© Wuhan University 2022

0 Introduction

Let ${\mathcal{K}}^n$ denote the set of convex bodies (compact, convex sets with non-empty interiors) in Euclidean space $R^n$. For the set of convex bodies containing the origin in their interiors, the set of origin-symmetric convex bodies, we write ${\mathcal{K}}_{\text{o}}^n$ and ${\mathcal{K}}_{\text{os}}^n$, respectively. Let V(K) denote the volume of K and $S^{n-1}$ the unit sphere.

Centroid bodies are a classical notion from geometry which have attracted increasing attention in recent years^[1-11]. In 1997, Lutwak and Zhang ^[12] introduced the concept of $L_p$ centroid body as follows: For each compact star-shaped about the origin $K\in R^n$ and $p>1$, the $L_p$ centroid body, ${\Gamma }_{p}K$, of K is the origin-symmetric convex body whose support function is defined by$h({\Gamma }_{p}K,u)=\frac{1}{\left(n+p\right)c_{n,p}V(K)}{{\displaystyle \underset{S^{n-1}}{\int }\left|u\cdot v\right|}}^p\rho {(K,v)}^{n+p}\text{d}S(v)$(1)for any $u\in S^{n-1}$. Refs. [13-19] had conducted a series of studies on the $L_p$ centroid body, and many scholars were attracted. The $L_p$ centroid body has got many results from these articles. Particularly, Refs. [20, 21] gave the Brunn-Minkowski inequality and monotonicity inequality for the $L_p$ centroid body. Grinberg and Zhang ^[22] gave the Shephard problems for the $L_p$ centroid body.

However, complex convex geometry has been studied in many works^[23-28]. In this paper, we mainly study the complex centroid body. First, we introduce some notations in complex vector space $C^n$. Let $\mathcal{C}(C^n)$ denote the set of compact convex subsets of complex vector space $C^n$. Let $\mathcal{K}(C^n)$, ${\mathcal{K}}_{\text{o}}(C^n)$ and ${\mathcal{K}}_{\text{os}}(C^n)$ denote the set of complex convex bodies, the set of complex convex bodies containing the origin in their interiors, and the set of origin symmetric complex convex bodies, respectively. Let $\mathcal{S}(C^n)$, ${\mathcal{S}}_{\text{o}}(C^n)$ and ${\mathcal{S}}_{\text{os}}(C^n)$ denote the set of complex star bodies, the set of complex star bodies containing the origin in their interiors, and the set of origin symmetric complex star bodies, respectively. $S^{2n-1}$ stands for the complex unit sphere.

Harberl^[29] firstly proposed the complex centroid body of K and established the Busemann-Petty centroid inequality. In 2021, Wu^[30] introduced the concept of the $L_p$ complex centroid body ${\Gamma }_{p,C}K$ as follows: If $p>1,K\in {\mathcal{K}}_{\text{o}}(C^n)$ and $C\in \mathcal{K}(C)$, the complex $L_p$ centroid body ${\Gamma }_{p,C}K$ is the convex body with support function$\begin{array}{l}h({\Gamma }_{p,C}K,u{)}^p\\ =\frac{1}{\left(2n+p\right)V(K)}{\displaystyle \underset{S^{n-1}}{\int }h(Cu,v{)}^p}\rho (K,v{)}^{2n+p}\text{d}S(v)\end{array}$(2)where the integration is with respect to the push forward of the Lebesgue measure under the canonical isomorphism $\eta$ and as for $\eta$, it is the canonical isomorphism between $C^n$ and $R^{2n}$, i.e.,$\eta \text{(}c\text{)=(}\mathcal{R}\text{[}{c}_1\text{],}\cdots \text{,}\mathcal{R}\text{[}{c}_n\text{],}\xi \text{[}{c}_1\text{],}\cdots \text{,}\xi \text{[}{c}_n\text{]),\hspace{0.17em}}c\in C^n$where $\mathcal{R}\text{,\hspace{0.17em}}\xi$ are the real part and imaginary part, respectively. It is obvious to get that if $p\ge 1,K\in {\mathcal{S}}_{\text{o}}(C^n)$, then${\Gamma }_{p,C}(\lambda K)=\lambda {\Gamma }_{p,C}K$(3)

In this article, associated with the definition of complex $L_p$ centroid body, we continuously study the complex $L_p$ centroid body. Let ${\Gamma }_{p,C}^{\text{*}}K$ denote the polar of ${\Gamma }_{p,C}K$ and ${\Gamma }_{p,\overline{C}}^{\text{*}}K$ denote the polar for complex conjugate of ${\Gamma }_{p,C}K$. First, we establish the Brunn-Minkowski type inequalities for complex $L_p$ centroid bodies.

