Let $ \mathcal{K}^n $ denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space $ \mathbb{R}^n $. The set of convex bodies containing the origin in their interiors, we write $ \mathcal{K}_o^n $. $ \mathcal{S}_o^n $ denotes the set of star bodies (about the origin) in $ \mathbb{R}^n $. The unit ball in $ \mathbb{R}^n $ and its surface will be denoted by $ B $ and $ S^{n-1} $, respectively. $ V(K) $ denotes the $ n $-dimensional volume of a body $ K $ and write $ V(B) = \omega_n $.

For $ K\in \mathcal{K}^n $, its support function, $ h_K $ = $ h(K, \cdot) $: $ \mathbb{R}^n\rightarrow\mathbb{R} $, is defined by (see [1, 2])

$ \begin{aligned} h(K, x) & = \max\{x\cdot y :y \in K\}, \ \ \ x\in\mathbb{R}^n, \end{aligned} $

where $ x\cdot y $ denotes the standard inner product of $ x $ and $ y $.

The conception of $ L_p $-centroid body was introduced by Lutwak and Zhang (see [3]). For each compact star-shaped (about the origin) $ K $ in $ \mathbb{R}^n $ and real $ p\geq1 $, the $ L_p $-centroid body, $ \Gamma_p K $, of $ K $ is an origin-symmetric convex body which support function is defined by

$ \begin{align} \begin{aligned} h_{\Gamma_p K}^{p}(u) & = \frac{1}{c_{n, p}V(K)}\int_K \mid u\cdot x\mid^p dx \\ & = \frac{1}{c_{n, p}(n+p)V(K)}\int_{S^{n-1}}\mid u\cdot v\mid^p \rho_K^{n+p}(v)dS(v) \end{aligned} \end{align} $

(1.1)

for any $ u\in S^{n-1} $, where the integration is in connection with Lebesgue measure on $ S^{n-1} $ and

$ \begin{align} \begin{aligned} c_{n, p} & = \frac{\omega_{n+p}}{\omega_2 \omega_n \omega_{p-1}}. \end{aligned} \end{align} $

(1.2)

In 2000, Lutwak, Yang and Zhang in [4] put forward the notion of $ L_p $-projection body. For $ K\in\mathcal{K}_o^n $ and real $ p\geq1 $, the $ L_p $-projection body, $ \Pi_p K $, of $ K $ is an origin-symmetric convex body whose support function is given by

$ \begin{align} \begin{aligned} h^p_{\Pi_p K}(u) & = \alpha_{n, p}\int_{S^{n-1}}\mid u\cdot v\mid^pdS_p(K, v) \end{aligned} \end{align} $

(1.3)

for all $ u\in S^{n-1} $. Here $ S_p(K, \cdot) $ is the $ L_p $-surface area measure of $ K $,

$ \begin{align} \begin{aligned} \alpha_{n, p} & = \frac{1}{n\omega_n c_{n-2, p}}, \end{aligned} \end{align} $

(1.4)

and $ c_{n-2, p} $ satisfies (1.2). At the same time, they (see [4]) proved the $ L_p $-Petty projection inequality and $ L_p $-Busemann-Petty centroid inequality. For the $ L_p $-centroid bodies and $ L_p $-projection bodies, some scholars made a series of researches and gained several results (see [5-15]). In particular, Wang, Lu and Leng in [12] established the following monotonic inequalities.

Theorem 1.A   Let $ K, L\in \mathcal{K}^n_o $ and $ p\geq1 $. If for any $ Q\in\mathcal{K}^n_o $, $ V_p(K, Q)\leq V_p(L, Q) $, then $ V(\Pi_p K) \leq V(\Pi_p L) $ with equality for $ p = 1 $ if and only if $ \Pi_p K $ and $ \Pi_p L $ are translates, for $ p>1 $ if and only if $ \Pi_p K = \Pi_p L $, here $ V_p(M, N) $ denotes the $ L_p $-mixed volume of $ M, N\in\mathcal{K}^n_o $.

Theorem 1.B   Let $ K, L\in \mathcal{K}^n_o $ and $ p\geq1 $. If for any $ Q\in\mathcal{K}^n_o $, $ V_p(K, Q)\leq V_p(L, Q) $, then $ V(\Pi_p^\ast K) \geq V(\Pi_p^\ast L) $ with equality if and only if $ \Pi_p K = \Pi_p L $, here $ \Pi_p^\ast M $ denotes the polar of $ \Pi_p M $.

Theorem 1.C   Let $ K, L\in \mathcal{S}^n_o $ and $ p\geq1 $. If for any $ Q\in\mathcal{S}^n_o $, $ \widetilde{V}_{-p}(K, Q)\leq \widetilde{V}_{-p}(L, Q), $ then

$ \begin{aligned} \frac{V(\Gamma_p K)^{-\frac{p}{n}}}{V(K)} &\geq\frac{V(\Gamma_p L)^{-\frac{p}{n}}}{V(L)} \end{aligned} $

with equality for $ p = 1 $ if and only if $ \Gamma_p K $ and $ \Gamma_p L $ are translates, for $ p>1 $ if and only if $ \Gamma_p K = \Gamma_p L $, here $ \widetilde{V}_{-p}(M, N) $ denotes the $ L_p $-dual mixed volume of $ M, N\in\mathcal{S}^n_o $.

