,hL are strictly positive functions on Sn−1.Givenϕ∈SLn and p≠0 (see [4]), we obtain that for s,t∈ℝ,
(63)s·ϕK+pt·ϕL=ϕs·K+pt·L.Ifs+t=1, then Equation (63) holds for p=0 as well.Forp∈ℝ\0 and K,L∈Kon, the Lp-mixed volume VpK,L is defined by the following:
(64)VpK,L=1n∫Sn−1hLpudSpK,u=pnlimt⟶0+VK+pt·L−VKt.From Equations (64) and (63), we get that for ϕ∈SLn (see [45]),
(65)VpϕK,ϕL=VpK,L.TheLp-surface area SpK of K∈Kon is given by SpK=nVpK,B.The following definition will be required.Definition 8 (see [4]).
Letp∈ℝ. If μ is a Borel measure on Sn−1 and ϕ∈SLn, then ϕp⊣μ, the Lp image of μ under ϕ, is a Borel measure such that
(66)∫Sn−1fudϕp⊣μu=∫Sn−1ϕ−1upfϕ−1udμu,for each Borel f:Sn−1⟶ℝ.Recall that theLp-mixed volume has a dual integral formulation (see [4]): If K,L∈Kon, then
(67)VpK,L=1n∫Sn−1hLhKpαKuρKnudu,where αK is the radial Gauss map of K.ForK,L∈Kon and p>0, we define the volume-normalized Lp-mixed volume by the following:
(68)V¯pK,L=VpK,LVK1/p=∫Sn−1hLhKpαKudV~Ku1/p.Note thatdV~K=1/nρKndu/VK is the normalized dual conical measure of K, it is a probability measure on suppS·. Let p⟶0. Then,
(69)V¯0K,L=exp∫Sn−1loghLhKαKudV~Ku=exp1nVK∫Sn−1loghLhKαKuρKnudu.The mixed entropyEK,L of K,L∈Kon is defined by the following:
(70)EK,L=1n∫Sn−1loghLhKαKuρKnudu.Note thatEK,L=V0¯K,L. As the case in Equation (63), for the dual mixed entropy, we have that for ϕ∈SLn,
(71)EϕK,ϕL=EK,L.


### 2.2. The Radial Gauss Map

The following results come from the articles [3, 4].SupposeK is a convex body in ℝn. For each v∈ℝn\0, the hyperplane
(72)HKv=x∈ℝn:x·v=hKvis called the supporting hyperplane to K with outer normal vector v.The spherical image ofσ⊂∂K is defined by the following:
(73)νKσ=v∈Sn−1:x∈HKvforsomex∈σ⊂Sn−1.The reverse spherical image ofη⊂Sn−1 is defined by the following:
(74)xKη=x∈∂K:x∈HKvforsomev∈η⊂∂K.SupposeσK⊂∂K is a set consisting of all x∈∂K, for which the set νKx, which we frequently abbreviate as νKx, contains more than a single element. It is a well-known fact that Hn−1σK=0 (see Schneider [46], p. 84). The function on the set of regular radial vectors of ∂K is precisely defined by the following:
(75)νK:∂K\σK⟶Sn−1,by making νKx be the unique element in νKx for each x∈∂K\σK, The function νK is called the spherical image map of K and is known to be continuous (see Schneider [40], Lemma 2.2.12). It will be very convenient to abbreviate ∂K\σK by ∂′K. Since Hn−1σK=0, when the integration is about Hn−1, it does not matter if the domain is over subsets of ∂′K or ∂K.The setηK⊂Sn−1 consisting of all v∈Sn−1, for which the set xKv contains more than a single element, is of Hn−1–measure 0 (see Schneider [40], Theorem 2.2.11). The function is precisely defined on the set of regular unit normal vectors of K:
(76)xK:Sn−1\ηK⟶∂K,by making xKv be the unique element in xKv, for each v∈Sn−1\ηK. The function xK is called the reverse spherical image map and is well known to be continuous (see Schneider [40], Lemma 2.2.12). By extending xK to be a homogeneous function of degree 0 in ℝn\0, we get a natural definition of xK on the set of all regular normal vectors on ∂K.Forω⊂Sn−1, the radial Gauss image of ω is defined by the following:
(77)αKω=v∈Sn−1:ρKuu∈HKvforsomeu∈ω.For a subsetη⊂Sn−1, the reverse radial Gauss image of η is defined by the following:
(78)αK∗η=u∈Sn−1:ρKuu∈HKvforsomev∈η.Thus,(79)αK∗η=x¯:x∈∂Kwherex∈HKvforsomev∈η.In particular, we can see that ifη contains only a single vector v∈Sn−1,
(80)αK∗v=x¯:x∈∂Kwherex∈HKv.Note that Equation (78), and hence for u∈Sn−1 and η⊂Sn−1, we see from Equation (77) that
(81)u∈αK∗η⇔αKu∩η≠∅.Thus, forη1,η2⊆Sn−1,
(82)η1⊆η2⇒αK∗η1⊆αK∗η2.We shall need to make use of the fact that foru,v∈Sn−1,
(83)u∈αK∗v⇔v∈αKu.Ifu∈ωK, then αKu=αKu, and Equation (77) becomes
(84)u∈αK∗η⇔αKu∈η,and hence Equation (84) holds for almost all u∈Sn−1, with respect to spherical Lebesgue measure.The following lemma will be used.Lemma 9 (see [4]).
IfK∈Kon, then
(85)αK∗η=αK∗η,for each η⊆Sn−1.SinceαK∗v=αK∗v for almost all v∈Sn−1 with respect to spherical Lebesgue measure, and αK∗v=αK∗v for almost all v∈Sn−1 with respect to spherical Lebesgue measure, Lemma 9 implies that if K∈Kon, then
(86)αK∗=αK∗,almost everywhere with respect to spherical Lebesgue measure.ForK∈Kon, the radial map of K is defined by the following:
(87)rK:Sn−1⟶∂KbyrKu=ρKuu∈∂K,for u∈Sn−1. Note that rK−1:∂K⟶Sn−1 is just the restriction to ∂K of the map ℝn\0⟶Sn−1.The radial Gauss map of the convex bodyK∈Kon is defined by the following:
(88)αK:Sn−1\ωK⟶Sn−1byαK=νK∘rK,where ωK=σK¯=rK−1σK. Since rK−1=·¯ is a bi-Lipschitz map between the spaces ∂K and Sn−1, so it follows that ωK has spherical Lebesgue measure 0. We observed that if u∈Sn−1/ωK, then αKu contains only the element αKu. Since both νK and rK are continuous, αK is continuous. Notice that for x∈∂′K,
(89)αKx¯=νKx,and hence for x∈∂′K,
(90)hKαKx¯=hKνKx=x·νKx.Ifu∈Sn−1/ωK, we see that x=ρKuu∈∂K/ωK with x¯=u from the definition of ωK. Hence from Equation (89) we have αKu=αKx¯=νKx and we get the following (see [4]):
(91)αKu=−∇ρKu∇ρKu=∇hK∗u∇hK∗u,u∈Sn−1\ωK.Combining with Equations (86) and (91), we have the following:
(92)αK∗v=∇hKv∇hKv,for almost all v with respect to spherical Lebesgue measure.The surface area measureSK,· of a convex body K can be defined, for Borel η⊆Sn−1, by the following:
(93)SK,η=Hn−1xKη,where xKη is the reverse spherical image of η⊂Sn−1.If the boundary of a convex bodyK, denoted by ∂K, is smooth with positive Gauss curvature, the surface area measure of K is absolutely continuous with respect to spherical Lebesgue measure. The density can be regarded as the reciprocal of Gauss curvature and expressed in terms of the support function and its Hessian matrix on Sn−1:
(94)dSK,·dS=det∇¯2hK+hKI,where ∇¯2hK denotes the Hessian matrix of hK and I is the identity matrix with respect to an orthonormal frame on Sn−1. See Schneider [46].Forp∈ℝ and K∈Kon, its Lp-surface area measure SpK,· introduced in [2] is defined by the following:
(95)dSpK,·=hK1−pdSK,·,or equivalently by the following:
(96)SpK,η=∫xKηx·νKx1−pdHn−1x,for each Borel η⊆Sn−1, where νK is the spherical image function of σ⊂∂K.Forλ>0, we easily see hλK=λhK and SλK,·=λn−1SK,·. Then, Equation (91) implies the following:
(97)SpλK,·=λn−qSpK,·.The following integral identity is established in [3].Lemma 10.
Ifq∈ℝ and K∈Kon, while f:Sn−1⟶ℝ is bounded and Lebesgue integrable, then
(98)∫Sn−1fuρKuqdu=∫∂′Kfx¯xq−nx·νKxdHn−1x.In [3], we see thatLemma 11.
IfK∈Kon is strictly convex, and f:Sn−1⟶ℝ and F:∂K⟶ℝ are both continuous, then
(99)∫Sn−1fuF∇hKuhKudSK,u=∫∂′Kx·νKxfνKxFxdHn−1x,where ∇hK is the gradient of hK in ℝn, and νK is defined only on ∂K\σK, the set σK has Hn−1 measure 0.We will require a slight extension of Equation (97). To be specific, if p∈ℝ, while K∈Kon is strictly convex, and f:Sn−1⟶ℝ and F:∂K⟶ℝ are both continuous, then (see [4])
(100)∫Sn−1fuF∇hKudSpK,u=∫∂′Kx·νKx1−pfνKxFxdHn−1x.The following lemma will be used.Lemma 12 (see [4]).
For eachp∈ℝ, the set
(101)chKp¯−chBp¯:K∈Kon,c>0is dense in CSn−1.