Theorem 1  　If $p\ge 1,K,L\in {\mathcal{S}}_{\text{os}}(C^n)$ and $C\in$$\mathcal{K}(C)$, then$V{\left({\Gamma }_{p,C}\left(K{\widehat{+}}_{p}L\right)\right)}^{\frac{p}{2n}}\ge V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}+V{\left({\Gamma }_{p,C}L\right)}^{\frac{p}{2n}}$(4)with equality if and only if K and L are real dilation.

Theorem 2  　If $p\ge 1,K,L\in {\mathcal{S}}_{\text{os}}(C^n)$ and $C\in$$\mathcal{K}(C)$, then$V{\left({\Gamma }_{p,C}^{\text{*}}\left(K{\widehat{+}}_{p}L\right)\right)}^{-\frac{p}{2n}}\ge V{\left({\Gamma }_{p,C}^{\text{*}}K\right)}^{-\frac{p}{2n}}+V{\left({\Gamma }_{p,C}^{\text{*}}L\right)}^{-\frac{p}{2n}}$(5)with equality if and only if L and K are real dilation.

Then we obtain monotonicity inequalities for complex $L_p$ centroid bodies.

Theorem 3  　For $p\ge 1,K,L\in {\mathcal{S}}_{\text{o}}(C^n)$, $C\in \mathcal{K}(C)$, if ${\tilde{V}}_{-p}\left(K,Q\right)\le {\tilde{V}}_{-p}\left(L,Q\right)$ for any $Q\in {\mathcal{S}}_{\text{o}}(C^n)$, then$\frac{V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}}{V{(K)}^{-1}}\le \frac{V{\left({\Gamma }_{p,C}L\right)}^{\frac{p}{2n}}}{V{(L)}^{-1}}$(6)with equality if and only if K=L.

Theorem 4  　For $p\ge 1,K,L\in {\mathcal{S}}_{\text{o}}(C^n)$, $C\in \mathcal{K}(C)$, if ${\tilde{V}}_{-p}\left(K,Q\right)\le {\tilde{V}}_{-p}\left(L,Q\right)$, for any $Q\in {\mathcal{S}}_{\text{o}}(C^n)$, then$\frac{V{\left({\Gamma }_{p,C}^{*}K\right)}^{\frac{p}{2n}}}{V\left(K\right)}\ge \frac{V{\left({\Gamma }_{p,\overline{C}}^{*}L\right)}^{\frac{p}{2n}}}{V\left(L\right)}$(7)with equality if and only if K=L.

Finally, we study the $L_p$ Shephard type problem of complex $L_p$ centroid bodies and give the negative form.

Theorem 5  　Let ${\mathcal{Z}}_{p,\overline{C}}^{*}$ denote the set of polar for complex conjugate of ${\Gamma }_{p,C}K$. For $K\in {\mathcal{S}}_{\text{o}}(C^n),$$L\in {\mathcal{Z}}_{p,\overline{C}}^{*},p\ge 1$, if ${\Gamma }_{p,C}K\subset {\Gamma }_{p,C}L$, then $V(K)\le$$V(L)$ with equality if and only if K=L.

Theorem 6  　For $p\ge 1,L\in {\mathcal{S}}_{\text{o}}(C^n)$, if L is not origin symmetric star body, then there exists $K\in {\mathcal{S}}_{\text{os}}(C^n)$ such that ${\Gamma }_{p,C}K\subset {\Gamma }_{p,C}L$, but $V(K)>V(L)$.

Throughout this paper, we assume that $\dim C>0$.

1 Preliminaries

In this section, we collect complex reformulations of well-known results from convex geometry. These complex versions can be directly deduced from their real counterparts by an appropriate application of $\eta$. For standard reference, the readers may consult the books of Gardner^[31] and Schneider^[32].

