The Minkowski problem based on the $(p,q)$-mixed quermassintegrals ^††thanks: Research is supported in part by the Natural Science Foundation of China (No.11871275; No.11371194).

Brunn and Minkowski pioneered the classical Brunn-Minkowski theory of convex bodies in $n$-dimensional Euclidean space. The Minkowski linear combination of vectors, mixed volumes, surface area measures and basic Brunn-Minkowski inequalities etc. are the essence in Brunn-Minkowski theory. The classical dual Brunn-Minkowski theory of star bodies was introduced by Lutwak ([18]) in 1975. The formation of dual Brunn-Minkowski theory was based on radial Minkowski combination and dual mixed volumes of star bodies. For more information about classical Brunn-Minkowski theory and dual Brunn-Minkowski theory, please refer to two excellent books [12, 29] in details.

In modern convex geometry, the $L_{p}$ Brunn-Minkowski theory and $L_{p}$ dual Brunn-Minkowski theory generalize and dualize the classical Brunn-Minkowski theory. The $L_{p}$ mixed quermassintegrals $W_{p,i}(M,N)$ of convex bodies introduced by Lutwak (see[19]) and $L_{p}$ dual mixed quermassintegrals $\widetilde{W}_{-p,i}(M,N)$ of star bodies introduced by Wang and Leng (see[32]) generalized the $L_{p}$ mixed volumes $V_{p}(M,N)$ introduced in [19] and $L_{p}$ dual mixed volumes $\widetilde{V}_{-p}(M,N)$ introduced in [20] are the main geometric measures in these two theories. In 2018, Lutwak, Young and Zhang established the $(p,q)$-mixed volume $V_{p,q}(M,N,Q)$ in [24]. Meanwhile, they demonstrated a surprising connection between the $L_{p}$ mixed volume and $L_{p}$ dual mixed volume. During the last three decades, scholars have done a lot of work on the research of $L_{p}$ Brunn-Minkowski theory and $L_{p}$ dual Brunn-Minkowski theory, see e.g., [2, 3, 4, 7, 11, 15, 16, 22, 25, 30, 33, 34, 35, 36].

The Minkowski problem in Brunn-Minkowski theory and dual Brunn-Minkowski theory is a hot topic of research. The classical Minkowski problem was first studied by Minkowski [27, 28], who proved both existence and uniqueness of solutions for the given measure. The $L_{p}$ Minkowski problem in $L_{p}$ Brunn-Minkowski theory was introduced in [19]. Existence of solutions for the dual Minkowski problem for even data within the class of origin-symmetric convex bodies was proved in [15]. Very recently, $L_{p}$ dual Minkowski problem for $L_{p}$ dual curvature measures was stated in [24], which a unified Minkowski problem of $L_{p}$ Minkowski problem when $q=n$ and dual Minkowski problem when $p=0$ and $N=B$. The Minkowski problem and dual Minkowski problem have already aroused considerable attention; see, e.g., [6, 7, 8, 13, 14, 21, 23, 26, 31, 37, 38, 39, 40, 41].

The $L_{p}$ dual curvature measure $\widetilde{C}_{p,q}(M,N,\cdot)$ of $M\in\mathcal{K}_{o}^{n}$ and $N\in\mathcal{S}_{o}^{n}$ was introduced in [24], which constructs a unified frame for $L_{p}$ surface area measures introduced in [19] and dual curvature measures introduced in [15]. Meanwhile, they demonstrated a surprising connection between the $L_{p}$ Brunn-Minkowski theory and $L_{p}$ dual Brunn-Minkowski theory by establishing geometric inequalities and variational integral formulas of $L_{p}$ dual curvature measure, respectively.