## 2.1. Basics in Convex Geometry

We work in then-dimensional Euclidean space ℝn. For x,y∈ℝn, we use x·y to denote the standard inner product of x and y, and x=x·x to denote the Euclidean norm of x. For x∈ℝn\0, we will use both x¯ and x to abbreviate x/x.We denote byCSn−1 the family of continuous functions defined on Sn−1 as endowed with the topology induced by the max-norm: f∞=maxv∈Sn−1fv, for f∈CSn−1.For the support function, we know that forλ>0 and x∈ℝn,
(44)hλKx=λhKx.Generally, forϕ∈GLn, the image ϕK=ϕx:x∈K satisfies that for x∈ℝn,
(45)hϕKx=hKϕtx,where ϕt denotes the transpose of ϕ.Since the support function is positive homogeneous of degree1, we can restricted it on the unit sphere. For convex bodies K,L∈Kn, their Hausdorff metric is given by the following:
(46)δHK,L≔hK−hL∞=maxu∈Sn−1hKu−hLu.At the pointv∈Sn−1 where hK is differentiable, the gradient of hK in ℝn is as follows:
(47)∇hKv=∇¯hKv+hKvv,where ∇¯hK denotes the gradient of hK on Sn−1 with respect to the standard metric of Sn−1.For the radial function, we see that forK∈Son, ϕ∈GLn and x∈ℝn\0,
(48)ρϕKx=ρKϕ−1x.Using the radial function, the volume ofK∈Son can be expressed as follows:
(49)VK=1n∫Sn−1ρKundSu.ForK∈Kon, the polar body K∗ of K is defined by the following:
(50)K∗=x∈ℝn:x·y≤1forally∈K.From this definition, we get that foru∈Sn−1,
(51)ρKu=1hK∗u,hKu=1ρK∗u,and for K∈Kon,
(52)K∗∗=K∗∗=K.ForK∈Kon, the Minkowski function of K is defined by the following:
(53)xK=mina≥0:x∈aK.Obviously, it is a continuous function onℝn, and
(54)xK∗=ρK∗x−1=hKx.In the whole process,Ω⊂Sn−1 will represent a closed set that cannot be contained in any of the closed hemispheres of Sn−1. Wulff shape h∈Kon, a continuous function h:Ω⟶0,∞, also known as h of the Aleksandrov body, is defined by the following:
(55)h=⋂v∈Ωx∈ℝn:x·v≤hv.IfK∈Kon, then it is easily seen that
(56)hK=K.Assume that the functionρ:Ω⟶0,∞ is continuous. Since Ω⊂Sn−1 is assumed to be closed, and ρ is continuous, we have ρuu:u∈Ω is a compact set in ℝn. The convex hull ρ generated by ρ,
(57)ρ=convρuu:u∈Ω,is compact as well (see Schneider [40], Theorem 1.1.11). Since Ω is not contained in any closed hemisphere of Sn−1, we get that ρ contains the origin in its interior; namely, ρ∈Kon. Obviously, if K∈Kon,
(58)ρK=K.The following lemma will be required.Lemma 7 (see [3]).
LetΩ⊂Sn−1 be a closed set that is not contained in any closed hemisphere of Sn−1. Let h:Ω⟶0,∞ be continuous. Then, the Wulff shape h determined by h and the convex hull 1/h generated by the function 1/h are polar reciprocals of each other; namely,
(59)h∗=1h.
LetK,L∈Kon and p≥1. The Lp-Minkowski combination s·K+pt·L is the convex body whose support function is given by the following (see [2]):
(60)hs·K+pt·L,·p=shK,·p+thL,·p.From Equation (53), we can extend the Lp-Minkowski combinations to the cases of p<1.Letp≠0. For K,L∈Kon, and s,t∈ℝ such that shKp+thLp is a strictly positive function on Sn−1, Lutwak et al. [4] defined the Lp-Minkowski combination s·K+pt·L∈Kon by the following:
(61)s·K+pt·L=shKp+thLp1/p.Whenp=0, define s·K+0t·L by the following:
(62)s·K+0t·L=hKshLt.Note thats·K+0t·L is defined for all s,t∈ℝ, since hK,hL are strictly positive functions on Sn−1.Givenϕ∈SLn and p≠0 (see [4]), we obtain that for s,t∈ℝ,
(63)s·ϕK+pt·ϕL=ϕs·K+pt·L.Ifs+t=1, then Equation (63) holds for p=0 as well.Forp∈ℝ\0 and K,L∈Kon, the Lp-mixed volume VpK,L is defined by the following:
(64)VpK,L=1n∫Sn−1hLpudSpK,u=pnlimt⟶0+VK+pt·L−VKt.From Equations (64) and (63), we get that for ϕ∈SLn (see [45]),
(65)VpϕK,ϕL=VpK,L.TheLp-surface area SpK of K∈Kon is given by SpK=nVpK,B.The following definition will be required.Definition 8 (see [4]).
Letp∈ℝ. If μ is a Borel measure on Sn−1 and ϕ∈SLn, then ϕp⊣μ, the Lp image of μ under ϕ, is a Borel measure such that
(66)∫Sn−1fudϕp⊣μu=∫Sn−1ϕ−1upfϕ−1udμu,for each Borel f:Sn−1⟶ℝ.Recall that theLp-mixed volume has a dual integral formulation (see [4]): If K,L∈Kon, then
(67)VpK,L=1n∫Sn−1hLhKpαKuρKnudu,where αK is the radial Gauss map of K.ForK,L∈Kon and p>0, we define the volume-normalized Lp-mixed volume by the following:
(68)V¯pK,L=VpK,LVK1/p=∫Sn−1hLhKpαKudV~Ku1/p.Note thatdV~K=1/nρKndu/VK is the normalized dual conical measure of K, it is a probability measure on suppS·. Let p⟶0. Then,
(69)V¯0K,L=exp∫Sn−1loghLhKαKudV~Ku=exp1nVK∫Sn−1loghLhKαKuρKnudu.The mixed entropyEK,L of K,L∈Kon is defined by the following:
(70)EK,L=1n∫Sn−1loghLhKαKuρKnudu.Note thatEK,L=V0¯K,L. As the case in Equation (63), for the dual mixed entropy, we have that for ϕ∈SLn,
(71)EϕK,ϕL=EK,L.


## 2.2. The Radial Gauss Map

The following results come from the articles [3, 4].SupposeK is a convex body in ℝn. For each v∈ℝn\0, the hyperplane
(72)HKv=x∈ℝn:x·v=hKvis called the supporting hyperplane to K with outer normal vector v.The spherical image ofσ⊂∂K is defined by the following:
(73)νKσ=v∈Sn−1:x∈HKvforsomex∈σ⊂Sn−1.The reverse spherical image ofη⊂Sn−1 is defined by the following:
(74)xKη=x∈∂K:x∈HKvforsomev∈η⊂∂K.SupposeσK⊂∂K is a set consisting of all x∈∂K, for which the set νKx, which we frequently abbreviate as νKx, contains more than a single element. It is a well-known fact that Hn−1σK=0 (see Schneider [46], p. 84). The function on the set of regular radial vectors of ∂K is precisely defined by the following:
(75)νK:∂K\σK⟶Sn−1,by making νKx be the unique element in νKx for each x∈∂K\σK, The function νK is called the spherical image map of K and is known to be continuous (see Schneider [40], Lemma 2.2.12). It will be very convenient to abbreviate ∂K\σK by ∂′K. Since Hn−1σK=0, when the integration is about Hn−1, it does not matter if the domain is over subsets of ∂′K or ∂K.The setηK⊂Sn−1 consisting of all v∈Sn−1, for which the set xKv contains more than a single element, is of Hn−1–measure 0 (see Schneider [40], Theorem 2.2.11). The function is precisely defined on the set of regular unit normal vectors of K:
(76)xK:Sn−1\ηK⟶∂K,by making xKv be the unique element in xKv, for each v∈Sn−1\ηK. The function xK is called the reverse spherical image map and is well known to be continuous (see Schneider [40], Lemma 2.2.12). By extending xK to be a homogeneous function of degree 0 in ℝn\0, we get a natural definition of xK on the set of all regular normal vectors on ∂K.Forω⊂Sn−1, the radial Gauss image of ω is defined by the following:
(77)αKω=v∈Sn−1:ρKuu∈HKvforsomeu∈ω.For a subsetη⊂Sn−1, the reverse radial Gauss image of η is defined by the following:
(78)αK∗η=u∈Sn−1:ρKuu∈HKvforsomev∈η.Thus,(79)αK∗η=x¯:x∈∂Kwherex∈HKvforsomev∈η.In particular, we can see that ifη contains only a single vector v∈Sn−1,
(80)αK∗v=x¯:x∈∂Kwherex∈HKv.Note that Equation (78), and hence for u∈Sn−1 and η⊂Sn−1, we see from Equation (77) that
(81)u∈αK∗η⇔αKu∩η≠∅.Thus, forη1,η2⊆Sn−1,
(82)η1⊆η2⇒αK∗η1⊆αK∗η2.We shall need to make use of the fact that foru,v∈Sn−1,
(83)u∈αK∗v⇔v∈αKu.Ifu∈ωK, then αKu=αKu, and Equation (77) becomes
(84)u∈αK∗η⇔αKu∈η,and hence Equation (84) holds for almost all u∈Sn−1, with respect to spherical Lebesgue measure.The following lemma will be used.Lemma 9 (see [4]).
IfK∈Kon, then
(85)αK∗η=αK∗η,for each η⊆Sn−1.SinceαK∗v=αK∗v for almost all v∈Sn−1 with respect to spherical Lebesgue measure, and αK∗v=αK∗v for almost all v∈Sn−1 with respect to spherical Lebesgue measure, Lemma 9 implies that if K∈Kon, then
(86)αK∗=αK∗,almost everywhere with respect to spherical Lebesgue measure.ForK∈Kon, the radial map of K is defined by the following:
(87)rK:Sn−1⟶∂KbyrKu=ρKuu∈∂K,for u∈Sn−1. Note that rK−1:∂K⟶Sn−1 is just the restriction to ∂K of the map ℝn\0⟶Sn−1.The radial Gauss map of the convex bodyK∈Kon is defined by the following:
(88)αK:Sn−1\ωK⟶Sn−1byαK=νK∘rK,where ωK=σK¯=rK−1σK. Since rK−1=·¯ is a bi-Lipschitz map between the spaces ∂K and Sn−1, so it follows that ωK has spherical Lebesgue measure 0. We observed that if u∈Sn−1/ωK, then αKu contains only the element αKu. Since both νK and rK are continuous, αK is continuous. Notice that for x∈∂′K,
(89)αKx¯=νKx,and hence for x∈∂′K,
(90)hKαKx¯=hKνKx=x·νKx.Ifu∈Sn−1/ωK, we see that x=ρKuu∈∂K/ωK with x¯=u from the definition of ωK. Hence from Equation (89) we have αKu=αKx¯=νKx and we get the following (see [4]):
(91)αKu=−∇ρKu∇ρKu=∇hK∗u∇hK∗u,u∈Sn−1\ωK.Combining with Equations (86) and (91), we have the following:
(92)αK∗v=∇hKv∇hKv,for almost all v with respect to spherical Lebesgue measure.The surface area measureSK,· of a convex body K can be defined, for Borel η⊆Sn−1, by the following:
(93)SK,η=Hn−1xKη,where xKη is the reverse spherical image of η⊂Sn−1.If the boundary of a convex bodyK, denoted by ∂K, is smooth with positive Gauss curvature, the surface area measure of K is absolutely continuous with respect to spherical Lebesgue measure. The density can be regarded as the reciprocal of Gauss curvature and expressed in terms of the support function and its Hessian matrix on Sn−1:
(94)dSK,·dS=det∇¯2hK+hKI,where ∇¯2hK denotes the Hessian matrix of hK and I is the identity matrix with respect to an orthonormal frame on Sn−1. See Schneider [46].Forp∈ℝ and K∈Kon, its Lp-surface area measure SpK,· introduced in [2] is defined by the following:
(95)dSpK,·=hK1−pdSK,·,or equivalently by the following:
(96)SpK,η=∫xKηx·νKx1−pdHn−1x,for each Borel η⊆Sn−1, where νK is the spherical image function of σ⊂∂K.Forλ>0, we easily see hλK=λhK and SλK,·=λn−1SK,·. Then, Equation (91) implies the following:
(97)SpλK,·=λn−qSpK,·.The following integral identity is established in [3].Lemma 10.
Ifq∈ℝ and K∈Kon, while f:Sn−1⟶ℝ is bounded and Lebesgue integrable, then
(98)∫Sn−1fuρKuqdu=∫∂′Kfx¯xq−nx·νKxdHn−1x.In [3], we see thatLemma 11.
IfK∈Kon is strictly convex, and f:Sn−1⟶ℝ and F:∂K⟶ℝ are both continuous, then
(99)∫Sn−1fuF∇hKuhKudSK,u=∫∂′Kx·νKxfνKxFxdHn−1x,where ∇hK is the gradient of hK in ℝn, and νK is defined only on ∂K\σK, the set σK has Hn−1 measure 0.We will require a slight extension of Equation (97). To be specific, if p∈ℝ, while K∈Kon is strictly convex, and f:Sn−1⟶ℝ and F:∂K⟶ℝ are both continuous, then (see [4])
(100)∫Sn−1fuF∇hKudSpK,u=∫∂′Kx·νKx1−pfνKxFxdHn−1x.The following lemma will be used.Lemma 12 (see [4]).
For eachp∈ℝ, the set
(101)chKp¯−chBp¯:K∈Kon,c>0is dense in CSn−1.