1.1 Complex Support Functions and Radial Functions

For a complex number $c\in C^n$, we write $\overline{c}$ for its conjugate and |c| for its norm. If $\varphi \in C^{m\times n}$, then ${\varphi }^{\text{*}}$ denotes the conjugate transpose of $\varphi$ and if $\varphi$ is invertible, ${\varphi }^{-1}$ denotes the inverse of $\varphi$. A complex convex body $K\in \mathcal{K}(C^n)$ is uniquely determined by its support function $h\left(K,x\right):C^n\to R$,$h(K,x)=\max \{\mathcal{R}[x\cdot y]:y\in K\}$where “∙” means the standard Hermitian inner production in $C^n$ and $\mathcal{R}\left[x·y\right]$ is the real part of $x\cdot y$. It is easy to see that $h_{\lambda K}\text{=}\lambda h_K$ for all $\lambda >0$ and $h_{\varphi K}\text{=}{\varphi }^{\text{*}}\circ h_K$ for all $\varphi \in \text{GL}(n,C)$. The complex radial function ${\rho }_K(x)=\rho (K,x):C^n\backslash \left\{0\right\}\to [0,\infty )$ of a compact star- shaped (about the origin) $K$ is defined, for x∈Cⁿ\{0}, by$\rho (K,x)\text{=}\max \{\lambda \ge 0:\lambda x\in K\}$It is easy to see that ${\rho }_{\lambda K}\text{=}\lambda {\rho }_K$ for all $\lambda >0$ and ${\rho }_{\varphi K}\text{=}{\varphi }^{-1}\circ {\rho }_K$ for all $\varphi \in \text{GL}(n,C)$. If ${\rho }_K$ is positive and continuous, K will be called a star body. Moreover, if $K\in {\mathcal{K}}_{\text{o}}(C^n)$, it is easy to certify that$h_{K^{*}}=\frac{1}{{\rho }_K},{\rho }_{K^{*}}=\frac{1}{h_K}$(8)

An application of polar coordinates to the volume of a complex star body $K\in {\mathcal{S}}_{\text{o}}(C^n)$ gives that$V(K)=\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }\rho {(K,u)}^{2n}\text{d}S(K,u)}$

1.2 Complex L_p Mixed Volume and Dual L_p Mixed Volume

For $p\ge 1,K,L\in {\mathcal{K}}_{\text{o}}(C^n)$ and $\alpha ,\beta \ge 0$(not both zero), the complex $L_p$ Minkowski combination $\alpha \cdot K{+}_p\beta \cdot L$ is defined by$h{\left(\alpha \cdot K{+}_p\beta \cdot L,u\right)}^p=\alpha h{(K,u)}^p+\beta h{(L,u)}^p$The complex $L_p$ mixed volume, $V_p(K,L)$ of $K,L\in {\mathcal{K}}_{\text{o}}(C^n)$ is defined by (see Ref.[33])$\frac{2n}{p}{V}_p(K,L)=\underset{\epsilon \to 0^{+}}{\lim }\frac{V\left(K{+}_p\epsilon \cdot L\right)-V(K)}{\epsilon }$(9)By (9) we have $V_p(K,L)=V_p(\eta K,\eta L)$ and for $\varphi \in \text{GL}\left(n,C\right)$,$V_p(\varphi K,\varphi L)\text{=}{\left|\varphi \right|}^2{V}_p(K,L)$For every Borel set $\varpi \subset S^{2n-1}$, the complex surface area measure $S_K$ of $K\in \mathcal{K}(C^n)$ is defined by$S_K(\overline{\omega })={\mathcal{H}}^{2n-1}\left(\eta \left\{x\in K,\exists u\in \overline{\omega },\mathcal{R}\left[x,y\right]=h_K(u)\right\}\right)$where ${\mathcal{H}}^{2n-1}$ stands for (2n-1)-dimensional Hausdorff measure on $R^{2n}$.