Based on the concepts introduced above, it is worth researching to find a concept that can unify the $L_{p}$ mixed quermassintegrals, $L_{p}$ dual mixed quermassintegrals and $(p,q)$-mixed volume. Motivated by ideas in [19] and [24], the main purpose of this paper is to introduce the concept of $(p,q)$-dual mixed curvature measures of convex bodies and star bodies. Next, by the concept of $(p,q)$-dual mixed curvature measures, we establish the integral formula of $(p,q)$-mixed quermassintegrals $\widetilde{W}_{p,q,i}(M,N,Q)$, which is a unified structure of $(p,q)$-mixed volumes, $L_{p}$ mixed quermassintegrals and $L_{p}$ dual mixed quermassintegrals. Our main work in this article can be illustrated in the block diagram below:

Let $M$ be a convex body if $M$ is a compact, convex subset in $n$-dimensional Euclidean space $\mathbb{R}^{n}$ with non-empty interior. The set of all convex bodies containing the origin in their interiors in $\mathbb{R}^{n}$ is written as $\mathcal{K}_{o}^{n}$. We write $u$ for a unit vector and $B$ for the unit ball centered at the origin, the surface of $B$ denoted by $S^{n-1}$. We shall use $V(M)$ for the $n$-dimensional volume of the body $M$ in $\mathbb{R}^{n}$, we write $V(B)=\omega_{n}$.

For $M\in\mathcal{K}^{n}$, the support function $h_{M}=h(M,\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$ is defined by

$$h(M,x)=\max\{x\cdot y:y\in M\},\ \ \ \ x\in\mathbb{R}^{n}.$$

where $x\cdot y$ denotes the standard inner product of $x$ and $y$ in $\mathbb{R}^{n}$ (see[12, 29]).

In the dual Brunn-Minkowski theory, for $M$ is a compact star shaped (about the origin) in $\mathbb{R}^{n}$, the radial function, $\rho_{M}=\rho(M,\cdot)$: $\mathbb{R}^{n}\backslash\{0\}\rightarrow[0,+\infty)$, is defined by

$$\rho(M,x)=\max\{\lambda\geq 0:\lambda x\in M\},\ \ \ \ x\in\mathbb{R}^{n}\backslash\{0\}.$$

If $\rho_{M}$ is positive and continuous, then call $M$ is a star body and write as $\mathcal{S}_{o}^{n}$ (see[12, 29]).

For $K,L\in\mathcal{K}_{o}^{n}$, $p\geq 1$ and $\lambda,\mu\geq 0$ (not both zero), the $L_{p}$-Minkowski linear combination, $\lambda\cdot K+_{p}\mu\cdot L\in\mathcal{K}_{o}^{n}$, of $K$ and $L$ satisfies (see[9])

$$h(\lambda\cdot K+_{p}\mu\cdot L,\cdot)^{p}=\lambda h(K,\cdot)^{p}+\mu h(L,\cdot)^{p}.$$

The $L_{p}$ mixed surface area measures $S_{p,i}(M,\cdot)$ of $M\in\mathcal{K}_{o}^{n}$ can be defined by the variational formula,

$$\frac{d}{dt}W_{p,i}(M+_{p}t\cdot N)\bigg{|}_{t=0^{+}}=\frac{1}{p}\int_{S^{n-1}}h_{N}^{p}(u)dS_{p,i}(M,u),$$

for $N\in\mathcal{K}_{o}^{n}$ and $0\leq i\leq n-1$.

For $q\in\mathbb{R}$ and $0\leq i\leq n-1$, the $q$-th dual mixed quermassintegral of $M,N\in\mathcal{S}_{o}^{n}$, is defined by

$\displaystyle\widetilde{W}_{q,i}(M,N)=\frac{1}{n}\int_{S^{n-1}}\rho_{M}^{q}(u)\rho_{N}^{n-q-i}(u)du.$

(1.1)

In this paper, the $(p,q)$-dual mixed curvature measures, $\widetilde{\mathcal{C}}_{p,q,j}$, is a three-parameter family of Borel measures on $S^{n-1}$. For $p,q\in\mathbb{R}$, $j\neq n$, and $M\in\mathcal{K}_{o}^{n}$, $Q\in\mathcal{S}_{o}^{n}$, we define the $\widetilde{\mathcal{C}}_{p,q,j}(M,Q,\cdot)$ by

$$\int_{S^{n-1}}g(v)d\widetilde{\mathcal{C}}_{p,q,j}(M,Q,v)=\frac{1}{n}\int_{S^{n-1}}g(\alpha_{M}(u))h_{M}^{-p}(\alpha_{M}(u))\rho_{M}^{q}(u)\rho_{Q}^{n-q-j}(u)du,$$

for a continuous function $g:S^{n-1}\rightarrow\mathbb{R}$, where $\alpha_{M}$ is the radial Gauss map (see Section 2) that associates to almost $u\in S^{n-1}$ the unique out unit normal vector at the point $\rho_{M}(u)u\in\partial M$.