Theorem 1.D   Let $ K, L\in \mathcal{S}^n_o $ and $ p\geq1 $. If for any $ Q\in\mathcal{S}^n_o $, $ \widetilde{V}_{-p}(K, Q)\leq \widetilde{V}_{-p}(L, Q) $, then

$ \begin{aligned} \frac{V(\Gamma_p^\ast K)^{\frac{p}{n}}}{V(K)} &\geq\frac{V(\Gamma_p^\ast L)^{\frac{p}{n}}}{V(L)} \end{aligned} $

with equality if and only if $ \Gamma_p K = \Gamma_p L $, here $ \Gamma_p^\ast M $ denotes the polar of $ \Gamma_p M $.

Ludwig (see [16]) introduced a function $ \varphi_\tau :\mathbb{R}\rightarrow[0, +\infty) $ given by $ \varphi_\tau(t) = |t|+\tau t $ for $ \tau\in[-1, 1] $. Using this function, Ludwig [16] defined general $ L_p $-projection bodies as follows: for $ K\in\mathcal{K}_o^n $, $ p\geq1 $ and $ \tau\in[-1, 1] $, general $ L_p $-projection body, $ \Pi_p^\tau K\in\mathcal{K}_o^n $, of $ K $ with support function by

$ \begin{align} \begin{aligned} h_{\Pi_p^\tau K}^p(u) & = \alpha_{n, p}(\tau)\int_{S^{n-1}}{\varphi_\tau(u\cdot v)}^pdS_p(K, v), \end{aligned} \end{align} $

(1.5)

where

$ \begin{align} \begin{aligned} \alpha_{n, p}(\tau) & = \frac{2\alpha_{n, p}}{(1+\tau)^p +(1-\tau)^p}, \end{aligned} \end{align} $

(1.6)

and $ \alpha_{n, p} $ satisfies (1.4). For every $ \tau\in[-1, 1] $, the normalization is chosen such that $ \Pi_p^\tau B = B $. Clearly, if $ \tau = 0 $, then $ \Pi_p^\tau K = \Pi_p^0 K = \Pi_pK $.

Regarding general $ L_p $-projection bodies, Wang and Wan (see [17]) studied the Shephard type problem. Wang and Feng (see [18]) established general $ L_p $-Petty affine projection inequality. Wang and Wang (see [19]) gave the extremums of quermassintegrals and dual quermassintegrals for general $ L_p $-projection bodies and their polar.

Subsequently, according to definition (1.1) of $ L_p $-centroid bodies, Feng, Wang and Lu (see [20]) imported the notion of general $ L_p $-centroid bodies. For $ K\in\mathcal{S} _o^n $, $ p\geq1 $ and $ \tau\in[-1, 1] $, the general $ L_p $-centroid body, $ \Gamma_p^\tau K\in\mathcal{K}_o^n $, of $ K $ which support function is defined by

$ \begin{align} \begin{aligned} h_{\Gamma_p^\tau K}^p(u) & = \frac{1}{c_{n, p}(\tau)V(K)}\int_K\varphi_\tau(u\cdot x)^pdx \\ & = \frac{\gamma_{n, p}(\tau)}{V(K)}\int_{S^{n-1}}\varphi_\tau(u\cdot v)^p \rho_K^{n+p}(v)dS(v), \end{aligned} \end{align} $

(1.7)

where

$ \begin{eqnarray} \gamma_{n, p}(\tau) & = &\frac{1}{(n+p)c_{n, p}(\tau)}, \\ c_{n, p}(\tau) & = &\frac{1}{2}c_{n, p}[(1+\tau)^p+(1-\tau)^p], \end{eqnarray} $

(1.8)

and $ c_{n, p} $ satisfies (1.2). The normalization is chosen such that $ \Gamma_p^\tau B = B $ for every $ \tau\in[-1, 1] $, and $ \Gamma_p^0K = \Gamma_pK $.

From the definition of $ L_p $-projection body, Wang and Leng (see [21]) gave the following concept of $ L_p $-mixed projection body. For each $ K\in\mathcal{K}^n_o $, real $ p\geq1 $ and $ i = 0, 1, \cdots, n-1 $, the $ L_p $-mixed projection body, $ \Pi_{p, i}K $, of $ K $ is an origin-symmetric convex body, which support function is defined by

$ \begin{align} \begin{aligned} h^p_{\Pi_{p, i} K}(u) & = \alpha_{n, p}\int_{S^{n-1}}\mid u\cdot v\mid^p dS_{p, i}(K, v) \end{aligned} \end{align} $

(1.9)

for any $ u\in S^{n-1} $, the positive Borel measure $ S_{p, i}(K, \cdot) $ on $ S^{n-1} $ is absolutely continuous with respect to $ S_i(K, \cdot) $, and has the Radon-Nikodym derivative

$ \begin{align} \begin{aligned} \frac{dS_{p, i}(K, \cdot)}{dS_i(K, \cdot)} & = h^{1-p}(K, \cdot). \end{aligned} \end{align} $

(1.10)

By definitions (1.9) and (1.3), we easily know that $ \Pi_{p, 0}K = \Pi_p K $.