## 3.Lp-Curvature Measures

For a star bodyQ∈Son, define ·Q:ℝn⟶0,∞ by letting (see [4])
(102)xQ=1ρQx,ifx≠0,0,ifx=0.Note that·Q is continuous and positively homogeneous of degree 1. If Q is an origin-symmetric convex body in ℝn, then ·Q is just an ordinary norm in ℝn, and ℝn,·Q is the n-dimensional Banach space whose unit ball is Q.Note that the definition (Equation (102)) is an extension of Minkowski functional (Equation (53)) of convex body K∈Kon.Definition 13.
Supposeq∈ℝ. For K,Q∈Kon, the q-th area measure SqK,Q,· is defined by the following:
(103)SqK,Q,ω=1n∫ωhQhKqαKuρKnudu,for each Lebesgue measurable ω⊆Sn−1, and the q-th curvature measure CqK,Q,· is defined by the following:
(104)CqK,Q,η=1n∫αK∗ηhQhKqαKuρKnudu,for each Borel η⊆Sn−1. Moreover, for each p∈ℝ, the Lp-curvature measure Cp,qK,Q,· is defined by the following:
(105)dCp,qK,Q,·=hQ−pdCqK,Q,·.Observe that(106)C0,qK,Q,·=CqK,Q,·.Note that from definition (Equation (104)) and the fact that Equation (84) holds off of the set ωK of spherical Lebesgue measure 0, so for each Borel η⊆Sn−1, we get the following:
(107)∫Sn−11ηvdCqK,Q,v=CqK,Q,η=1n∫αK∗ηhQhKqαKvρKnvdv=1n∫Sn−11αK∗ηvhQhKqαKvρKnvdv=1n∫Sn−11ηαKuhQqαKuhK−qαKuρKnudu.That is,(108)∫Sn−11ηvdCqK,Q,v=1n∫Sn−11ηαKuhQqαKuhK−qαKuρKnudu.We observed thatCqK,Q,· is absolutely continuous with respect to spherical Lebesgue measure. Then, from Equation (108), we deduce thatLemma 14.
LetK∈Kon and q∈ℝ. If each function f:Sn−1⟶ℝ is bounded and Borel, then
(109)∫Sn−1fvdCqK,Q,v=1n∫Sn−1fαKuhQqαKuhK−qαKuρKnudu.Proof.
Because Equation (109) is shown by Equation (108) as an indicator function of the Borel set, we see that Equation (109) holds for a linear combination of the indicator functions of the Borel set, namely, simple functions ϕ:Sn−1⟶ℝ, is given by the following:
(110)ϕ=∑i=1mci1ηi,where ci∈ℝ and Borel ηi⊂Sn−1. Now let us choose a sequence of simple functions ϕk:Sn−1⟶ℝ converging to the bounded Borel function f:Sn−1⟶ℝ. Note that f is bounded, ϕk can be selected as uniformly bounded. Then, ϕk∘αK converges pointwise to f∘αK on Sn−1\ωK. Since f:Sn−1⟶ℝ is a Borel function and the radial Gauss map αK:Sn−1\ωK⟶Sn−1 is continuous; thus, f∘αK is a Borel function on Sn−1\ωK. Because f is bounded, and ωK has spherical Lebesgue measure 0, we can infer that f is CqK,q,· integrable, and f∘αK is spherical Lebesgue integrable in Sn−1. Since CqK,q,· is a finite measure, by taking the limit k⟶∞, we obtain Equation (109).☐Proposition 15.
Letp,q∈ℝ. If K,Q∈Kon, then
(111)Cp,qK,Q,η=1n∫αK∗ηhQq−pαKuhK−qαKuρKnudu,for each Borel set η⊆Sn−1.Proof.
From Equations (105), (109), and (84), we have for each Borel η⊆Sn−1,
(112)Cp,qK,Q,η=∫Sn−11ηudCp,qK,Q,u=∫Sn−11ηuhQu−pdCqK,Q,u=1n∫Sn−11ηαKuhQ−pαKuhQqαKuhK−qαKuρKnudu=1n∫Sn−11αK∗ηuhQq−pαKuhK−qαKuρKnudu=1n∫αK∗ηhQq−pαKuhK−qαKuρKnudu.☐Obviously, the total measures of theq-th curvature measure and the q-th area measure are the q-th mixed volume, i.e.,
(113)VqQ,K=SqK,Q,Sn−1=CqK,Q,Sn−1.It follows immediately from Equations (103) and (104) that
(114)CqK,Q,η=SqK,Q,αK∗η.TheLp-curvature measures have the following properties.Property 16.
Letp,q∈ℝ. If K,Q∈Kon. Then, for each Borel set η⊆Sn−1 and each bounded Borel function g:Sn−1⟶ℝ, we have the following:
(115)∫Sn−1gvdCp,qK,Q,v=1n∫Sn−1gαKuhQq−pαKuhK−qαKuρKnudu,(116)∫Sn−1gvdCp,qK,Q,v=1n∫∂′KgνKxx·νKx1−qνKxQ∗q−pdHn−1x,(117)Cp,qK,Q,η=1n∫x∈xKηx·νKx1−qνKxQ∗q−pdHn−1x.Proof.
BecausehQ−p:Sn−1⟶ℝ is a bounded Borel function, from Equation (109) with f=ghQ−p, we have the following:
(118)∫Sn−1gvhQ−pvdCqK,Q,v=1n∫Sn−1gαKuhQq−pαKuhK−qαKuρKnudu.Thus, in light of Equation (105) is the desired result (Equation (115)).
By Equations (115), (89), and (90), and letting f=g∘αKhQq−p∘αKhK−q∘αK and q=n in Equation (98), we have the following:
(119)∫Sn−1gvdCp,qK,Q,v=1n∫Sn−1gαKuhQq−pαKuhK−qαKuρKnudu=1n∫∂′KgαKx¯hQq−pαKx¯hK−qαKx¯x·νKxdHn−1x=1n∫∂′KgνKxx·νKx1−qhQq−pνKxdHn−1x=1n∫∂′KgνKxx·νKx1−qνKxQ∗q−pdHn−1x.☐This yields Equation (116).Takeg=1η in Equation (116). Notice that νKx∈η⇔x∈xKη for almost all x with respect to spherical Lebesgue measure. So, we immediately obtain Equation (117).Remark 17.
Equation (115) tells us the rationality for Definition 1 of the Lp-curvature measure Cp,qK,Q,·.Example 18 (Lp-curvature measures of polytopes).
SupposeP∈Kon be a polytope with outer unit normal vectors v1,v2,⋯,vm. If Δi is a cone consisting of all rays emanating from the origin and passing through the face of P whose outer normal is vi. Remember that we abbreviate αP∗vi by αP∗vi, and from Equation (80), we get the following:
(120)αP∗vi=Sn−1∩Δi,αPu=vi,foralmostallu∈Δi∩Sn−1.
Ifη⊂Sn−1 is a Borel set such that v1,v2,⋯,vm∩η=∅, then αP∗η has spherical Lebesgue measure 0. So, the Lp-curvature measure Cp,qP,Q,· is discrete and concentrated on v1,v2,⋯,vm. From Proposition 15 and Equation (120), we have the following:
(121)Cp,qP,Q·=∑i=1mdiδvi,where δvi represents the delta measure centered on vi, and
(122)di=1nhQq−pvihP−qvi∫Sn−1∩ΔiρPnudu.Example 19 (Lp-curvature measures of strictly convex bodies).
LetK,Q∈Kon are strictly convex. Suppose g:Sn−1⟶ℝ is continuous, then we start with Equations (116) and (100)(taking Fx=νKxQ∗q−p) and combine the fact that ∂K/∂′K has measure 0, it follows that
(123)∫Sn−1gvdCp,qK,Q,v=1n∫∂′Kx·νKx1−qgνKxνKxQ∗q−pdHn−1x=1n∫Sn−1gvνK∇hKvQ∗q−pdSqK,v.Using Equation (95), this shows that