The $L_{p}$ mixed surface area measures and $L_{p}$ dual curvature measures are special cases of the $(p,q)$-dual mixed curvature measures. For $p,q\in\mathbb{R}$, $M\in\mathcal{K}_{o}^{n}$ and $M\in\mathcal{S}_{o}^{n}$

$$\widetilde{\mathcal{C}}_{p,q,0}(M,Q,\cdot)=\widetilde{\mathcal{C}}_{p,q}(M,Q,\cdot),\ \ \ \ \widetilde{\mathcal{C}}_{p,q,0}(M,M,\cdot)=\frac{1}{n}S_{p}(M,\cdot),$$

$$\widetilde{\mathcal{C}}_{p,n,0}(M,B,\cdot)=\frac{1}{n}S_{p}(M,\cdot),\ \ \ \ \widetilde{\mathcal{C}}_{0,q,0}(M,B,\cdot)=\frac{1}{n}\widetilde{\mathcal{C}}_{q}(M,\cdot).$$

Recall that $L_{p}$ Minkowski linear combination, we give the variational formulas of $q$-th dual mixed quermassintegral: Suppose $p,q\in\mathbb{R}$, $j\neq n$. For $M,N\in\mathcal{K}_{o}^{n}$ and $Q\in\mathcal{S}_{o}^{n}$,

$\displaystyle\frac{d}{dt}\widetilde{W}_{q,j}(M+_{p}t\cdot N,Q)\bigg{|}_{t=0^{+}}=\frac{q}{p}\int_{S^{n-1}}h_{N}^{p}(u)d\widetilde{C}_{p,q,j}(M,Q,u).$

(1.2)

It will be shown that the $(p,q)$-mixed quermassintegral $\widetilde{W}_{p,q,j}$ has a integral representation using the $(p,q)$-dual mixed curvature measure: For $p,q\in\mathbb{R}$, and $j=0,1,...,n-1$, there exists a regular Borel measure $\mathcal{\widetilde{C}}_{p,q,j}(M,N,\cdot)$ on $S^{n-1}$, such that the $(p,q)$-mixed quermassintegral $\widetilde{W}_{p,q,j}$ has the following integral representation:

$\displaystyle\widetilde{W}_{p,q,j}(M,N,Q)=\int_{S^{n-1}}h_{N}^{p}(v)d\widetilde{C}_{p,q,j}(M,Q,v).$

(1.3)

The $L_{p}$ mixed quermassintegrals and the $L_{p}$ dual mixed quermassintegrals will be the special cases of $(p,q)$-mixed quermassintegrals,

$\displaystyle\widetilde{W}_{p,q,j}(M,N,M)=W_{p,j}(M,N),$

(1.4)

$\displaystyle\widetilde{W}_{p,q,j}(M,M,Q)=\widetilde{W}_{q,j}(M,N).$

(1.5)

The following inequality for $(p,q)$-mixed quermassintegral is a generalization of the $L_{p}$ Minkowski inequality: Suppose $p,q\in\mathbb{R}$ satisfies $1\leq\frac{q}{n-j}\leq p$, $j\neq n$. If $M,N\in\mathcal{K}_{o}^{n}$ and $Q\in\mathcal{S}_{o}^{n}$, then

$$\widetilde{W}_{p,q,j}(M,N,Q)\geq W_{j}(M)^{\frac{q-p}{n-j}}W_{j}(N)^{\frac{p}{n-j}}\widetilde{W}_{j}(Q)^{\frac{n-q-j}{n-j}},$$

with equality if and only if $M,N,Q$ are dilates when $1<\frac{q}{n-j}<p$, while only $M$ and $N$ are dilates when $q=n$ and $p>1$, and $M$ and $N$ are homotheic when $q=n$ and $p=1$.