In addition, the complex surface area measures are translation invariant and $S_{cK}\left(\varpi \right)=S_K\left(\overline{c}\varpi \right)$ for all $c\in S^{n-1}$ and each Borel set $\varpi \subset S^{2n-1}$. If $p\ge 1$, we define the complex $L_p$ surface area measure $S_p(K,\cdot )$ of $K\in \mathcal{K}(C^n)$ as$S_p(K,\varpi )={{\displaystyle \underset{\varpi }{\int }h(K,v)}}^{1-p}\text{d}S(K,v)$For $K,L\in {\mathcal{K}}_{\text{o}}(C^n)$, there is the $L_p$ surface area measure $S_{p,K}$ of K on $S^{2n-1}$ such that$V_p(K,L)=\frac{1}{2n}{{\displaystyle \underset{S^{2n-1}}{\int }h(L,u)}}^p\text{d}{S}_p(K,u)$(10)It turns out that the measure $S_{p,K}$ is absolutely continuous with respect to $S_K$ and has Radon Nikodym derivative $\text{d}{S}_{p,K}/\text{d}{S}_K=h_K^{1-p}$. There is the complex $L_p$ Minkowski inequality for complex convex body: If $p\ge 1,K,L\in {\mathcal{K}}_{\text{o}}(C^n)$, then$V_p{\left(K,L\right)}^{2n}\ge V{\left(K\right)}^{2n-p}V{\left(L\right)}^p$(11)with equality if and only if K and L are real dilation. The real $L_p$ Minkowski inequality and its proof are shown in Ref . [32] .

For $p\ge 1,K,L\in {\mathcal{S}}_{\text{o}}(C^n)$ and $\alpha ,\beta \ge 0$(not both zero), the complex $L_p$ harmonic radial combination $\alpha \cdot K{+}_{-p}\beta \cdot L$ is defined by$\rho {\left(\alpha \cdot K{\widetilde{+}}_{-p}\beta \cdot L\right)}^{-p}=\alpha \rho {(K,u)}^{-p}+\beta \rho {(L,u)}^{-p}$Then the dual complex $L_p$ mixed volume ${\tilde{V}}_{-p}(K,L)$ is defined by (see Ref.[33])${\tilde{V}}_{-p}(K,L)=-\frac{p}{2n}\underset{\epsilon \to 0^{+}}{\lim }\frac{\tilde{V}\left(K{\widetilde{+}}_{-p}\epsilon \cdot L\right)-\tilde{V}(K)}{\epsilon }$The polar coordinate formula for volume yields${\tilde{V}}_{-p}(K,L)=\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }\rho {(K,u)}^{2n+p}}\rho {(L,u)}^{-p}\text{d}S(u)$(12)Particularly, ${\tilde{V}}_{-p}(K,K)=V(K)$.

The integral representation (12), together with the Hӧlder inequality^[34] immediately gives that${\tilde{V}}_{-p}{(K,L)}^{2n}\ge V{(K)}^{2n+p}V{(L)}^{-p}$(13)with equality if and only if K and L are real dilation. For the real $L_p$ harmonic radial combination and real $L_p$ dual Minkowski inequality, we refer to Ref. [35].

1.3 The Complex L_p Harmonic Blaschke Combination

The notion of real $L_p$ harmonic Blaschke combination was given by Lu and Leng^[36]. Then, we extend real $L_p$ harmonic Blaschke combination to the complex case.

For $p\ge 1,K,L\in {\mathcal{S}}_{\text{o}}(C^n)$ and $\lambda ,\mu \ge 0$(not both zero), the complex $L_p$ harmonic Blaschke combination $\lambda \ast K{\widehat{+}}_p\mu \ast L$ of K and L is defined by$\frac{\rho {\left(\lambda \ast K{\widehat{+}}_p\mu \ast L,\cdot \right)}^{2n+p}}{V\left(\lambda \ast K{\widehat{+}}_p\mu \ast L\right)}=\lambda \frac{\rho {(K,\cdot )}^{2n+p}}{V(K)}+\mu \frac{\rho {(L,\cdot )}^{2n+p}}{V(L)}$(14)where $\lambda \ast K$ is $L_p$ harmonic Blaschke scalar multiplication and $\lambda \ast K={\lambda }^{\frac{1}{p}}K.$ Taking $\lambda \text{=}\mu \text{=}\frac{1}{2},K=L$ in $\lambda \ast K{\widehat{+}}_p\mu \ast L$, then the complex $L_p$ harmonic Blaschke body ${\stackrel{⌢}{\nabla }}_{p,C}K$ is introduced by${\stackrel{⌢}{\nabla }}_{p,C}K=\frac{1}{2}\ast K{\widehat{+}}_p\frac{1}{2}\ast (-K)$(15)Obviously, ${\stackrel{⌢}{\nabla }}_{p,C}K$ is origin symmetric.