(124)dCp,qK,Q,·=1nνK∇hKQ∗q−pdSqK,·=1nhK1−qνK∘∇hKQ∗q−pdSK,·.Example 20 (Lp-curvature measures of smooth convex bodies).
LetQ∈Kon has a C2 boundary with everywhere positive Gauss curvature. Because in this case, SQ,· is absolutely continuous for the spherical Lebesgue measure; therefore, Cp,qK,Q,· is absolutely continuous for the spherical Lebesgue measure, and from Equations (124), (94), and (47), we get the following:
(125)dCp,qK,Q,vdv=1nhK1−qvνK∇¯hKv+hKvvQ∗q−pdet∇¯2hKv+hKvI,where ∇¯hKv represents the gradient of hK on Sn−1 at v and ∇¯2hK represents the Hessian matrix of hK with respect to an orthonormal frame on Sn−1. We write Equation (125) as 1/nfp,qK,Q,v, that is,
(126)1nfp,qK,Q,v=dCp,qK,Q,vdv.We say convex body Q with respect to a fixed convex body K as a parameter have a positive continuous p,q-curvature function fp,qK,Q,·.The weak convergence ofLp-curvature measure is an important property contained in the following propositions.Proposition 21.
Letp,q∈ℝ and Q∈Kon. If Ki∈Kon with Ki⟶K0∈Kon, then Cp,qKi,Q,·⟶Cp,qK0,Q,·, weakly.Proof.
Letg:Sn−1⟶ℝ is continuous. From Equation (115) we know that
(127)∫Sn−1gvdCp,qKi,Q,v=1n∫Sn−1gαKiuhQq−pαKiuhKi−qαKiuρKinudu,for all i. Since Ki⟶K0, with respect to the Hausdorff metric, we have that hKi⟶hK0, uniformly on Sn−1, and the surface area measure has the following property (see [2, 7, 23]):
(128)Ki⟶K0⇒SKi⟶SK0weakly.
Thus,(129)∫Sn−1gαKiuhQq−pαKiuhKi−qαKiuρKinudu,⟶∫Sn−1gαK0uhQq−pαK0uhK0−qαK0uρK0nudu.
It follows thatCp,qKi,Q,·⟶Cp,qK0,Q,·, weakly. ☐The following statement contains the absolute continuity ofLp-curvature measure with respect to surface area measure.Proposition 22.
Letp,q∈ℝ. If K,Q∈Kon, then Lp-curvature measure Cp,qK,Q,· is absolutely continuous with respect to the surface area measure SK,·.Proof.
Letη⊂Sn−1 be such that SK,η=0, or equivalently by definition (Equation (96)), Hn−1xKη=0. Then, Equation (117) states that
(130)Cp,qK,Q,η=1n∫x∈xKηx·νKx1−qνKxQ∗q−pdHn−1x=0.Thus, the integration is over a set of measure 0.☐The following proposition shows that theLp-curvature measure including the classical surface area measures and the Lp-surface area measures. Therefore, the classical surface area measures and the Lp-surface area measures are special cases of the Lp-curvature measures.Proposition 23.
SupposeK,Q∈Kon and p,q∈ℝ. Then,
(131)Cq,qK,Q,·=1nSqK,·,(132)Cp,qK,K,·=1nSpK,·,(133)Cp,qK,B,·=1nSqK,·,(134)Cp,0K,K,·=1nSpK,·,(135)C0,qK,B,·=1nSqK,·.Proof.
Letη⊂Sn−1 be a Borel set. From Equations (117) and (96), we have the following:
(136)Cq,qK,Q,η=∫x∈xKηx·νKx1−qdHn−1x=Cp,qK,B,η=1nSqK,η.
Therefore, we get Equations (131) and (133).
From Equations (117), (54), (90), and (96), we have the following:
(137)Cp,qK,K,η=1n∫x∈xKηx·νKx1−qνKxK∗q−pdHn−1x=1n∫x∈xKηx·νKx1−qhKνKxq−pdHn−1x=1n∫x∈xKηx·νKx1−pdHn−1x=1nSpK,η.
Therefore, we get Equation (132). Similarly, we can get the rest.☐Recall that the concept of the valuation. A functionΦ defined on the space Kn of convex bodies and taking values in an abelian semigroup is called a valuation if
(138)ΦK∪L+ΦK∩L=ΦK+ΦL,whenever K,L,K∩L,K∪L∈Kn.The set of Borel measures onSn−1 is represented by MSn−1. We are going to prove that now, for fixed indices p,q∈ℝ, and a fixed convex body K∈Kon, the functional Kon⟶MSn−1, defined by Q↦Cp,qK,Q,· is a valuation; namely, if K,L∈Kon, are such that K∪L∈Kon then
(139)Cp,qK∪L,Q,·+Cp,qK∩L,Q,·=Cp,qK,Q,·+Cp,qL,Q,·.To prove the valuation ofLp-curvature measure, we shall employ Weil’s approximation lemma (see [4]):Lemma 24.
IfK,L∈Kon are such that K∪L is convex, then K and L may be approximated by sequences of bodies Ki,Li∈Kon that are both strictly convex and smooth and such that Ki∪Li∈Kon.We appeal to Proposition21 together with Weil’s approximation lemma in order to complete our proof.Theorem 25.
Supposep,q∈ℝ and Q∈Kon. Then, the functional
(140)Cp,q·,Q,·:Kon⟶MSn−1,defined by K↦Cp,qK,Q,·, is a valuation.Proof.
We will use the fact that ifK,L∈Kon are such that K∪L∈Kon, then hK∪L=maxhK,hL and hK∩L=minhK,hL. We will also take advantage of the fact that νK and νL are defined Hn−1 almost everywhere on the boundaries of K and L, respectively.
First of all, let us assume thatK and L are both strictly convex. For a fixed θ⊂Sn−1, write θ as the union of three disjoint pieces θ=θ0∪θK∪θL, where
(141)θK=u∈θ:hKu>hLu,θL=u∈θ:hKu<hLu,while
(142)θ0=u∈θ:hKu=hLu.
In this case, we have the following:(143)∫x∈xK∪LθKx·νK∪Lx1−qhQq−pνK∪LxdHn−1x=∫x∈xKθKx·νKx1−qhQq−pνKxdHn−1x,while
(144)∫x∈xK∩LθKx·νK∩Lx1−qhQq−pνK∩LxdHn−1x=∫x∈xLθKx·νLx1−qhQq−pνLxdHn−1x.
Alternatively, using Equation (117), this has
(145)Cp,qK∪L,Q,θK=Cp,qK,Q,θK,Cp,qK∩L,θK=Cp,qL,Q,θK.
Similarly,(146)Cp,qK∪L,Q,θL=Cp,qL,Q,θL,Cp,qK∩L,Q,θL=Cp,qK,Q,θL.
It is also the case that(147)Cp,qK∪L,Q,θ0=Cp,qK,Q,θ0,Cp,qK∩L,Q,θ0=Cp,qL,Q,θ0.
In order to see the fact that the last one, we observe that the strict convexity ofK and L forces xK∪Lθ0=xK∩Lθ0.
Using the fact thatCp,qK,·,· is a measure in the third argument on Sn−1, combined with the fact that the union θ=θ0∪θK∪θL is disjoint, by adding Equations (145), (146), and (147) we obtain that
(148)Cp,qK∪L,Q,θ+Cp,qK∩L,Q,θ=Cp,qK,Q,θ+Cp,qL,Q,θ,which is the desired result.
For anyK,L∈Kon, we resort to Proposition 21 in order to use the weak continuity of Cp,q·,Q,· in the first argument.☐