Therefore, we can see that the $L_{p}$ Brunn-Minkowski theory and $L_{p}$ dual Brunn-Minkowski theory can be finally unified by definitions of $(p,q)$-dual mixed curvature measures and $(p,q)$-mixed quermassintegrals.

The another goal of this article is to study the Minkowski problem for $(p,q)$-dual mixed curvature measures, which is a general Minkowski problem that unify the $L_{p}$ Minkowski problem and the $L_{p}$ dual Minkowski problem. To prove the existence of Minkowski problems, the key is to convert the Minkowski problem to maximization problem.

The existence problem for $(p,q)$-dual mixed curvature measures is: For $p,q\in\mathbb{R}$, $j\neq n$ and $Q\in\mathcal{S}_{o}^{n}$. Given a Borel measure $\mu$, what are necessary and sufficient conditions on $\mu$ so that there exists a convex body $M\in\mathcal{K}_{o}^{n}$ whose dual curvature measure $\widetilde{\mathcal{C}}_{p,q,j}(M,Q,\cdot)$ is the given measure $\mu$?

The uniqueness problem for $(p,q)$-dual mixed curvature measures is: For $p,q\in\mathbb{R}$, $j\neq n$ and $Q\in\mathcal{S}_{o}^{n}$. If $M_{1},M_{2}\in\mathcal{K}_{o}^{n}$, are such that

$$\widetilde{\mathcal{C}}_{p,q,j}(M_{1},Q,\cdot)=\widetilde{\mathcal{C}}_{p,q,j}(M_{2},Q,\cdot),$$

then how is $M_{1}$ related to $M_{2}$?

The organization is as follows. The corresponding background materials and some results are introduced in Section 2. In Section 3, we define the concept of $(p,q)$-dual mixed curvature measures on the basis of $L_{p}$ dual curvature measures. In Section 4, we then establish the variational formulas for $q$-th dual mixed quermassintegrals, and obtain the integral formula for $(p,q)$-mixed quermassintegrals via the concept of $(p,q)$-dual mixed curvature measures. In Section 5, we derive at some important geometric inequalities involving Minkowski inequality etc. Finally, we further study the Minkowski problem for $(p,q)$-dual mixed curvature measures in Section 6.

   In this section, we give the interrelated background materials and some results. Gardner’s book [12] and Schneider’s book [29] are our standard references for the basics regarding convex bodies and star bodies.

2.1  Support function, radial function, Wulff shape and convex hull

For $M\in\mathcal{K}^{n}$, the support function $h_{M}=h(M,\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$ is given by

$$h(M,x)=\max\{x\cdot y:y\in M\},\ \ \ \ x\in\mathbb{R}^{n}.$$

where $x\cdot y$ denotes the standard inner product of $x$ and $y$ in $\mathbb{R}^{n}$. For $\phi\in GL(n)$ (which denotes the general linear transformation group), the image of $M$ under $\phi$, we have

$$h(\phi M,x)=h(M,\phi^{t}x),\ \ \ \ x\in\mathbb{R}^{n},$$

where $\phi^{t}$ denotes the transpose of $\phi$.

For $M\in\mathcal{S}_{o}^{n}$, the radial function, $\rho_{M}=\rho(M,\cdot)$: $\mathbb{R}^{n}\backslash\{0\}\rightarrow[0,+\infty)$, is defined by

$$\rho(M,x)=\max\{\lambda\geq 0:\lambda x\in M\},\ \ \ \ x\in\mathbb{R}^{n}\backslash\{0\}.$$