Just as the definition of the $ L_p $-mixed projection body, $ L_p $-mixed centroid body was introduced by Wang, Leng and Lu (see [11]). If $ K\subset\mathbb{R}^n $ is compact star-shaped about the origin, $ p\geq1 $, $ i\in\mathbb{\mathbb{R}} $, then the $ L_p $-mixed centroid body, $ \Gamma_{p, i}K $, of $ K $ is the origin-symmetric convex body whose support function is given by

$ \begin{aligned} h_{\Gamma_{p, i}K}^p(u) & = \frac{1}{(n+p)c_{n, p}V(K)}\int_{S^{n-1}}\mid u\cdot v\mid^p\rho_K^{n+p-i}(v)dS(v) \end{aligned} $

for every $ u\in S^{n-1}. $ From this and definition (1.1), we have $ \Gamma_{p, 0}K = \Gamma_p K. $

For the studies of $ L_p $-mixed projection bodies and $ L_p $-mixed centroid bodies, Wang and Leng [21] demonstrated the Petty projection inequality for $ L_p $-mixed projection bodies, and then, Wang, Leng and Lu [11] obtained the forms of quermassintegrals and dual quermassintegrals of Theorem 1.A and Theorem 1.B. Moreover, on one hand, associated with the definition of quermassintegrals, Wang and Leng [10] extended Theorem 1.C to the quermassintegrals; on the other hand, Wang, Lu and Leng [13] gave the dual quermassintegrals form for Theorem 1.D. In regard to the studies of the $ L_p $-mixed projection bodies and the $ L_p $-mixed centroid bodies, see also [22-25].

According to definitions (1.5) and (1.9), general $ L_p $-mixed projection bodies were raised by Wan and Wang [26]. For $ K\in\mathcal{K}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $ and $ i = 0, 1, \cdots, n-1 $, the general $ L_p $-mixed projection bodies, $ \Pi_{p, i}^\tau K\in\mathcal{K}_o^n $, whose support function is provided by

$ \begin{align} \begin{aligned} h_{\Pi_{p, i}^\tau K}^p(u) & = \alpha_{n, p}(\tau)\int_{S^{n-1}}\varphi_\tau(u\cdot v)^pdS_{p, i}(K, v). \end{aligned} \end{align} $

(1.11)

From (1.11) and (1.5), if $ i = 0 $, then $ \Pi_{p, 0}^\tau K = \Pi_p^\tau K $.

Similar to Wan and Wang's idea, we define general $ L_p $-mixed centroid bodies as follows: for $ K\in\mathcal{S}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $ and $ i $ is any real, the general $ L_p $-mixed centroid body, $ \Gamma_{p, i}^\tau K\in\mathcal{K}_o^n $, of $ K $ is presented by

$ \begin{align} \begin{aligned} h_{\Gamma_{p, i}^\tau K}^p(u) & = \frac{\gamma_{n, p}(\tau)}{V(K)}\int_{S^{n-1}}\varphi_\tau(u\cdot v)^p \rho_K^{n+p-i}(v)dS(v), \end{aligned} \end{align} $

(1.12)

where $ \gamma_{n, p}(\tau) $ is the same as (1.8). Especially, if $ i = 0 $, by definitions (1.12) and (1.7), we easily get $ \Gamma_{p, 0}^\tau K = \Gamma_p^\tau K $.

In this article, we first extend Theorem 1.A and Theorem 1.B to quermassintegrals and dual quermassintegrals, which can be stated as follows.

Theorem 1.1   Let $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $ and $ i, j = 0, 1, \cdots, n-1 $. If for any $ Q\in\mathcal{K}_o^n $,

$ \begin{aligned} W_{p, j}(K, Q) &\leq W_{p, j}(L, Q), \end{aligned} $

then

$ \begin{align} \begin{aligned} W_i(\Pi_{p, j}^\tau K) &\leq W_i(\Pi_{p, j}^\tau L). \end{aligned} \end{align} $

(1.13)

Equality holds in (1.13) for $ p = 1 $ if and only if $ \Pi_{p, j}^\tau K $ and $ \Pi_{p, j}^\tau L $ are translates; for $ p > 1 $ if and only if $ \Pi_{p, j}^\tau K = \Pi_{p, j}^\tau L $. Here $ W_{p, j}(M, N) $ $ (j = 0, 1, \cdots, n-1) $ denotes the $ L_p $-mixed quermassintegrals of $ M, N\in\mathcal{K}_o^n $.

Theorem 1.2   Let $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $, real $ i\neq n $ and $ j = 0, 1, \cdots, n-1 $. If for any $ Q\in \mathcal{K}_o^n $,

$ \begin{aligned} W_{p, j}(K, Q) &\leq W_{p, j}(L, Q), \end{aligned} $

then for $ i<n $,

$ \begin{align} \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K) &\geq\widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L); \end{aligned} \end{align} $

(1.14)

for $ n<i<n+p $ or $ i>n+p $,

$ \begin{align} \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K) &\leq\widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L). \end{aligned} \end{align} $

(1.15)

Equality holds in (1.14) or (1.15) for $ i\neq n+p $ if and only if $ \Pi_{p, j}^\tau K = \Pi_{p, j}^\tau L $. For $ i = n+p $, inequality (1.15) is identic.

Moreover, we establish the following inequalities of quermassintegrals and dual quermassintegrals for general $ L_p $-mixed centroid bodies, which is regarded as a generalization of Theorem 1.C and Theorem 1.D.