2 Proofs of Theorems

In this section, we will prove Theorem 1-Theorem 6.

Proof of Theorem 1  　For $p\ge 1$ and $C\in \mathcal{K}(C)$, the $L_p$ harmonic Blaschke combination (14) together with (2) yields$\begin{array}{l}h{\left({\Gamma }_{p,C}\left(\lambda \ast K{\widehat{+}}_p\mu \ast L\right),u\right)}^p\\ =\lambda h{\left({\Gamma }_{p,C}K,u\right)}^p+\mu h{\left({\Gamma }_{p,C}L,u\right)}^p\end{array}$(16)

From (10) and for any $Q\in {\mathcal{S}}_{\text{o}}(C^n)$, we obtain$\begin{array}{l}{V}_p\left(Q,{\Gamma }_{p,C}\left(K{\widehat{+}}_{p}L\right)\right)\\ =\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }h{\left({\Gamma }_{p,C}\left(K{\widehat{+}}_{p}L\right),u\right)}^p}\text{d}{S}_p\left(Q,u\right)\\ =\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }(h{({\Gamma }_{p,C}K,u)}^p+h{({\Gamma }_{p,C}K,u)}^p})\text{d}{S}_p(Q,u)\\ =V_p\left(Q,{\Gamma }_{p,C}K\right)+V_p\left(Q,{\Gamma }_{p,C}L\right)\end{array}$Therefore, by (11), we get$\begin{array}{l}{V}_p\left(Q,{\Gamma }_{p,C}\left(K{\widehat{+}}_{p}L\right)\right)\\ \ge V{\left(Q\right)}^{\frac{2n+p}{2n}}\left(V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}+V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}\right)\end{array}$(17)with equality if and only if $Q,{\Gamma }_{p,C}K$ and ${\Gamma }_{p,C}L$ are real dilation. Taking $Q={\Gamma }_{p,C}(K{\widehat{+}}_{p}L)$ in (17), one has$V{\left({\Gamma }_{p,C}\left(K{\widehat{+}}_{p}L\right)\right)}^{\frac{p}{2n}}\ge V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}+V{\left({\Gamma }_{p,C}L\right)}^{\frac{p}{2n}}$Together (16) with the equality condition of (17), we know that the equality holds if and only if K and L are real dilation.

Proof of Theorem 2  　From (8) and (16), one has$\begin{array}{l}\rho {\left(\left({\Gamma }_{p,C}^{\text{*}}\lambda \ast K{\widehat{+}}_p\mu \ast L\right),u\right)}^{-p}\\ =\lambda \rho {\left({\Gamma }_{p,C}^{\text{*}}K,u\right)}^{-p}+\mu \rho {\left({\Gamma }_{p,C}^{\text{*}}L,u\right)}^{-p}\end{array}$(18)

Then by (12) and the inverse Minkowski’s integral inequality^[34], we obtain$$\begin{array}{l}{V}_{-p}{\left(Q,{\Gamma }_{p,C}^{\text{*}}\left(K{\widehat{+}}_{p}L\right)\right)}^{-\frac{p}{2n}}\\ ={\left(\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }{\left(\rho {\left({\Gamma }_{p,C}^{\text{*}}\left(K{\widehat{+}}_{p}L\right),u\right)}^{-p}\right)}^{-\frac{2n}{p}}}\text{d}{S}_p\left(Q,u\right)\right)}^{-\frac{p}{2n}}\\ ={\left(\frac{1}{2n}{{\displaystyle \underset{S^{2n-1}}{\int }\left(\rho {\left({\Gamma }_{p,C}^{\text{*}}K,u\right)}^{-p}+\rho {\left({\Gamma }_{p,C}^{\text{*}}L,u\right)}^{-p}\right)}}^{-\frac{2n}{p}}\text{d}{S}_p\left(Q,u\right)\right)}^{-\frac{p}{2n}}\\ \ge {\left(\frac{1}{2n}{{\displaystyle \underset{S^{2n-1}}{\int }\rho \left({\Gamma }_{p,C}^{\text{*}}K,u\right)}}^{2n}\text{d}{S}_p\left(Q,u\right)\right)}^{-\frac{p}{2n}}\\ +{\left(\frac{1}{2n}{{\displaystyle \underset{S^{2n-1}}{\int }\rho \left({\Gamma }_{p,C}^{\text{*}}L,u\right)}}^{2n}\text{d}{S}_p\left(Q,u\right)\right)}^{-\frac{p}{2n}}\\ \text{=}{V}_{-p}{\left(Q,{\Gamma }_{p,C}^{\text{*}}K\right)}^{-\frac{p}{2n}}+V_{-p}{\left(Q,{\Gamma }_{p,C}^{\text{*}}L\right)}^{-\frac{p}{2n}}\end{array}$$(19)Taking $Q={\Gamma }_{p,C}^{*}(K{\widehat{+}}_{p}L)$ in (19) and by (13), one yields the inequality (6). According to the equality conditions of Minkowski’s integral inequalities, we see that equality holds in (19) if and only if K and L are real dilation.