Similarly, for $\phi\in GL(n)$, we have

$$\rho(\phi M,x)=\rho(M,\phi^{-1}x),\ \ \ \ x\in\mathbb{R}^{n}\backslash\{0\}.$$

For $M\in\mathcal{K}_{o}^{n}$, the polar body $M^{\ast}$ of $M$ is the convex body in $\mathbb{R}^{n}$, and defined by

$$M^{\ast}=\{x\in\mathbb{R}^{n}:x\cdot y\leq 1,\ for\ all\ y\in M\}.$$

From this definition, we easily see that on $\mathbb{R}^{n}\backslash\{0\}$,

$\displaystyle\rho_{M}=\frac{1}{h_{M^{\ast}}}\ \ \ \ and\ \ \ \ h_{M}=\frac{1}{\rho_{M^{\ast}}}.$

(2.1)

It follows that for $M\in\mathcal{K}_{o}^{n}$

$$(M^{\ast})^{\ast}=M.$$

Let $\Omega\subset S^{n-1}$ denote a set that is closed and cannot be contained in any closed hemisphere of $S^{n-1}$. The Wulff shape $[h]\in\mathcal{K}_{o}^{n}$, of a continuous function $h:\Omega\rightarrow(0,\infty)$, also known as the Aleksandrov body of $h$, is define by

$$[h]=\bigcap_{u\in\Omega}\{x\in\mathbb{R}^{n}:x\cdot u\leq h(u)\}.$$

Because of the restrictions placed on $\Omega$, we see that $[h]\in\mathcal{K}_{o}^{n}$. If $M\in\mathcal{K}_{o}^{n}$, then

$$[h_{M}]=M.$$

For the radial function of the Wulff shape, we have

$$\rho_{[h]}(u)^{-1}=\max_{u\in\Omega}(u\cdot v)h(v)^{-1}.$$

Let $\rho:\Omega\rightarrow(0,\infty)$ be continuous and $\{\rho(u)u:u\in S^{n-1}\}$ is a compact set in $\mathbb{R}^{n}$. The convex hull $\langle\rho\rangle$ is defined by

$$\langle\rho\rangle=conv\{\rho(u)u:u\in S^{n-1}\},$$

is compact as well (see Schneider [29] Theorem 1.1.11). If $M\in\mathcal{K}_{o}^{n}$, then

$$\langle\rho_{M}\rangle=M.$$

For the support function of the convex hulls, we have

$$h_{\langle\rho\rangle}(v)\max_{u\in\Omega}(u\cdot v)\rho(u),$$

for all $v\in S^{n-1}$.

Using the concept of Wulff shape, the definition of $L_{p}$ Minkowski combination can be extended to $p<1$ and even negative $k$ or $l$: Fix a real $p\neq 0$. For $M,N\in\mathcal{K}_{o}^{n}$, and $k,l\in\mathbb{R}$ such that $kh_{M}^{p}+lh_{N}^{p}$ is a strictly positive function on $S^{n-1}$, define the $L_{p}$ Minkowski combination, $k\cdot M+_{p}l\cdot N\in\mathcal{K}_{o}^{n}$, by

$\displaystyle k\cdot M+_{p}l\cdot N=[(kh_{M}^{p}+lh_{N}^{p})^{\frac{1}{p}}].$

(2.2)

For $\phi\in GL(n)$ and $p\neq 0$,

$\displaystyle k\cdot\phi M+_{p}l\cdot\phi N=\phi(k\cdot M+_{p}l\cdot N).$

(2.3)

2.2  The radial Gauss map

Suppose $M\in\mathcal{K}^{n}$ in $\mathbb{R}^{n}$. For $v\in\mathbb{R}^{n}\backslash\{0\}$, the hyperplane

$$H_{M}(v)=\{x\in\mathbb{R}^{n}:x\cdot v=h_{M}(v)\}$$

is called the supporting hyperplane to $M$ with outer normal $v$.