Theorem 1.3   Let $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $, real $ i\neq n $ and $ j = 0, 1, \cdots, n-1 $. If for any $ Q\in \mathcal{S}_o^n $,

$ \begin{aligned} \widetilde{W}_{-p, i}(K, Q) &\leq\widetilde{W}_{-p, i}(L, Q), \end{aligned} $

then

$ \begin{align} \begin{aligned} \frac{W_j(\Gamma_{p, i}^\tau K)^{-\frac{p}{n-j}}}{V(K)} &\geq\frac{W_j(\Gamma_{p, i}^\tau L)^{-\frac{p}{n-j}}}{V(L)}. \end{aligned} \end{align} $

(1.16)

Equality holds in (1.16) for $ p = 1 $ if and only if $ \Gamma_{p, i}^\tau K $ and $ \Gamma_{p, i}^\tau L $ are translates, for $ p>1 $ if and only if $ \Gamma_{p, i}^\tau K = \Gamma_{p, i}^\tau L $. Here $ \widetilde{W}_{-p, j}(M, N) $ $ (j\neq n) $ denotes the $ L_p $-dual mixed quermassintegrals of $ M, N\in\mathcal{S}_o^n $.

Theorem 1.4   Let $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $, real $ i, j\neq n $. If for any $ Q\in\mathcal{S}^n_o $,

$ \begin{aligned} \widetilde{W}_{-p, i}(K, Q) &\leq\widetilde{W}_{-p, i}(L, Q), \end{aligned} $

then

$ \begin{align} \begin{aligned} \frac{\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}K)^{\frac{p}{n-j}}}{V(K)} &\geq\frac{\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}L)^{\frac{p}{n-j}}}{V(L)}. \end{aligned} \end{align} $

(1.17)

Equality holds in (1.17) for $ j\neq n+p $ if and only if $ \Gamma_{p, i}^\tau K = \Gamma_{p, i}^\tau L $. For $ j = n+p $, inequality (1.17) is identic.

Obviously, taking $ i = j = \tau = 0 $ in Theorems 1.1–1.4, then inequalities (1.13)–(1.17) reduce to Theorems 1.A–1.D, respectively.

This paper is organized as follows. In section 2, we provide some basic notions and results. Section 3 gives the proofs of Theorems 1.1–1.4.

If $ K $ is a compact star-shaped (about the origin) set in $ \mathbb{R}^n, $ then its radial function, $ \rho_K = \rho(K, \cdot):\mathbb{R}^n\setminus\{0\}\longrightarrow[0, +\infty), $ is defined by (see [2])

$ \begin{aligned} \rho(K, u) & = \max\{\lambda\geq0:\lambda\cdot u\in K\}, \ \ \ u\in S^{n-1}. \end{aligned} $

If $ \rho_K $ is positive and continuous, then $ K $ is viewed as a star body (about the origin). Two star bodies $ K $ and $ L $ will be dilates (of one another) if $ \rho_K(u)/\rho_L(u) $ is independent of $ u\in S^{n-1}. $

If $ K $ is a nonempty subset of $ \mathbb{R}^n $, then the polar set $ K^\ast $ of $ K $ is defined by (see [1, 2])

$ \begin{aligned} K^\ast & = \{x\in\mathbb{R}^n:x\cdot y\leq1, y\in K\}. \end{aligned} $

If $ K\in\mathcal{K}_o^n, $ it follows that $ (K^{\ast})^\ast = K $ and

$ \begin{align} h_{K^\ast} = \frac{1}{\rho_K}, \ \ \ \ \rho_{K^\ast} = \frac{1}{h_K}. \end{align} $

(2.1)

For $ K, L\in\mathcal{K}_o^n $, real $ p\geq1 $ and $ \lambda, \mu\geq0 $ (not both zero), the $ L_p $-Minkowski combination (also called the Firey $ L_p $-combination), $ \lambda\cdot K+_p\mu\cdot L\in\mathcal{K}_o^n $, of $ K $ and $ L $ is defined by (see [27])

$ \begin{aligned} h(\lambda\cdot K+_p\mu\cdot L, \cdot)^p & = \lambda h(K, \cdot)^p+\mu h(L, \cdot)^p, \end{aligned} $

where the operation $ \lambda\cdot K $ denotes Firey scalar multiplication. Obviously, Firey scalar multiplication and usual scalar multiplication are related by $ \lambda\cdot K = \lambda^{\frac{1}{p}}K $.

For $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, $ \lambda, \mu\geq0 $ (not both zero), the $ L_p $-harmonic radial combination, $ \lambda\star K+_{-p}\mu\star L\in\mathcal{S}_o^n $, of $ K $ and $ L $ is defined by (see [28])

$ \begin{aligned} \rho(\lambda\star K+_{-p}\mu\star L, \cdot)^{-p} & = \lambda\rho(K, \cdot)^{-p}+\mu\rho(L, \cdot)^{-p}. \end{aligned} $

Here $ \lambda\star K $ denotes $ L_p $-harmonic radial scalar multiplication, and we can see $ \lambda\star K = \lambda^{-\frac{1}{p}}K. $ Note that for convex bodies, the $ L_p $-harmonic radial combination was investigated by Firey (see [29]).