Next, we turn to prove Theorem 3 and Theorem 4. Lemma 1 provides a connection of ${\Gamma }_{p,C}K$ and ${\Pi }_{p,\overline{C}}^{\text{*}}K$ in terms of mixed volumes and their dual.

Lemma 1  　If $C\in \mathcal{K}(C)$ and $K,L\in {\mathcal{S}}_{\text{o}}(C^n)$, then,$V_p\left(K,{\Gamma }_{p,C}L\right)=\frac{1}{(2n+p)V(L)}{\tilde{V}}_{-p}\left(L,{\Pi }_{p,\overline{C}}^{*}K\right)$(20)

Proof  　From (3), (8), (10) and definition of $L_p$ projection body^[29], we have$\begin{array}{l}{V}_p\left(K,{\Gamma }_{p,C}L\right)\\ =\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }h{\left({\Gamma }_{p,C}L,u\right)}^p}\text{d}{S}_p(K,u)\\ =\frac{1}{2n(2n+p)V(L)}{{\displaystyle \underset{S^{2n-1}}{\int }{\displaystyle \underset{S^{2n-1}}{\int }\rho (L,u)}}}^{2n\text{+}p}\\ \times h(Cu,v{)}^p\text{d}S(v)\text{d}{S}_p(K,u)\\ =\frac{1}{2n(2n+p)V(L)}{{\displaystyle \underset{S^{2n-1}}{\int }\rho (L,u)}}^{2n\text{+}p}\\ \times h{\left({\Pi }_{p,\overline{C}}K,u\right)}^p\text{d}S(v)\\ =\frac{1}{2n(2n+p)V(L)}{{\displaystyle \underset{S^{2n-1}}{\int }\rho (L,u)}}^{2n\text{+}p}\\ \times \rho {\left({\Pi }_{p,\overline{C}}^{\text{*}}K,u\right)}^{-p}\text{d}S(v)\\ =\frac{1}{(2n+p)V(L)}{\tilde{V}}_{-p}\left(L,{\Pi }_{p,\overline{C}}^{*}K\right)\end{array}$which ends the proof of Lemma 1.

Lemma 2  　 If $p\ge 1,K,L\in {\mathcal{S}}_{\text{o}}(C^n)$, then$\frac{{\tilde{V}}_{-p}\left(K,{\Gamma }_{p,\overline{C}}^{\text{*}}L\right)}{V(K)}=\frac{{\tilde{V}}_{-p}\left(L,{\Gamma }_{p,C}^{*}K\right)}{V(L)}$(21)

Proof  　 From (3), (8) and (12), it easily gets$\begin{array}{l}{\tilde{V}}_{-p}\left(L,{\Gamma }_{p,C}^{\text{*}}K\right)\\ =\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }\rho {(L,u)}^{2n+p}\rho {\left({\Gamma }_{p,C}^{\text{*}}K,u\right)}^{-p}}\text{d}S(u)\\ =\frac{1}{2n}{\displaystyle \underset{S^{2n-1}}{\int }\rho {(L,u)}^{2n+p}h{\left({\Gamma }_{p,C}K,u\right)}^p}\text{d}S(u)\\ =\frac{1}{2n(2n+p)V(K)}{\displaystyle \underset{S^{2n-1}}{\int }{\displaystyle \underset{S^{2n-1}}{\int }\rho {(L,u)}^{2n+p}}}\\ \times \rho (K,u{)}^{2n+p}h(Cu,v{)}^p\text{d}S(v)\text{d}S(u)\\ =\frac{V(L)}{2nV(K)}{{\displaystyle \underset{S^{2n-1}}{\int }\rho (K,v)}}^{2n\text{+}p}\rho {\left({\Gamma }_{p,\overline{C}}^{\text{*}}L,u\right)}^{-p}\text{d}S(v)\\ =\frac{V(L)}{V(K)}{\tilde{V}}_{-p}\left(K,{\Gamma }_{p,\overline{C}}^{\text{*}}L\right)\end{array}$That is to say,$\frac{{\tilde{V}}_{-p}\left(L,{\Gamma }_{p,C}^{\text{*}}K\right)}{V(L)}=\frac{{\tilde{V}}_{-p}\left(K,{\Gamma }_{p,\overline{C}}^{\text{*}}L\right)}{V(K)}$which yields (21).