The spherical image of $\sigma\subset\partial M$ is defined by

$$\mathbb{V}_{M}(\sigma)=\{v\in S^{n-1}:x\in H_{M}(v)\ for\ some\ x\in\sigma\}\subset S^{n-1}.$$

The reverse spherical image of $\eta\subset S^{n-1}$ is defined by

$$\mathbb{X}_{M}(\eta)=\{x\in\partial M:x\in H_{M}(v)\ for\ some\ v\in\eta\}\subset\partial M.$$

Let $\sigma_{M}\subset\partial M$ be the set consisting of all $x\in\partial M$, for which the set $\mathbb{V}_{M}(\{x\})$, which we frequently abbreviate as $\mathbb{V}_{M}(x)$, contains more than a single element. It is well known that $\mathcal{H}^{n-1}(\sigma_{M})=0$ (see p. 84 of Schneider [29]). On precisely the set of regular radial vectors of $\partial M$ is defined the function

$$\nu_{M}:\partial M\backslash\sigma_{M}\rightarrow S^{n-1},$$

by letting $\nu_{M}(x)$ be the unique element in $\mathbb{V}_{M}(x)$ for each $x\in\partial M\backslash\sigma_{M}$. The function $\nu_{M}$ is called the spherical image map of $M$ and is known to be continuous (see Lemma 2.2.12 of Schneider [29]). It will occasionally be convenient to abbreviate $\partial M\backslash\sigma_{M}$ by $\partial^{\prime}M$. Since $\mathcal{H}^{n-1}(\sigma_{M})=0$, when the integration is with respect to $\mathcal{H}^{n-1}$, it will be immaterial if the domain is over subsets of $\partial^{\prime}M$ or $\partial M$.

For $M\in\mathcal{K}_{o}^{n}$ and $u\in S^{n-1}$, the radial map of $M$ is defined as,

$$r_{M}:S^{n-1}\rightarrow\partial M\ \ by\ \ r_{M}(u)=\rho_{M}(u)u\in\partial M.$$

For $\omega\subset S^{n-1}$, the radial Gauss image of $\omega$ is defined as

$$\Re_{M}(\omega)=\{v\in S^{n-1}:r_{M}(u)\in H_{M}(v)\ for\ some\ u\in\omega\},$$

and thus, for $u\in S^{n-1}$,

$\displaystyle\Re_{M}(u)=\{v\in S^{n-1}:r_{M}(u)\in H_{M}(v)\}.$

(2.4)

For $M\in\mathcal{K}_{o}^{n}$, the radial Gauss map is defined by

$$\alpha_{M}:S^{n-1}\backslash\omega_{M}\rightarrow S^{n-1},\ \ by\ \ \alpha_{M}=\nu_{M}\circ r_{M}$$

where $\omega_{M}=r_{M}^{-1}(\sigma_{M})$. Since $r_{M}^{-1}$ is a bi-Lipschitz map between the space $\partial M$ and $S^{n-1}$ it follows that $\omega_{M}$ has spherical Lebesgue measure zero. Observe that if $u\in S^{n-1}\backslash\omega_{M}$, then $\Re_{M}(u)$ contains only the element $\alpha_{M}(u)$. For $x\in\mathbb{R}^{n}\backslash\{0\}$, define $\overline{x}\in S^{n-1}$ by $\overline{x}=x/|x|$. Since both $\nu_{M}$ and $r_{M}$ are continuous, then $\alpha_{M}$ is continuous. Note that for $x\in\partial^{\prime}M$,

$\displaystyle\alpha_{M}(\overline{x})=\nu_{M}(x),$

(2.5)

and hence, for $x\in\partial^{\prime}M$,

$\displaystyle h_{M}(\alpha_{M}(\overline{x}))=h_{M}(\nu_{M}(x))=x\cdot\nu_{M}(x).$

(2.6)

For $\eta\subset S^{n-1}$, the reverse radial Gauss image of $\eta$ is defined by

$$\Re_{M}^{\ast}(\eta)=r_{M}^{-1}(\mathbb{X}_{M}(\eta))=\langle\mathbb{X}_{M}(\eta)\rangle.$$

Thus,

$$\Re_{M}^{\ast}(\eta)=\{\overline{x}:x\in\partial M\ \ where\ x\in H_{M}(v)\ for\ some\ v\in\eta\}.$$