## 4. Variational Formulas forLp,q-Mixed Volumes

SupposeΩ is a closed subset of Sn−1 that is not contained in any closed hemisphere. Let h0:Ω⟶0,∞ and f:Ω⟶ℝ be consecutive, and δ>0. Let ht:Ω⟶0,∞ be a positive continuous function defined as follows:
(149)loghtv=logh0v+tfv+ot,v,for each t∈−δ,δ, where ot,·:Ω⟶ℝ is continuous and limt⟶0ot,·/t=0, uniformly on Ω. And denote by
(150)ht=x∈ℝn:x·v≤htvforallv∈Ω,Wulff shape determined byht. We call ht the logarithmic Wulff shape family generated by h0,f. If h0 is the support function hK of convex body K, we also put ht written K,f,t.Letρ0:Ω⟶0,∞ and g:Ω⟶ℝ be continuous, and δ>0. Let ρt:Ω⟶0,∞ be a positive continuous function defined by the following:
(151)logρtu=logρ0u+tgu+ot,u,for each t∈−δ,δ, where again ot,·:Ω⟶ℝ is continuous and limt⟶0ot,·/t=0 uniformly on Ω. And denote by
(152)ρt=convρtuu:u∈Sn−1,the convex hull generated by ρt. We call ρt the logarithmic family of convex hull generated by ρ0,g. If ρ0 is the radial function ρK of convex body K, we also put ρt as K,g,t.The following lemma shows that the support functions of a logarithmic family of the polar of convex hulls are differentiable with respect to the variational variable.Lemma 26.
SupposeΩ⊂Sn−1 be a closed set that is not contained in any closed hemisphere of Sn−1. Let ρ0:Ω⟶0,∞ and g:Ω⟶ℝ be continuous. If ρt is a logarithmic family of convex hulls of ρ0,g and q∈ℝ, then
(153)limt⟶0hρt∗qv−hρ0∗qvt=−qρ0−qvgv,for all v∈Sn−1/ηρ0∗; namely, for all regular normals v of ρ0∗, where Equation (153) holds a.e. with respect to spherical Lebesgue measure. Moreover, there exist δ>0 and M>0 so that
(154)loghρt∗qv−loghρ0∗qv≤Mt,for all v∈Sn−1 and all t∈−δ,δ.Proof.
Obviously,(155)limt⟶0ρt−qv−ρ0−qvt=−pρ0−qvlimt⟶0logρtv−logρ0vt=−qρ0−qvgv.
Therefore,(156)limt⟶0hρt∗qv−hρ0∗qvt=limt⟶0ρt−qv−ρ0−qvt=−qρ0−qvgv.
Sinceρ0 and ρ0∗ are two convex bodies in Kon, and ρt∗⟶ρ0∗ as t⟶0, there exist m0,m1∈0,∞ and δ0>0 such that
(157)0<m0<hρt∗<m1onSn−1,for each t∈−δ0,δ0. From this, it follows that there exists M1>1 so that
(158)0<hρt∗−qhρ0∗−q<M1onSn−1.
It is easily seen thats−1≥logs whenever s∈0,1, whereas s−1≤M1logs whenever s∈1,M1. Thus,
(159)s−1≤M1logswhens∈0,M1.
It follows that(160)hρt∗−qhρ0∗−q−1≤M1loghρt∗−qhρ0∗−qwhens∈0,M1,that is
(161)hρt∗−q−hρ0∗−q≤hρ0∗−qM1loghρt∗−loghρ0∗≤M1minm0q,m1qloghρt∗−loghρ0∗=M1minm0q,m1qlogρt−logρ0on Sn−1, whenever t∈−δ0,δ0.
LetM0=maxu∈Ωgu. Since ot,·/t⟶0 as t⟶0 uniformly on Ω, we may choose δ1>0 so that for all t∈−δ1,δ1, we have ot,·≤t on Ω. From Equation (151) and the definition of M0, we immediately see that
(162)logρt−logρ0≤M0+1ton Sn−1, whenever t∈−δ1,δ1. Let −δ,δ=−δ0,δ0∩−δ1,δ1. Together with Equations (161) and (162), we give Equation (154).☐The following theorem gives variational formulas for theLp-mixed volume and Lp-mixed entropy for a family of logarithmic convex hulls.Theorem 27.
LetΩ⊂Sn−1 is a closed set not contained in any closed hemisphere of Sn−1. If ρ0:Ω⟶0,∞ and g:Ω⟶ℝ are continuous, and ρt is a logarithmic family of convex hulls of ρ0,g, then for K∈Kon and q≠0,
(163)limt⟶0VqK,ρt∗−VqK,ρ0∗t=−q∫ΩgudCqK,ρ0∗,u,for q=0,
(164)limt⟶0EK,ρt∗−EK,ρ0∗t=−∫ΩgudC0K,ρ0∗,u,limt⟶0logV¯0K,ρt∗−logV¯0K,ρ0∗t=−1VK∫ΩgudC0K,ρ0∗,u.Proof.
AbbreviateηK∗ by η0. Recall that η0 is the set of spherical Lebesgue measure zero that consists of the complement, in Sn−1, of the regular normal vectors of the convex body K∗. Note that the continuous function
(165)αK∗∗:Sn−1\η0⟶Sn−1is well defined by αK∗∗v∈αK∗∗v=αK∗∗v for all v∈Sn−1\η0.
Letv∈Sn−1/η0. To see that αK∗∗v⊂Ω, let
(166)hK∗v=maxu∈ΩρK∗uu·v=ρK∗u0u0·v,for some u0∈Ω. This means that
(167)ρK∗u0u0∈HK∗v,
and henceρK∗u0u0∈∂K∗. Because in addition to ρK∗u0u0 obviously belonging to K∗, it also belongs to HK∗v. But v is a regular normal vector of K∗, and therefore, αK∗∗v=u0∈Ω. Then,
(168)αK∗∗Sn−1\η0⊂Ω.
From this, Equation (168), Equation (52), and Lemma 9 yield the following facts:
(169)αKSn−1\η0=αK∗∗Sn−1\η0⊂Ω.
AsΩ is closed, by using the Tietze extension theorem, extend the continuous function g:Ω⟶ℝ to a continuous function g^:Sn−1⟶ℝ. Therefore, using Equation (169) we see that
(170)gαKv=g^1KαKv,for v∈Sn−1\η0.
Using Equation (22), the fact that η0 has measure zero, Equation (51), Equation (154), the dominated convergence theorem, Lemma 26, Equation (86), Equation (170), Lemma 14, and again Equation (170), we have the following:
(171)limt⟶0VqK,ρt∗−VqK,ρ0∗t=limt⟶01n∫Sn−1hρt∗qαKv−hρ0∗qαKvhK−qαKvρKnvdvt=limt⟶01n∫Sn−1/η0hρt∗qαKv−hρ0∗qαKvhK−qαKvρKnvdvt=−qn∫Sn−1/η0gαKvhρ00∗qαKvhK−qαKvρKnvdv=−qn∫Sn−1g^1ΩαKvhρ0∗qαKvhK−qαKvρKnvdv=−q∫Sn−1g^1ΩudCqK,ρ0∗,u=−q∫ΩgudCqK,ρ0∗,u.
☐According to Equations (70) and (51), the fact that η0 has measure zero, the dominated convergence theorem, Equation (151), together with Equations (170) and (86), Lemma 14, and again Equation (170), we have the following:
(172)limt⟶0EK,ρt∗−EK,ρ0∗t=limt⟶01n∫Sn−1loghρt∗/hKαKv−loghρ0∗/hKαKvtρKnvdv=limt⟶01n∫Sn−1loghρt∗αKv−loghρ0∗αKvtρKnvdv=−limt⟶01n∫Sn−1logρtαKv−logρ0αKvtρKnvdv=−limt⟶01n∫Sn−1/η0logρtαKv−logρ0αKvtρKnvdv=−1n∫Sn−1/η0gαKvρKnvdv=−1n∫Sn−1g^1ΩαKvρKnvdv=−∫Sn−1g^1ΩudC0K,ρ0∗,u=−∫ΩgudC0K,ρ0∗,u.Using the same argument as in the second part of the proof, we get that(173)limt⟶0logV¯0K,ρt∗−logV¯0K,ρ0∗t=1nVKlimt⟶0∫Sn−1loghρt∗αKv−loghρ0∗αKvtρKnvdv=−1VK∫ΩgudC0K,ρ0∗,u.The following theorem gives the variational formulas for theLp-mixed volumes and mixed entropy of the logarithmic family of Wulff shapes.Theorem 28.
SupposeΩ⊂Sn−1 is a closed set not contained in any closed hemisphere of Sn−1. Let h0:Ω⟶0,∞ and f:Ω⟶ℝ be continuous, and ht be a logarithmic family of Wulff shapes associated with h0,f. If K∈Kon, then for q≠0,
(174)limt⟶0VqK,ht−VqK,h0t=q∫ΩfvdCqK,h0,v,for q=0,
(175)limt⟶0EK,ht−EK,h0t=∫ΩfvdC0K,h0,v,limt⟶0logV¯0K,ht−logV¯0K,h0t=1VQ∫ΩfvdC0K,h0,v.Proof.
The logarithmic family of Wulff shapeht is defined as the Wulff shape of ht, where ht is given by the following:
(176)loght=loght+tf+ot,·.
Letρt=ht−1. Then,
(177)logρt=logρt−tf−ot,·.
Letρt be the logarithmic family of convex hulls associated with ρ0,−f. Then from Lemma 7, we obtain that
(178)ht=ρt∗,and the desired conclusions now follow from Theorem 27.We describe the special cases of Theorem27 and Theorem 28 for logarithmic families of convex hull and Wulff shape generated by convex bodies.Theorem 29.
IfK,Q∈Kon and g:Sn−1⟶ℝ is continuous, then for q≠0,
(179)limt⟶0VqK,Q∗,g,t∗−VqK,Qt=−q∫Sn−1gvdCqK,Q,v,for q=0,
(180)limt⟶0EK,Q∗,g,t∗−EK,Qt=−∫Sn−1gvdC0K,Q,v,limt⟶0logV¯0K,Q∗,g,t∗−logV¯0K,Qt=−1VK∫Sn−1gvdC0K,Q,v.Proof.
In Theorem27, let ρ0=1/hQ=ρQ∗. Then, ρt∗=Q∗,g,t∗. In particular, from (53) we have ρ0∗=ρQ∗∗=Q∗∗=Q.☐Above variational formulas for convex hulls imply variational formulas for Wulff shapes.Theorem 30.
IfK,Q∈Kon and f:Sn−1⟶ℝ is continuous, then for q≠0,
(181)limt⟶0VqK,Q,f,t−VqK,Qt=q∫Sn−1fvdCqK,Q,v,for q=0,
(182)limt⟶0EK,Q,f,t−EK,Qt=∫Sn−1fvdC0K,Q,v,limt⟶0logV¯0K,Q,f,t−logV¯0K,Qt=1VQ∫Sn−1fvdC0K,Q,v.Proof.
The logarithmic family of Wulff shapesQ,f,o,t is defined by the Wulff shape ht, where
(183)loght=loghQ+tf+ot,·.
This, and the fact that1/hQ=ρQ∗, allows us to define
(184)logρt∗=logρQ∗−tf−ot,·,and ρt∗ will generate a logarithmic family of convex hulls Q∗,−f,−o,t. Since ρt∗=1/ht, Lemma 7 gives the following:
(185)Q,f,o,t=Q∗,−f,−o,t∗.
Therefore, Theorem30 now follows directly from Theorem 29.☐The following theorem gives the variational formulas ofLp-mixed volumes and mixed entropies with respect to Lp Minkowski combinations.Theorem 31.
Ifp,q∈ℝn and K,L,Q∈Kon, then for p≠0,q≠0,
(186)limt⟶0VqK,Q+pt·L−VqK,Qt=qp∫Sn−1hLpvdCp,qK,Q,v,for p=0 and q≠0,
(187)limt⟶0VqK,Q+0t·L−VqK,Qt=q∫Sn−1loghLvdCqK,Q,v,for p≠0 and q=0,
(188)limt⟶0EK,Q+pt·L−EK,Qt=1p∫Sn−1hLpvdCp,0K,Q,v,(189)limt⟶0logV¯0K,Q+pt·L−logV¯0K,Qt=1pVK∫Sn−1hLpvdCp,0K,Q,u,and if p=q=0,
(190)limt⟶0EK,Q+0t·L−EK,Qt=∫Sn−1loghLvdC0K,Q,v,limt⟶0logV¯0K,Q+0t·L−logV¯0K,Qt=1VK∫Sn−1loghLvdC0K,Q,u.Proof.
For smallt, ht is defined by the following:
(191)htp=hQp+thLpforp≠0,ht=hQhLtforp=0.
From Equations (61) and (62), the Wulff shape ht=Q+pt·L. For sufficiently small t, it follows from Equation (191) that
(192)loght=loghQ+tphLphQp+ot,·,p≠0,loght=loghQ+tloghL,p=0.
Letf=1/phLp/hQp when p≠0, and let f=loghL when p=0. The required formulas now follow Theorem 30 and Equation (105).☐We use the normalized power function, and we can write the formula in Theorem31 as a single formula.Theorem 32.
Supposep,q∈ℝ. For K,L,Q∈Kon,
(193)ddtVq¯K,Q+pt·Lt=0=∫Sn−1hLp¯vdCp,qK,Q,v.ForL0 Minkowski linear combinations, it would help to have an affine version of Theorem 31. This is contained inTheorem 33.
Supposeq≠0. If K,L,Q∈Kon, then
(194)limt⟶0VqK,1−t·Q+0t·L−VqK,Qt=q∫Sn−1loghLvhQvdCqK,Q,v,(195)limt⟶0EK,1−t·Q+0t·L−EK,Qt=∫Sn−1loghLvhQvdC0K,Q,v,(196)limt⟶0logV¯0K,1−t·Q+0t·L−logV¯0K,Qt=1VK∫Sn−1loghLvhQvdC0K,Q,v.Proof.
Let(197)ht=hQ1−thLt.
From Equation (58) we know the Wulff space ht=1−t·Q+0t·L. From the above definition of ht, it follows immediately that for sufficiently small t,
(198)loght=loghQ+tloghLhQ.
Letf=loghL/hQ. The desired formulas now follow directly from Theorem 30.☐Theorem 34.
Ifp≠0 and q≠0, then for all K,L,Q∈Kon and ϕ∈SLn,
(199)Cp,qϕK,ϕQ,·=ϕpt⊣Cp,qK,Q,·,(200)Cp,0ϕK,ϕQ,·=ϕpt⊣Cp,0K,Q,·,(201)CqϕK,ϕQ,·=ϕ0t⊣CqK,Q,·,(202)C0ϕK,ϕQ,·=ϕ0t⊣C0K,Q,·.Proof.
Obviously, the casep≠0 and q=0 is handled by Equation (200). The case p=0 and q≠0 is handled by Equation (201), while the case p=0 and q=0 is handled by Equation (202).
We adopt the methods and techniques of paper [4]. Recall that Haberl and Parapatits refer to the [9] classified measure-valued operators on Kon, which are SLn-inverse degree p and corresponding to the transformation behavior in Theorem 34. From Equations (63), (65), and (186), we see that for all K,L,Q∈Kon and all ϕ∈SLn,
(203)∫Sn−1hϕLpvdCp,qϕK,ϕQ,v=∫Sn−1hLpvdCp,qK,Q,v,or equivalently for all K,L,Q∈Kon and all ϕ∈SLn,
(204)∫Sn−1hLpvdCp,qϕK,ϕQ,v=∫Sn−1hϕ−1LpvdCp,qK,Q,v.
By Definition8, and note the important fact that support functions are positively homogeneous of degree 1, from Equations (45) and (204), we have the following:
(205)∫Sn−1hLpvdϕpt⊣Cp,qK,Q,v=∫Sn−1hLpϕ−tvdCp,qK,Q,v=∫Sn−1hϕ−1LpvdCp,qK,Q,v=∫Sn−1hLpvdCp,qϕK,ϕQ,v.
This shows that the measuresϕpt⊣Cp,qK,Q,· and Cp,qϕK,ϕQ,· when integrated against the p-th power of support functions of bodies in Kon are identical; thus, Lemma 12 now indicates that
(206)Cp,qϕK,ϕQ,·=ϕpt⊣Cp,qK,Q,·,it can be concluded that Equation (199).
The proof for Equation (200) is the same as the proof for Equation (199): As long as p≠0, it will be the case that Equation (204) continues to hold even if q=0 provided we appeal to Equations (188) and (71) when previously we had turned to Equations (188) and (65).
From Equations (63), (65), and (194), we know that for all K,L,Q∈Kon and all ϕ∈SLn,
(207)∫Sn−1loghϕ−1LvhKvdCqK,Q,v=∫Sn−1loghLvhϕKvdCqϕK,ϕQ,v.
In Equation (207), choose L=B. Then, by Equation (45), we see that hϕ−1Lv=hLϕ−tv=ϕ−tv, and (6.15) becomes the following form:
(208)∫Sn−1loghKvdCqK,Q,v=∫Sn−1logϕ−tvdCqK,Q,v+∫Sn−1loghϕKvdCqϕK,ϕQ,v,for all ϕ∈SLn and all K,Q∈Kon. Together with Equations (207) and (208), we have the following:
(209)∫Sn−1loghϕ−1Lvϕ−tvdCqK,Q,v=∫Sn−1loghLvdCqϕK,ϕQ,v,this and Equation (45) give that for all ϕ∈SLn and all K,L,Q∈Kon,
(210)∫Sn−1loghLϕ−tvdCqK,Q,v=∫Sn−1loghLvdCqϕK,ϕQ,v.
Equivalently,(211)∫Sn−1loghLvdϕ0t⊣CqK,Q,v=∫Sn−1loghLvdCqϕK,ϕQ,v,for all ϕ∈SLn and all K,Q∈Kon. Using Lemma 12, we see that Equation (211) yields
(212)CqϕK,ϕQ,·=ϕ0t⊣CqϕK,ϕQ,v,for all ϕ∈SLn and all K,Q∈Kon. This establishes Equation (201).
The proof of Equation (202) is identical to the proof of Equation (201) except that instead of appealing to Equations (194) and (65) we appeal to Equations (195) and (71).☐