If $ K\in\mathcal{K}^n $, the quermassintegrals $ W_i(K) $ $ (i = 0, 1, \cdots, n-1) $ of $ K $ are defined by (see [1, 2])

$ \begin{align} \begin{aligned} W_i(K) & = \frac{1}{n}\int_{S^{n-1}}h_K(u)dS_i(K, u), \end{aligned} \end{align} $

(2.2)

where $ S_i(K, \cdot) $ $ (i = 0, 1, \cdots, n-1) $ is the mixed surface area measure of $ K\in\mathcal{K}^n $, $ S_0(K, \cdot) $ is the surface area measure of $ K $. In particular, we easily see that

$ \begin{align} \begin{aligned} W_0(K) & = V(K). \end{aligned} \end{align} $

(2.3)

In [30], Lutwak defined the $ L_p $-mixed quermassintegrals and showed that for $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $ and $ i = 0, 1, \cdots, n-1 $, the $ L_p $-mixed quermassintegrals $ W_{p, i}(K, L) $ has the following integral representation

$ \begin{align} \begin{aligned} W_{p, i}(K, L) & = \frac{1}{n}\int_{S^{n-1}}h_L^p(u)dS_{p, i}(K, u). \end{aligned} \end{align} $

(2.4)

Here $ S_{p, i}(K, \cdot) $ $ (i = 0, 1, \cdots, n-1) $ satisfies (1.10). The case $ i = 0 $, $ S_{p, 0}(K, \cdot) $ is just the $ L_p $-surface area measure $ S_p(K, \cdot) $ of $ K\in\mathcal{K}_o^n. $

From (2.2), (2.4) and (1.10), it follows immediately that for each $ K\in\mathcal{K}_o^n $ and $ p\geq1 $,

$ \begin{align} \begin{aligned} W_{p, i}(K, K) & = W_i(K). \end{aligned} \end{align} $

(2.5)

For the $ L_p $-mixed quermassintegrals $ W_{p, i}(K, L) $, Lutwak [30] established the following Minkowski inequality

Theorem 2.A   If $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $ and $ i = 0, 1, \cdots, n-1 $, then

$ \begin{align} \begin{aligned} W_{p, i}(K, L) &\geq W_i(K)^{\frac{n-p-i}{n-i}}W_i(L)^{\frac{p}{n-i}} \end{aligned} \end{align} $

(2.6)

with equality for $ p = 1 $ if and only if $ K $ and $ L $ are homothetic, for $ p>1 $ if and only if $ K $ and $ L $ are dilates.

For $ K\in\mathcal{S}^n_o $ and real $ i $, the dual quermassintegrals, $ \widetilde{W}_i(K) $, of $ K $ are defined by (see [31])

$ \begin{align} \begin{aligned} \widetilde{W}_i(K) & = \frac{1}{n}\int_{S^{n-1}}\rho(K, u)^{n-i}dS(u). \end{aligned} \end{align} $

(2.7)

Obviously,

$ \begin{align} \begin{aligned} \widetilde{W}_0(K) & = \frac{1}{n}\int_{S^{n-1}}\rho(K, u)^ndS(u) = V(K). \end{aligned} \end{align} $

(2.8)

In 2005, Wang and Leng [32] introduced the $ L_p $-dual mixed quermassintegrals as follows: for $ K, L\in\mathcal{S}^n_o $, $ p\geq1 $ and real $ i\neq n $, the $ L_p $-dual mixed quermassintegrals, $ \widetilde{W}_{-p, i}(K, L) $, of $ K $ and $ L $ are given by

$ \begin{align} \begin{aligned} \widetilde{W}_{-p, i}(K, L) & = \frac{1}{n}\int_{S^{n-1}}\rho_K^{n+p-i}(u)\rho_L^{-p}(u)dS(u). \end{aligned} \end{align} $

(2.9)

From formula (2.9) and definition (2.7), we get

$ \begin{align} \begin{aligned} \widetilde{W}_{-p, i}(K, K) & = \widetilde{W}_i(K). \end{aligned} \end{align} $

(2.10)

For the $ L_p $-dual mixed quermassintegrals, Wang and Leng (see [32]) proved the following Minkowski inequality.

Theorem 2.B   If $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, real $ i\neq n $, then for $ i<n $ or $ n<i<n+p $,

$ \begin{align} \begin{aligned} \widetilde{W}_{-p, i}(K, L) &\geq\widetilde{W}_i(K)^{\frac{n+p-i}{n-i}}\widetilde{W}_i(L)^{-\frac{p}{n-i}}; \end{aligned} \end{align} $

(2.11)

for $ i>n+p $,

$ \begin{align} \begin{aligned} \widetilde{W}_{-p, i}(K, L) &\leq\widetilde{W}_i(K)^{\frac{n+p-i}{n-i}}\widetilde{W}_i(L)^{-\frac{p}{n-i}}. \end{aligned} \end{align} $

(2.12)

Equality holds in each inequality if and only if $ K $ and $ L $ are dilates.

In this section, we prove Theorems 1.1–1.4. First, the following lemmas are necessary.