If $\eta$ contains only the single vector $v\in S^{n-1}$, then

$$\Re_{M}^{\ast}(v)=\{\overline{x}:x\in\partial M\ \ where\ x\in H_{M}(v)\}.$$

Note that the set $\eta\subset S^{n-1}$,

$$\Re_{M}^{\ast}(\eta)=\{u\in S^{n-1}:r_{M}(u)\in H_{M}(v)\ for\ some\ v\in\eta\},$$

and for $u\in S^{n-1}$ and $\eta\subset S^{n-1}$, we see that from (2.4) that

$\displaystyle u\in\Re_{M}^{\ast}(\eta)\ \ \Leftrightarrow\ \ \Re_{M}(u)\cap\eta\neq\varnothing.$

(2.7)

If $u\not\in\omega_{M}$, then $\Re_{M}(u)=\{\alpha_{M}(u)\}$, then (2.7) becomes

$\displaystyle u\in\Re_{M}^{\ast}(\eta)\ \Leftrightarrow\ \alpha_{M}(u)\in\eta,$

(2.8)

and hence (2.8) holds for almost all $u\in S^{n-1}$, with respect to spherical Lebesgue measure.

The reverse radial Gauss image of a convex body and the radial Gauss image of its polar body are related (see[15]).

For almost all $v\in S^{n-1}$, then $\Re_{M}^{\ast}(v)=\{\alpha_{M}^{\ast}(v)\}$, and $\Re_{M^{\ast}}(v)=\{\alpha_{M^{\ast}}(v)\}$. Therefore,

$$\alpha_{M}^{\ast}=\alpha_{M^{\ast}}.$$

2.3  The surface area measure

By using the spherical image and reverse spherical image, one can define the surface area measures, and their $L_{p}$ extensions.

For Borel set $\eta\subset S^{n-1}$ and $M\in\mathcal{K}_{o}^{n}$. the surface area measures $S(M,\cdot)$ can be defined by

$$S(M,\eta)=\mathcal{H}^{n-1}(\mathbb{X}_{M}(\eta)).$$

For convex bodies $M,N$ in $\mathbb{R}^{n}$, the mixed quermassintegral $W_{i}(M,N)$ has the following integral representation:

$$W_{i}(M,N)=\frac{1}{n}\int_{S^{n-1}}h_{N}(u)dS_{i}(M,u),$$

where $dS_{i}(M,u)/dS(u)=f_{i}(M,u):S^{n-1}\rightarrow[0,\infty)$ denote the $i$-th curvature function (see[19]).

For $p\in\mathbb{R}$ and $M,N\in\mathcal{K}_{o}^{n}$, the $L_{p}$ mixed quermassintegral $W_{p,i}(M,N)$ is defined by

$\displaystyle W_{p,i}(M,N)=\frac{1}{n}\int_{S^{n-1}}h_{N}^{p}(u)dS_{p,i}(M,u),$

(2.9)

where the $S_{p,i}(M,\cdot)$ is the $L_{p}$ surface area measure.

For $p\geq 1$, the $L_{p}$ Minkowski inequality of $L_{p}$ mixed quermassintegral is obtained

$\displaystyle W_{p,i}(M,N)\geq W_{i}(M)^{\frac{n-p-i}{n-i}}W_{i}(N)^{\frac{p}{n-i}},$

(2.10)

with equality if and only if $M$ and $N$ are dilates when $p>1$, and $M$ and $N$ are homothets when $p=1$.

The following important integral identity was established in [15]: For $q\in\mathbb{R}$, $j\neq n$ and $M\in\mathcal{K}_{o}^{n}$. If $f:S^{n-1}\rightarrow\mathbb{R}$ is bounded and Lebesgue integrable, then

$\displaystyle\int_{S^{n-1}}f(u)\rho_{M}^{q}(u)du=\int_{\partial M}f(\overline{x})|x|^{q-n}(x\cdot\nu_{M}(x))d\mathcal{H}^{n-1}(x).$

(2.11)

   In this section, we introduce the concept of $(p,q)$-dual mixed curvature measures on the basis of $L_{p}$ dual curvature measures.