## 5. TheLp,q-Mixed Volumes

ForK,L∈Kon, the Lp-mixed volume VpK,L has the integral representation
(213)VpK,L=1n∫Sn−1hLvpdSpK,v.From Equation (115), with q=p and g=hLp, we have that
(214)∫Sn−1hLpvdCp,pK,Q,v=1n∫Sn−1hLpαKuhK−pαKuρKnudu.By Equation (131), the Lp-mixed volume VpK,L has a dual integral formulation. If K,L∈Kon, then
(215)VpK,L=1n∫Sn−1hLhKpαKuρKnudu.The dual integral formulation ofLp-mixed volume was first introduced by Lutwak et al. in [4]. This leads us to define following Lp,q-mixed volumes.Definition 35.
Letp,q∈ℝ and K,L,Q∈Kon. The Lp,q-mixed volume Vp,qK,L,Q is defined by the following:
(216)Vp,qK,L,Q=∫Sn−1hLpvdCp,qK,Q,v.Using Equation (115) with g=hLp, Equation (216) can be written as follows:
(217)Vp,qK,L,Q=1n∫Sn−1hLhQpαKuhQhKqαKuρKnudu.From Equations (216) and (124), the Lp-mixed volume Vp,qK,L,Q can be written as follows:
(218)Vp,qK,L,Q=1n∫Sn−1hLpvhKv1−qhQvhQq−pνK∇hKvdSK,v,where the function νK:∇hKv:v∈Sn−1⊂∂′K⟶Sn−1.FromLp,q-mixed volume (Equation (30)) (or Equation (217)), the Lp-mixed volume (Equation (9)) (or Equation (22)) will be shown to be the special cases.Proposition 36.
Supposep,q∈ℝ. If K,L,Q∈Kon, then
(219)Vp,qK,L,K=VpK,L,(220)Vp,pK,L,Q=VpK,L,(221)Vp,qK,L,L=VqK,L,(222)Vp,pK,L,B=VpK,L,(223)V0,qK,L,Q=VqK,Q.Proof.
Identity (Equations (219)–(221)) follow from Equation (22) and Equation (34) (or Equation (217)). Similarly, we can prove Equations (222) and (223).Proposition 37.
TheLp,q-mixed volume Vp,q is SLn-invariant. That is, for p,q∈ℝ, K,L,Q∈Kon, and ϕ∈SLn,
(224)Vp,qϕK,ϕL,ϕQ=Vp,qK,L,Q.Proof.
Forp=0, the conclusion follows from Equation (223) and the SLn-invariance of Lp-mixed volumes (Equation (65)). We assume p≠0. By Definition 35, Equation (199), and Equation (200), the fact that support functions are positively homogeneous of degree 1, Equation (45), and Definition 8, we have the following:
(225)Vp,qϕK,ϕL,ϕQ=∫Sn−1hϕLpvdCp,qϕK,ϕQ,v=∫Sn−1hϕLpvdϕpt⊣Cp,qK,Q,v=∫Sn−1hϕLpϕ−tvdCp,qK,Q,v=∫Sn−1hLpvdCp,qK,Q,v=Vp,qK,L,Q.
From the dual Equation (217) of Lp,q-mixed volume and Equation (44), we have for real λ>0,
(226)Vp,qλK,λL,λQ=λnVp,qK,L,Q.Proposition37, together with Equations (216) and (226), shows that for ϕ∈GLn,
(227)Vp,qϕK,ϕL,ϕQ=ϕVp,qK,L,Q.ForLp,q-mixed volume, the following inequality is a generalization of the Lp-Minkowski inequality for Lp-mixed volume.Theorem 38.
Supposep,q are such that q≥1 and p<0. If K,L,Q∈Kon, then
(228)Vp,qK,L,Qn≥VKn−qVLpVQq−p,with equality if and only if K,L,Q are dilates when q>1 and K,Q are homothetic when q=1.Proof.
From Equations (21) and (217), we have the following:
(229)Vp,qK,L,Q=1n∫Sn−1hLαKuphQαKuq−phKαK−qρKnudu=1n∫Sn−1hLαKuphQαKuq−pdSqK,u=1n∫Sn−1hLαKuqp/qhQαKuqq−p/qdSqK,u.
From this, by the Hölder inequality (see [47]), the dual integral formulation (Equation (22)) of Lp-mixed volume and Lp-Minkowski inequality (Equation (11)), we have the following:
(230)Vp,qK,L,Q≥1n∫Sn−1hLαKuqdSqK,up/q1n∫Sn−1hQαKuqdSqK,uq−p/q=1n∫Sn−1hLhKqαKuρKundup/q1n∫Sn−1hQhKqαKuρKunduq−p/q=VqK,Lp/qVqK,Qq−p/q≥VKn−qp/nqVLp/nVKn−qq−p/nqVQq−p/n=VQq−p/nVLp/nVKn−q/n.The equality conditions follow from the equality conditions of Hölder inequality and theLp-Minkowski inequality (Equation (11)) for Lp-mixed volumes. Namely, the equality for the above inequality holds if and only if K,L,Q are dilates when q>1 and K,Q are homothetic when q=1.Over the past three decades, valuation theory has become an ever more important part of convex body geometry. See, e.g., [11–13, 18, 48–53]. The convex Lp,q-mixed volume is the valuation for each entry.Proposition 39.
TheLp,q-mixed volume Vp,qK,L,Q is a valuation over Kon with respect to all K,L, and Q.Proof.
TheLp,q-mixed volume Vp,qK,L,Q is a valuation on Kon respect to the third argument can be seen easily by writing Equation (216) as follows:
(231)Vp,qK,L,Q=1n∫Sn−1hLpudCp,qK,Q,u,and from Equation (139) (or Theorem 25), observing that for K1,K2∈Kon, we have the following:
(232)dCp,qK1,Q,·+dCp,qK2,Q,·=dCp,qK1∪K2,Q,·+dCp,qK1∩K2,Q,·.
Together with Equations (216) and (232), we have the following:
(233)Vp,qK1∪K2,L,Q+Vp,qK1∩K2,L,Q=Vp,qK1,L,Q+Vp,qK2,L,Q.
Namely,Vp,qK,L,Q is a valuation in the third argument.
Observing that forL1,L2∈Kon such that L1∪L2∈Kon. Then, we have the following:
(234)hL1∪L2p+hL1∩L2p=hL1p+hL2p,onSn−1.
Note thathL1∪L2=maxhL1,hL2 and hL1∩L2=minhL1,hL2. Together with Equations (216) and (234), we see that Vp,qK,L,Q is a valuation in the second argument, i.e,
(235)Vp,qK,L1∪L2,Q+Vp,qK,L1∩L2,Q=Vp,qK,L1,Q+Vp,qK,L2,Q.
Note that ifQ1,Q2∈Kon are such that Q1∪Q2∈Kon, then we have the following:
(236)hQ1∪Q2q−p+hQ1∩Q2q−p=hQ1q−p+hQ2q−p,onSn−1.
Together with Equations (218) and (236), we see that Vp,qK,L,Q is a valuation in the first argument, i.e,
(237)Vp,qK,L,Q1∪Q2+Vp,qK,L,Q1∩Q2=Vp,qK,L,Q1+Vp,qK,L,Q2.
LetK,Q∈Kon. The q-th mixed cone-volume measure CqK,Q,ω of K and Q is a Borel measure on the unit sphere Sn−1 is defined by for a Borel ω⊆Sn−1 and u∈ω,
(238)dCqK,Q,ω=1nhQhKqαKuρKundu.
Since theq-th mixed volume, VqK,Q has a dual integral formulation:
(239)VqK,Q=1n∫Sn−1hQhKqαKuρKnudu.
We can turn theq-th mixed cone-volume measure into the probability measure on the unit sphere by normalizing it by q-th mixed volume of the bodies. The q-th mixed cone-volume probability measure C¯qK,Q;· of K and Q is defined by the following:
(240)dC¯qK,Q,ω=1VqK,QdCqK,Q,ω.
IfK,L,Q∈Kon, then for each real p,q∈ℝ, we define the normalized Lp,q-mixed volume by the following:
(241)V¯p,qK,L,Q=Vp,qK,L,QVqK,Q1/p=∫Sn−1hLuhQupαKudC¯qK,Q,u1/p.
Letp⟶0. We give the following:
(242)V¯0,qK,L,Q=exp∫Sn−1loghLhQαKudC¯qK,Q,u.
Theq-th mixed entropy EqK,L,Q of convex bodies K,L,Q∈Kon is defined by the following:
(243)EqK,L,Q=∫Sn−1loghLhQαKudCqK,Q,u.
In particular,(244)V¯0,0K,L,K=V¯0K,L,EqQ,L,Q=EQ,L.
☐


## 6. TheLp,q-Minkowski Problems

The existence and uniqueness ofLp,q-Minkowski problem is the central problem to be investigated here. Its existence problem can be expressed as follows:Problem 40.
Letp,q∈ℝ, and Q∈Kon is fixed. Given a Borel measure μ∈MSn−1, what are necessary and sufficient conditions on μ such that there exists a K∈Kon whose Lp-curvature measures Cp,qK,Q,· is the given measure μ?Lp-Minkowski problem when q=p. When the given data measure μ has a density f, it follows from Equation (125) that Lp,q-Minkowski problem is equivalent to solving the following Monge-Ampère-type equation on Sn−1:
(245)h1−qν∘∇hQ∗q−pdet∇¯2h+hI=f,where h is the unknown function on Sn−1, and ∇h is the gradient vector function in ℝn of the extension from h to ℝn as a vector function that is positively homogeneous of degree 1. If we assume that the range of the gradient function ∇h is D, then ν:D⟶Sn−1 is also an unknown function related to h.Our uniqueness result for theLp,q-Minkowski problem is presented in the following:Problem 41.
For fixedp,q∈ℝ and Q∈Kon, if K,L∈Kon such that
(246)Cp,qK,Q,·=Cp,qL,Q,·,then how is K related to L?Now, we establish uniqueness of the solution to the problem withq≥n for the case of polytopes.Theorem 42.
LetP,P′∈Kon be polytopes and let Q∈Kon. Suppose
(247)Cp,qP,Q,·=Cp,qP′,Q,·.Then,P=P′ when q>n and P′ is a dilate of P when q=n.Proof.
According to Equations (121) and (122), we get that the curvature measures of polytopes are discrete, and that Cp,qP,Q,·=Cp,qP′,Q,· implies that P and P′ must have the same outer unit normal vectors v1,v2,⋯,vm and
(248)Cp,qP,Q·=Cp,qP′,Q,·=∑i=1mdiδvi,where δvi denotes the delta measure concentrated at vi, and
(249)di=1nhQq−pvihP−qvi∫Sn−1∩ΔiρPnudu=1nhQq−pvihP′−qvi∫Sn−1∩Δ′iρP′nudu.
HereΔi and Δ′i are the cones formed by the origin and the facets of P and P′ with vector vi, respectively.
Assume thatP≠P′. Tt is easy to see that P⊆P′ is not possible. Set λ be the maximal number with λP⊆P′. This has λ<1. Since λP and P′ have the same outer unit normal vectors, there is a facet of λP which is contained in a facet of P′. The outer unit normal vector of those facets is denoted by vi1. It follows that
(250)hλPvi1=hP′vi1,Δi1⊆Δ′i1,ρλPu=ρP′uforallu∈Δi1.
Thus,(251)1nhQq−pvi1hλP−qvi1∫Sn−1∩Δi1ρλPnudu≤1nhQq−pvi1hP′−qvi1∫Sn−1∩Δ′i1ρP′nudu.with equality if and only if Δi1=Δ′i1. By this and Equation (249), we can obtain that
(252)λn−q≤1.
Butλ<1 implies that λn−q>1 if q>n. Obviously, this is a contradiction.☐Ifq=n, then Equation (249) forces equality in Equation (251). So, Δi1=Δ′i1, and the facets of λP and P′ with outer unit normal vector vi1 are the same. Let vi2 is the outer unit normal vector to a facet, which is adjacent to the facet whose outer unit normal vector is vi1. Thus, the facet of λP with outer unit normal vector vi2 is contained in the facet of P′ with outer unit normal vector vi2. A similar argument holds that the two facets are the same. Continuing in this manner, it follows that λP=P′.


## 7. Several Other Problems

Here, we present several issues that need to be discussed in the future. Some of the definitions and problems below are different from the paper [40, 43, 44, 54].


### 7.1.Lp,q-Mixed Affine Surface Areas

In [7], Lutwak defined the Lp-affine surface area ΩpK for p≥1 by the following:
(253)n−p/nΩpKn+p/n=infnVpK,L∗VLp/n:L∈Son.Hug in [55] observed that the Lp-affine surface area is well defined for 0<p<1.The following affine isoperimetric inequality was established in [7] for p≥1, and in [56] for 0<p<1. If K∈Kcn, then
(254)ΩpKn+p≤nn+pωn2pVKn−p,p>0,with equality if and only if K is an ellipsoid. Here, ωn is the volume of the n dimensional unit sphere.Definition 43.
Supposeq∈ℝ. For K∈Kon and Q∈Son, the q-th curvature measure CqK,Q∗,· of K (related to star body Q) is defined by the following:
(255)CqK,Q∗,η=1n∫αK∗ηρKnuρQhKqαKudu,for each Borel η⊆Sn−1, and Lp-curvature measure Cp,qK,Q∗,· of K is defined by the following:
(256)Cp,qK,Q∗,η=1n∫αK∗ηρQpαKu·ρKnuρQhKqαKudu=1n∫αK∗ηρQpαKudCqK,Q∗,u,for each Borel η⊆Sn−1.It follows from Definition43 that
(257)dCp,qK,Q∗,·=ρQpdCqK,Q∗,·.Definition 44.
Supposeq∈ℝ. If K∈Kon,L∈Son, the q-th mixed volume VqK,L∗ is defined by the following:
(258)VqK,L∗=1n∫Sn−1ρKnuρLhKqαKudu=1n∫Sn−1ρL−qudSqK,u.Definition 45.
Supposep,q∈ℝ. If K∈Kon and Q,L∈Son, the Lp,q-mixed volume Vp,qK,L∗,Q∗ of K and L∗ (with respect to Q) is defined by the following:
(259)Vp,qK,L∗,Q∗=∫Sn−1ρLv−pdCp,qK,Q∗,v.Inspired by [40, 54], from Equations (258) and (259) we define Lp,q-mixed affine surface area as follows:Definition 46.
Forp∈ℝ,q>0 and K∈Kon,Q∈Son, the Lp,q-mixed affine surface area Ωp,qK,Q of K (relate to Q) is defined by the following:
(260)n−q/nΩp,qK,Qn+q/n=infnVp,qK,L∗,Q∗VqL,Qq/n:L∈Son.WhenQ=L, from Equation (219) we have the following:
(261)Vp,qK,L∗,L∗=VqK,L∗.Ωp,qK,L is the Lq-affine surface area ΩqK.Problem 47.
For theLp,q-mixed affine surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?