Lemma 3.1   If $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $ and $ i, j = 0, 1, \cdots, n-1 $, then

$ \begin{align} \begin{aligned} W_{p, i}(K, \Pi_{p, j}^\tau L) & = W_{p, j}(L, \Pi_{p, i}^\tau K). \end{aligned} \end{align} $

(3.1)

Proof   According to definitions (2.4) and (1.11), and using Fubini theorem, we get

$ \begin{aligned} W_{p, i}(K, \Pi_{p, j}^\tau L) & = \frac{1}{n}\int_{S^{n-1}}h(\Pi_{p, j}^\tau L, u)^pdS_{p, i}(K, u)\\ & = \frac{1}{n}\int_{S^{n-1}}\alpha_{n, p}(\tau) \int_{S^{n-1}}\varphi_\tau(u\cdot v)^pdS_{p, j}(L, v)dS_{p, i}(K, u)\\ & = \frac{1}{n}\int_{S^{n-1}}\alpha_{n, p}(\tau) \int_{S^{n-1}}\varphi_\tau(u\cdot v)^pdS_{p, i}(K, u)dS_{p, j}(L, v)\\ & = \frac{1}{n}\int_{S^{n-1}}h(\Pi_{p, i}^\tau K, v)^pdS_{p, j}(L, v)\\ & = W_{p, j}(L, \Pi_{p, i}^\tau K). \end{aligned} $

Lemma 3.2   If $ K\in\mathcal{K}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $, real $ i\neq n $ and $ j = 0, 1, \cdots, n-1 $, then for any $ M\in\mathcal{S}_o^n $,

$ \begin{align} \begin{aligned} W_{p, j}(K, \Gamma_{p, i}^\tau M) & = \frac{2\omega_n}{V(M)}\widetilde{W}_{-p, i}(M, \Pi_{p, j}^{\tau, \ast}K). \end{aligned} \end{align} $

(3.2)

Proof   From definitions (2.4), (2.9) and (1.12), and using $ nc_{n-2, p} = (n+p)c_{n, p} $, we have

$ \begin{aligned} W_{p, j}(K, \Gamma_{p, i}^\tau M) & = \frac{1}{n}\int_{S^{n-1}}h_{\Gamma_{p, i}^\tau M}^p(v)dS_{p, j}(K, v)\\ & = \frac{\gamma_{n, p}(\tau)}{nV(M)}\int_{S^{n-1}} \int_{S^{n-1}}\varphi_\tau(u\cdot v)^p\rho_M^{n+p-i}(u)dS(u)dS_{p, j}(K, v)\\ & = \frac{2\omega_n}{nV(M)}\int_{S^{n-1}}\rho_M^{n+p-i}(u) \rho_{\Pi_{p, j}^{\tau, \ast}K}^{-p}(u)dS(u)\\ & = \frac{2\omega_n}{V(M)}\widetilde{W}_{-p, i}(M, \Pi_{p, j}^{\tau, \ast}K). \end{aligned} $

Lemma 3.3   If $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, $ \tau\in[-1, 1] $ and reals $ i, j\neq n $, then

$ \begin{align} \begin{aligned} \frac{\widetilde{W}_{-p, j}(K, \Gamma_{p, i}^{\tau, \ast}L)}{V(K)} & = \frac{\widetilde{W}_{-p, i}(L, \Gamma_{p, j}^{\tau, \ast}K)}{V(L)}. \end{aligned} \end{align} $

(3.3)

Proof   Due to considerations (2.9), (1.12), (2.1) and Fubini theorem, we obtain

$ \begin{aligned}&\frac{\widetilde{W}_{-p, j}(K, \Gamma_{p, i}^{\tau, \ast}L)}{V(K)}\\ = &\frac{1}{nV(K)}\int_{S^{n-1}}\rho_K^{n+p-j}(u) \rho_{\Gamma_{p, i}^{\tau, \ast}L}^{-p}(u)dS(u)\\ = &\frac{1}{nV(K)}\int_{S^{n-1}} \rho_K^{n+p-j}(u)h_{\Gamma_{p, i}^\tau L}^p(u)dS(u)\\ = &\frac{\gamma_{n, p}(\tau)}{nV(K)V(L)}\int_{S^{n-1}} \rho_K^{n+p-j}(u)\int_{S^{n-1}}\varphi_\tau(u\cdot v)^p \rho_L^{n+p-i}(v)dS(v)dS(u)\\ = &\frac{\gamma_{n, p}(\tau)}{nV(K)V(L)}\int_{S^{n-1}} \rho_L^{n+p-i}(v)\int_{S^{n-1}}\varphi_\tau(u\cdot v)^p \rho_K^{n+p-j}(u)dS(u)dS(v)\\ = &\frac{1}{nV(L)}\int_{S^{n-1}} \rho_L^{n+p-i}(v)h_{\Gamma_{p, j}^\tau K}^p(v)dS(v)\\ = &\frac{1}{nV(L)}\int_{S^{n-1}} \rho_L^{n+p-i}(v)\rho_{\Gamma_{p, j}^{\tau, \ast} K}^{-p}(v)dS(v)\\ = &\frac{\widetilde{W}_{-p, i}(L, \Gamma_{p, j}^{\tau, \ast}K)}{V(L)}. \end{aligned} $

Proof of Theorem 1.1   Since $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $, $ j = 0, 1, \cdots, n-1 $, and for any $ Q\in\mathcal{K}_o^n $,

$ \begin{align} \begin{aligned} W_{p, j}(K, Q)&\leq W_{p, j}(L, Q), \end{aligned} \end{align} $

(3.4)

thus for any $ M\in\mathcal{K}_o^n $, let $ Q = \Pi_{p, i}^\tau M $, where $ \tau\in[-1, 1] $ and $ i = 0, 1, \cdots, n-1 $, then (3.4) gives

$ \begin{align} \begin{aligned} W_{p, j}(K, \Pi_{p, i}^\tau M)&\leq W_{p, j}(L, \Pi_{p, i}^\tau M). \end{aligned} \end{align} $