Remark 1  　If $p\ge 1,K\in {\mathcal{S}}_{\text{o}}(C^n)$, then ${\tilde{V}}_{-p}(K,$${\Gamma }_{p,\overline{C}}^{*}{\Gamma }_{p,C}^{*}K)\ge V(K).$ If K is a central ellipsoid or an Hermitian ellipsoid, then the equality holds.

Now we are in a position to prove Theorem 3 and Theorem 4.

Proof of Theorem 3  　Since $K,L\in {\mathcal{S}}_{\text{o}}(C^n)$ and ${\tilde{V}}_{-p}(K,Q)\le {\tilde{V}}_{-p}(L,Q)$ for any $Q\in {\mathcal{K}}_{\text{o}}(C^n)$, then taking $Q\text{=}{\Gamma }_{p,\overline{C}}^{*}M$ for any $M\in {\mathcal{S}}_{\text{o}}(C^n)$, we have${\tilde{V}}_{-p}\left(K,{\Gamma }_{p,\overline{C}}^{*}M\right)\le {\tilde{V}}_{-p}\left(L,{\Gamma }_{p,\overline{C}}^{*}M\right)$(22)with equality if and only if K = L. By Lemma 1, we obtain$V(K)V_p\left(M,{\Gamma }_{p,C}K\right)\le V(L)V_p\left(M,{\Gamma }_{p,C}L\right)$(23)

Taking $M={\Gamma }_{p,C}L$ in (23) and by (11), one has$\begin{array}{l}V(L)V\left({\Gamma }_{p,C}L\right)\\ \ge V(K)V_p\left({\Gamma }_{p,C}L,{\Gamma }_{p,C}K\right)\\ \ge V(K)V{\left({\Gamma }_{p,C}L\right)}^{\frac{2n-p}{2n}}V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}\end{array}$(24)with equality in the second inequality of (24) if and only if ${\Gamma }_{p,C}K$ and ${\Gamma }_{p,C}L$ are real dilation. Thus, it follows from (24) that we have$V(L)V{\left({\Gamma }_{p,C}L\right)}^{\frac{p}{2n}}\ge V(K)V{\left({\Gamma }_{p,C}K\right)}^{\frac{p}{2n}}$i.e.,$\frac{V({\Gamma }_{p,C}K{)}^{\frac{p}{2n}}}{V{(K)}^{-1}}\le \frac{V({\Gamma }_{p,C}L{)}^{\frac{p}{2n}}}{V{(L)}^{-1}}$(25)

From Lemma 1, we see that inequalities (22) and (23) are equivalent. Thus, equality holds in (25) if and only if K = L.