### 7.2.Lp,q-Mixed Geominimal Surface Area

In [7], Lutwak defined the Lp-geominimal surface area GpK by the following:
(262)ωnp/nGpK=infnVpK,LVL∗p/n:L∈Kon,and proved the following affine isoperimetric inequality: IfK∈Kon, then
(263)GpKn≤nnωnpVKn−p,with equality if and only if K is an ellipsoid.Motivated by theLp-mixed geominimal surface area (Equation (257)), we define Lp,q-mixed geominimal surface area, Gp,qK,Q, of K relative to Q as follows:Definition 48.
Forp∈ℝ,q≥1, and K,Q∈Kon, the Lp,q-mixed geominimal surface area Gp,qK,Q of K relative to Q is defined by the following:
(264)ωnq/nGp,qK,Q=infnVp,qK,L,QVqL∗,Q∗q/n:L∈Kon.WhenQ=L, from Equation (219) we have the following:
(265)Vp,qK,L,L=VqK,L.Gp,qK,L is the Lq-geominimal surface area GqK.Problem 49.
For theLp,q-mixed geominimal surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?


### 7.3.Lp,q-John Ellipsoids

Supposep∈0,∞ and K is a convex body in ℝn with the origin in its interior. Among all origin-symmetric ellipsoids E, the unique ellipsoid that solves the constrained maximization problem:
(266)maxEVEωn1/nsubjecttoV¯pK,E≤1is called theLp-John ellipsoid of K which defined in [45] and denoted by EpK. Clearly, EpB=B. Here,
(267)V¯pK,E=1nVK∫Sn−1hEuhKuphKudSK,u1/n,0<p<∞is the normalized Lp-mixed volume of K and E. In the case p=∞, we define the following:
(268)V¯∞K,E=suphEuhKu:u∈suppSK,·.In general, theLp-John ellipsoid EpK is not contained in K (except when p=∞). However, when 1≤p≤∞, it has VEpK≤VK. In reverse, for 0<p≤∞, the Lp version of ball’s volume-ratio inequality [45] states that
(269)VKVEpK≤2nωn,with equality if and only if K is a parallelotope.We know that from Equation (241), for 0<p<∞,q∈ℝ, the normalized Lp,q-mixed volume is calculated by the following:
(270)V¯p,qK,L,Q=∫Sn−1hLαKuhQαKupdC¯qK,Q,u1/p.In the casep=∞, define the following:
(271)V¯∞,qK,L,Q=maxhLαKuhQαKu:u∈suppCqK,Q,·.By Equation (271), we have the following:
(272)V¯∞,qK,L,Q≤1ifandonlyifL⊆Q.LetEn denote the class of origin-symmetric ellipsoids in ℝn. Inspired by the constrained maximization problem (Equation (266)), the reader may consider its Lp,q-version.Problem 50.
Let0<p≤∞,q∈ℝ. For K,Q∈Kon, find an ellipsoid, among all origin-symmetric ellipsoids, which solves the following constrained maximization problem:
(273)maxE∈EnVEωn1/nsubjecttoV¯p,qK,E,Q≤1.
An ellipsoid that solves the constrained maximization problem will be calledLp,q-John ellipsoid for K,Q and denoted by Ep,qK,Q.In particular, whenQ=K, from Equations (219) and (22), we have the following:
(274)Vp,qK,E,K=VpK,E,VqK,K=VK.Thus,V¯p,qK,E,K=V¯pK,E. So, Problem 50 degenerates into the problem.


## 7.1.Lp,q-Mixed Affine Surface Areas

In [7], Lutwak defined the Lp-affine surface area ΩpK for p≥1 by the following:
(253)n−p/nΩpKn+p/n=infnVpK,L∗VLp/n:L∈Son.Hug in [55] observed that the Lp-affine surface area is well defined for 0<p<1.The following affine isoperimetric inequality was established in [7] for p≥1, and in [56] for 0<p<1. If K∈Kcn, then
(254)ΩpKn+p≤nn+pωn2pVKn−p,p>0,with equality if and only if K is an ellipsoid. Here, ωn is the volume of the n dimensional unit sphere.Definition 43.
Supposeq∈ℝ. For K∈Kon and Q∈Son, the q-th curvature measure CqK,Q∗,· of K (related to star body Q) is defined by the following:
(255)CqK,Q∗,η=1n∫αK∗ηρKnuρQhKqαKudu,for each Borel η⊆Sn−1, and Lp-curvature measure Cp,qK,Q∗,· of K is defined by the following:
(256)Cp,qK,Q∗,η=1n∫αK∗ηρQpαKu·ρKnuρQhKqαKudu=1n∫αK∗ηρQpαKudCqK,Q∗,u,for each Borel η⊆Sn−1.It follows from Definition43 that
(257)dCp,qK,Q∗,·=ρQpdCqK,Q∗,·.Definition 44.
Supposeq∈ℝ. If K∈Kon,L∈Son, the q-th mixed volume VqK,L∗ is defined by the following:
(258)VqK,L∗=1n∫Sn−1ρKnuρLhKqαKudu=1n∫Sn−1ρL−qudSqK,u.Definition 45.
Supposep,q∈ℝ. If K∈Kon and Q,L∈Son, the Lp,q-mixed volume Vp,qK,L∗,Q∗ of K and L∗ (with respect to Q) is defined by the following:
(259)Vp,qK,L∗,Q∗=∫Sn−1ρLv−pdCp,qK,Q∗,v.Inspired by [40, 54], from Equations (258) and (259) we define Lp,q-mixed affine surface area as follows:Definition 46.
Forp∈ℝ,q>0 and K∈Kon,Q∈Son, the Lp,q-mixed affine surface area Ωp,qK,Q of K (relate to Q) is defined by the following:
(260)n−q/nΩp,qK,Qn+q/n=infnVp,qK,L∗,Q∗VqL,Qq/n:L∈Son.WhenQ=L, from Equation (219) we have the following:
(261)Vp,qK,L∗,L∗=VqK,L∗.Ωp,qK,L is the Lq-affine surface area ΩqK.Problem 47.
For theLp,q-mixed affine surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?


## 7.2.Lp,q-Mixed Geominimal Surface Area

In [7], Lutwak defined the Lp-geominimal surface area GpK by the following:
(262)ωnp/nGpK=infnVpK,LVL∗p/n:L∈Kon,and proved the following affine isoperimetric inequality: IfK∈Kon, then
(263)GpKn≤nnωnpVKn−p,with equality if and only if K is an ellipsoid.Motivated by theLp-mixed geominimal surface area (Equation (257)), we define Lp,q-mixed geominimal surface area, Gp,qK,Q, of K relative to Q as follows:Definition 48.
Forp∈ℝ,q≥1, and K,Q∈Kon, the Lp,q-mixed geominimal surface area Gp,qK,Q of K relative to Q is defined by the following:
(264)ωnq/nGp,qK,Q=infnVp,qK,L,QVqL∗,Q∗q/n:L∈Kon.WhenQ=L, from Equation (219) we have the following:
(265)Vp,qK,L,L=VqK,L.Gp,qK,L is the Lq-geominimal surface area GqK.Problem 49.
For theLp,q-mixed geominimal surface area, does it maintain affine invariance and continuity? How to establish its affine isoperimetric inequality?


## 7.3.Lp,q-John Ellipsoids

Supposep∈0,∞ and K is a convex body in ℝn with the origin in its interior. Among all origin-symmetric ellipsoids E, the unique ellipsoid that solves the constrained maximization problem:
(266)maxEVEωn1/nsubjecttoV¯pK,E≤1is called theLp-John ellipsoid of K which defined in [45] and denoted by EpK. Clearly, EpB=B. Here,
(267)V¯pK,E=1nVK∫Sn−1hEuhKuphKudSK,u1/n,0<p<∞is the normalized Lp-mixed volume of K and E. In the case p=∞, we define the following:
(268)V¯∞K,E=suphEuhKu:u∈suppSK,·.In general, theLp-John ellipsoid EpK is not contained in K (except when p=∞). However, when 1≤p≤∞, it has VEpK≤VK. In reverse, for 0<p≤∞, the Lp version of ball’s volume-ratio inequality [45] states that
(269)VKVEpK≤2nωn,with equality if and only if K is a parallelotope.We know that from Equation (241), for 0<p<∞,q∈ℝ, the normalized Lp,q-mixed volume is calculated by the following:
(270)V¯p,qK,L,Q=∫Sn−1hLαKuhQαKupdC¯qK,Q,u1/p.In the casep=∞, define the following:
(271)V¯∞,qK,L,Q=maxhLαKuhQαKu:u∈suppCqK,Q,·.By Equation (271), we have the following:
(272)V¯∞,qK,L,Q≤1ifandonlyifL⊆Q.LetEn denote the class of origin-symmetric ellipsoids in ℝn. Inspired by the constrained maximization problem (Equation (266)), the reader may consider its Lp,q-version.Problem 50.
Let0<p≤∞,q∈ℝ. For K,Q∈Kon, find an ellipsoid, among all origin-symmetric ellipsoids, which solves the following constrained maximization problem:
(273)maxE∈EnVEωn1/nsubjecttoV¯p,qK,E,Q≤1.
An ellipsoid that solves the constrained maximization problem will be calledLp,q-John ellipsoid for K,Q and denoted by Ep,qK,Q.In particular, whenQ=K, from Equations (219) and (22), we have the following:
(274)Vp,qK,E,K=VpK,E,VqK,K=VK.Thus,V¯p,qK,E,K=V¯pK,E. So, Problem 50 degenerates into the problem.



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*Source: 3787529-2022-08-18.xml*