(3.5)

By (3.1), we see that (3.5) can be written as the following inequality

$ \begin{align} \begin{aligned} W_{p, i}(M, \Pi_{p, j}^\tau K)&\leq W_{p, i}(M, \Pi_{p, j}^\tau L). \end{aligned} \end{align} $

(3.6)

Taking $ M = \Pi_{p, j}^\tau L $ in (3.6), and using (2.5) and inequality (2.6), we get

$ \begin{aligned} W_i(\Pi_{p, j}^\tau L)\geq W_{p, i}(\Pi_{p, j}^\tau L, \Pi_{p, j}^\tau K) \geq W_i(\Pi_{p, j}^\tau L)^{\frac{n-p-i}{n-i}}W_i(\Pi_{p, j}^\tau K)^{\frac{p}{n-i}}, \end{aligned} $

namely,

$ \begin{align} \begin{aligned} W_i(\Pi_{p, j}^\tau L)^{\frac{p}{n-i}}&\geq W_i(\Pi_{p, j}^\tau K)^{\frac{p}{n-i}}. \end{aligned} \end{align} $

(3.7)

Notice that $ 0\leq i<n $ and $ p\geq1 $, then inequality (3.7) can be expressed by

$ \begin{aligned} W_i(\Pi_{p, j}^\tau K)&\leq W_i(\Pi_{p, j}^\tau L), \end{aligned} $

this is just inequality (1.13).

According to the equality conditions of inequality (2.6), we see that equality holds in inequality (1.13) for $ p = 1 $ if and only if $ \Pi_{p, j}^\tau K $ and $ \Pi_{p, j}^\tau L $ are translates, for $ p>1 $ if and only if $ \Pi_{p, j}^\tau K = \Pi_{p, j}^\tau L $.

Proof of Theorem 1.2   For $ K, L\in\mathcal{K}_o^n $, $ p\geq1 $, $ j = 0, 1, \cdots, n-1 $, and for any $ Q\in\mathcal{K}_o^n $,

$ \begin{aligned} W_{p, j}(K, Q)&\leq W_{p, j}(L, Q), \end{aligned} $

so, let $ Q = \Gamma_{p, i}^\tau M $ for any $ M\in\mathcal{S}_o^n $, where $ \tau\in[-1, 1] $ and real $ i\neq n, n+p $. We get

$ \begin{aligned} W_{p, j}(K, \Gamma_{p, i}^\tau M)&\leq W_{p, j}(L, \Gamma_{p, i}^\tau M). \end{aligned} $

From (3.2), we know that

$ \begin{align} \begin{aligned} \widetilde{W}_{-p, i}(M, \Pi_{p, j}^{\tau, \ast} K)&\leq \widetilde{W}_{-p, i}(M, \Pi_{p, j}^{\tau, \ast} L). \end{aligned} \end{align} $

(3.8)

For $ i<n $ or $ n<i<n+p $, taking $ M = \Pi_{p, j}^{\tau, \ast}L $ in inequality (3.8), and using (2.10) and inequality (2.11), we obtain that

$ \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L)&\geq \widetilde{W}_{-p, i}(\Pi_{p, j}^{\tau, \ast}L, \Pi_{p, j}^{\tau, \ast}K)\\ &\geq\widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L)^{\frac{n+p-i}{n-i}} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)^{-\frac{p}{n-i}}, \end{aligned} $

that is

$ \begin{align} \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)^{-\frac{p}{n-i}}&\leq \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L)^{-\frac{p}{n-i}}. \end{aligned} \end{align} $

(3.9)

Therefore, for $ i<n $, inequality (3.9) has the following simple form

$ \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)&\geq \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L), \end{aligned} $

this yields inequality (1.14); for $ n<i<n+p $, inequality (3.9) shows

$ \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)&\leq \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L), \end{aligned} $

i.e., inequality (1.15) is obtained.

Similarly, for $ i>n+p $, taking $ M = \Pi_{p, j}^{\tau, \ast}K $ in (3.8), and utilizing (2.10) and inequality (2.12), we easily obtain that

$ \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)&\leq \widetilde{W}_{-p, i}(\Pi_{p, j}^{\tau, \ast}K, \Pi_{p, j}^{\tau, \ast}L)\\ &\leq\widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)^{\frac{n+p-i}{n-i}} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L)^{-\frac{p}{n-i}}, \end{aligned} $

namely,

$ \begin{aligned} \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}K)^{-\frac{p}{n-i}}&\leq \widetilde{W}_i(\Pi_{p, j}^{\tau, \ast}L)^{-\frac{p}{n-i}}, \end{aligned} $

notice that $ i>n+p $, we get inequality (1.15).

According to equality conditions of inequalities (2.11) and (2.12), we know that for $ i\neq n+p $, equality holds in (1.14) or (1.15) if and only if $ \Pi_{p, j}^{\tau, \ast}K = \Pi_{p, j}^{\tau, \ast}L $, i.e., $ \Pi_{p, j}^{\tau}K = \Pi_{p, j}^{\tau}L $. For $ i = n+p $, by (3.8) and (2.9) we know that inequality (1.15) still holds.