Let $Q\in\mathcal{S}_{o}^{n}$, define a continuous and positively function $\|\cdot\|_{Q}:\mathbb{R}^{n}\rightarrow[0,\infty)$ by

$\displaystyle\|x\|_{Q}=\bigg{\{}_{0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0.}^{\rho_{Q}^{-1}(x),\ \ \ \ \ \ \ \ \ \ x\neq 0}$

(3.1)

Note that $\|\cdot\|_{Q}$ is a continuous, and a positively homogeneous function of degree one. When $Q$ is an origin-symmetric convex bodies in $\mathbb{R}^{n}$, then $\|\cdot\|_{Q}$ is an ordinary norm in $\mathbb{R}^{n}$, and $(\mathbb{R}^{n},\|\cdot\|_{Q})$ is the $n$-dimensional Banach space whose unit ball is $Q$.

Then, we are ready to establish the integral representation of $(p,q)$-dual mixed curvature measures. But we also need a tool as follows. Let $\varphi:S^{n-1}\rightarrow\mathbb{R}$ be a simple function on $S^{n-1}$ by letting

$$\varphi=\sum_{i=1}^{k}c_{i}\mathbb{I}_{\eta_{i}}$$

for $c_{i}\in\mathbb{R}$ and Borel set $\eta_{i}\subset S^{n-1}$.

Proof.  From (3) and the fact (2.8), for any Borel sets $\eta\subset S^{n-1}$, we get

	$\displaystyle\int_{S^{n-1}}\varphi(u)d\widetilde{\mathcal{C}}_{q,j}(M,Q,u)$	$\displaystyle=\int_{S^{n-1}}\sum_{i=1}^{k}c_{i}\mathbb{I}_{\eta_{i}}(u)d\widetilde{\mathcal{C}}_{q,j}(M,Q,u)$
		$\displaystyle=\sum_{i=1}^{k}c_{i}\int_{S^{n-1}}\mathbb{I}_{\eta_{i}}(u)d\widetilde{\mathcal{C}}_{q,j}(M,Q,u)$
		$\displaystyle=\sum_{i=1}^{k}c_{i}\widetilde{\mathcal{C}}_{q,j}(M,Q,\eta_{i})$
		$\displaystyle=\frac{1}{n}\sum_{i=1}^{k}c_{i}\int_{\Re_{M}^{\ast}(\eta_{i})}\rho_{M}^{q}(v)\rho_{Q}^{n-q-j}(v)dv$		(3.5)
		$\displaystyle=\frac{1}{n}\int_{S^{n-1}}\sum_{i=1}^{k}c_{i}\mathbb{I}_{\Re_{M}^{\ast}(\eta_{i})}(v)\rho_{M}^{q}(v)\rho_{Q}^{n-q-j}(v)dv$
		$\displaystyle=\frac{1}{n}\int_{S^{n-1}}\sum_{i=1}^{k}c_{i}\mathbb{I}_{{\eta_{i}}}(\alpha_{M}(v))\rho_{M}^{q}(v)\rho_{Q}^{n-q-j}(v)dv$
		$\displaystyle=\frac{1}{n}\int_{S^{n-1}}\varphi(\alpha_{M}(u))\rho_{M}^{q}(u)\rho_{Q}^{n-q-j}(u)du.$

In (3), we choose a sequence of simple functions $f_{k}$ that converge uniformly to $f$. Then $f_{k}\circ\alpha_{M}$ converges to $f\circ\alpha_{M}$ a.e. with respect to spherical Lebesgue measure. Since $f$ is a Borel function on $S^{n-1}$ and the radial Gauss map $\alpha_{M}$ is continuous on $S^{n-1}\backslash\omega_{M}$, then composite function $f\circ\alpha_{M}$ is a Borel function on $S^{n-1}\backslash\omega_{M}$ as well. Since $f$ is bounded and $\omega_{M}$ has Lebesgue measure zero. Thus, $f$ and $S^{n-1}\backslash\omega_{M}$ are Lebesgue integrable on $S^{n-1}$. Taking the $k\rightarrow\infty$, we can establish (3.4). ${\square}$