Proof of Theorem 1.3   For $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, real $ i\neq n $ and any $ Q\in\mathcal{S}_o^n $, since $ \widetilde{W}_{-p, i}(K, Q)\leq\widetilde{W}_{-p, i}(L, Q) $, therefore, for any $ M\in\mathcal{K}_o^n $, $ \tau\in[-1, 1] $ and $ j = 0, 1, \cdots, n-1 $, let $ Q = \Pi_{p, j}^{\tau, \ast}M $, we get

$ \begin{aligned} \widetilde{W}_{-p, i}(K, \Pi_{p, j}^{\tau, \ast}M)&\leq \widetilde{W}_{-p, i}(L, \Pi_{p, j}^{\tau, \ast}M). \end{aligned} $

Together with (3.2), we obtain

$ \begin{align} \begin{aligned} V(K)W_{p, j}(M, \Gamma_{p, i}^\tau K)&\leq V(L)W_{p, j}(M, \Gamma_{p, i}^\tau L). \end{aligned} \end{align} $

(3.10)

Taking $ M = \Gamma_{p, i}^\tau L $ in inequality (3.10), and using (2.4) and inequality (2.6), we have

$ \begin{aligned} V(L)W_j(\Gamma_{p, i}^\tau L)&\geq V(K)W_{p, j}(\Gamma_{p, i}^\tau L, \Gamma_{p, i}^\tau K)\\ &\geq V(K)W_j(\Gamma_{p, i}^\tau L)^{\frac{n-p-j}{n-j}}W_j(\Gamma_{p, i}^\tau K)^{\frac{p}{n-j}}, \end{aligned} $

namely,

$ \begin{aligned} \frac{W_j(\Gamma_{p, i}^\tau K)^{-\frac{p}{n-j}}}{V(K)}&\geq \frac{W_j(\Gamma_{p, i}^\tau L)^{-\frac{p}{n-j}}}{V(L)}, \end{aligned} $

this is just inequality (1.16).

According to the condition of equality in (2.6), we know that equality holds in inequality (1.16) for $ p = 1 $ if and only if $ \Gamma_{p, i}^\tau K $ and $ \Gamma_{p, i}^\tau L $ are translates, for $ p>1 $ if and only if $ \Gamma_{p, i}^\tau K = \Gamma_{p, i}^\tau L $.

Proof of Theorem 1.4   For $ K, L\in\mathcal{S}_o^n $, $ p\geq1 $, real $ i\neq n $ and any $ Q\in\mathcal{S}_o^n $, because $ \widetilde{W}_{-p, i}(K, Q)\leq \widetilde{W}_{-p, i}(L, Q) $, thus let $ Q = \Gamma_{p, j}^{\tau, \ast}M $ for any $ M\in\mathcal{S}_o^n $, where $ \tau\in[-1, 1] $ and real $ j\neq n $, then

$ \begin{aligned} \widetilde{W}_{-p, i}(K, \Gamma_{p, j}^{\tau, \ast}M)&\leq \widetilde{W}_{-p, i}(L, \Gamma_{p, j}^{\tau, \ast}M). \end{aligned} $

From (3.3), we get

$ \begin{align} \begin{aligned} V(K)\widetilde{W}_{-p, j}(M, \Gamma_{p, i}^{\tau, \ast}K)&\leq V(L)\widetilde{W}_{-p, j}(M, \Gamma_{p, i}^{\tau, \ast}L). \end{aligned} \end{align} $

(3.11)

For $ j<n $ or $ n<j<n+p $, taking $ M = \Gamma_{p, i}^{\tau, \ast}L $ in (3.11), and together with inequality (2.11), we have

$ \begin{aligned} V(L)\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}L)&\geq V(K)\widetilde{W}_{-p, j}(\Gamma_{p, i}^{\tau, \ast}L, \Gamma_{p, i}^{\tau, \ast}K)\\ &\geq V(K)\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}L)^{\frac{n+p-j}{n-j}} \widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}K)^{-\frac{p}{n-j}}, \end{aligned} $

i.e.,

$ \begin{aligned} \frac{\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}K)^{\frac{p}{n-j}}}{V(K)} &\geq \frac{\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}L)^{\frac{p}{n-j}}}{V(L)}. \end{aligned} $

This is inequality (1.17).

For $ j>n+p $, let $ M = \Gamma_{p, i}^{\tau, \ast}K $ in (3.11), and together with inequality (2.12), we have

$ \begin{aligned} V(K)\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}K)&\leq V(L)\widetilde{W}_{-p, j}(\Gamma_{p, i}^{\tau, \ast}K, \Gamma_{p, i}^{\tau, \ast}L)\\ &\leq V(L)\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}K)^{\frac{n+p-j}{n-j}} \widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}L)^{-\frac{p}{n-j}}, \end{aligned} $

namely,

$ \begin{aligned} \frac{\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}K)^{\frac{p}{n-j}}}{V(K)} &\geq \frac{\widetilde{W}_j(\Gamma_{p, i}^{\tau, \ast}L)^{\frac{p}{n-j}}}{V(L)}, \end{aligned} $

this yields inequality (1.17).

According to equality conditions of inequalities (2.11) and (2.12), we see that for $ j\neq n+p $, equality holds in (1.17) if and only if $ \Gamma_{p, i}^{\tau, \ast}K = \Gamma_{p, i}^{\tau, \ast}L $, i.e., $ \Gamma_{p, i}^{\tau}K = \Gamma_{p, i}^{\tau}L $. For $ j = n+p $, by (3.11) and (2.9), we see that inequality (1.17) is still true.