Proof of Theorem 4  　Since ${\tilde{V}}_{-p}(K,Q)\le$${\tilde{V}}_{-p}(L,Q)$ for any $Q\in {\mathcal{K}}_{\text{o}}(C^n)$, then, taking $Q\text{=}{\Gamma }_{p,C}^{*}M$ for any $M\in {\mathcal{S}}_{\text{o}}(C^n)$, we get${\tilde{V}}_{-p}(K,{\Gamma }_{p,C}^{*}M)\le {\tilde{V}}_{-p}(L,{\Gamma }_{p,C}^{*}M)$(26)with equality if and only if K = L. Combining (21) and (26), we obtain$\frac{V(K){\tilde{V}}_{-p}\left(M,{\Gamma }_{p,\overline{C}}^{*}K\right)}{V(M)}\le \frac{V(L){\tilde{V}}_{-p}\left(M,{\Gamma }_{p,\overline{C}}^{*}L\right)}{V(M)}$(27)Taking $M={\Gamma }_{p,\overline{C}}^{*}L$ and by (13), it yields$\begin{array}{l}V(L)V\left({\Gamma }_{p,\overline{C}}^{*}L\right)\\ \ge V(K){\tilde{V}}_{-p}\left({\Gamma }_{p,\overline{C}}^{*}L,{\Gamma }_{p,C}^{*}K\right)\\ \ge V(K)V{\left({\Gamma }_{p,\overline{C}}^{*}L\right)}^{\frac{2n+p}{2n}}V{\left({\Gamma }_{p,C}^{*}K\right)}^{-\frac{p}{2n}}\end{array}$(28)with equality in the second inequality of (30) if and only if ${\Gamma }_{p,\overline{C}}^{*}L$ and ${\Gamma }_{p,C}^{*}K$ are real dilation. Thus, it follows from (28) that we have$\frac{V({\Gamma }_{p,C}^{*}K{)}^{\frac{p}{2n}}}{V(K)}\ge \frac{V({\Gamma }_{p,\overline{C}}^{*}L{)}^{\frac{p}{2n}}}{V(L)}$(29)with equality if and only if K = L.

Now, we are dedicated to proving Theorem 5 and Theorem 6.

Proof of Theorem 5  　For $p\ge 1$ and $M\in$${\mathcal{S}}_{\text{o}}(C^n)$, it follows from the Lemma 2,$\begin{array}{l}\frac{{\tilde{V}}_{-p}\left(K,{\Gamma }_{p,\overline{C}}^{\text{*}}M\right)}{V(K)}=\frac{{\tilde{V}}_{-p}\left(M,{\Gamma }_{p,C}^{*}K\right)}{V(M)},\\ \frac{{\tilde{V}}_{-p}\left(L,{\Gamma }_{p,\overline{C}}^{\text{*}}M\right)}{V(L)}=\frac{{\tilde{V}}_{-p}\left(M,{\Gamma }_{p,C}^{*}L\right)}{V(M)}\end{array}$(30)Since ${\Gamma }_{p,C}K\subset {\Gamma }_{p,C}L$, then ${\Gamma }_{p,C}^{*}L\subset {\Gamma }_{p,C}^{*}K$, hence for all $u\in S^{2n-1}$, we have$\rho {\left({\Gamma }_{p,C}^{*}L\right)}^{-p}\subset \rho {\left({\Gamma }_{p,C}^{*}K\right)}^{-p}$(31)Combining (30) and (31), we get$\frac{{\tilde{V}}_{-p}\left(K,{\Gamma }_{p,\overline{C}}^{\text{*}}M\right)}{V(K)}\le \frac{{\tilde{V}}_{-p}\left(L,{\Gamma }_{p,\overline{C}}^{*}M\right)}{V(L)}$(32)For $L\in {\mathcal{Z}}_{p,\overline{C}}^{*}$ and taking ${\Gamma }_{p,\overline{C}}^{\text{*}}M$ for L in (32), then from (13), we get $V(K)\le V(L)$ with equality if and only if K = L.

Proof of Theorem 6  　By (3), (15) and (16), we have$\begin{array}{l}h{\left({\Gamma }_{p,C}\left({\widehat{\nabla }}_{p,C}K\right),u\right)}^p\\ =h{\left({\Gamma }_{p,C}\left(\frac{1}{2}\ast K{\widehat{+}}_p\frac{1}{2}\ast \left(-K\right)\right),u\right)}^p\\ =\frac{1}{2}h{\left({\Gamma }_{p,C}K,u\right)}^p+\frac{1}{2}h{\left({\Gamma }_{p,C}\left(-K\right),u\right)}^p\\ =h{\left({\Gamma }_{p,C}K,u\right)}^p\end{array}$(33)Meanwhile, according to (12), (13) and (14), it yields$$\begin{array}{l}\frac{{\tilde{V}}_{-p}\left(\lambda \ast K{\widehat{+}}_p\mu \ast L,Q\right)}{V\left(\lambda \ast K{\widehat{+}}_p\mu \ast L\right)}\\ =\lambda \frac{{\tilde{V}}_{-p}\left(K,Q\right)}{V(K)}+\mu \frac{{\tilde{V}}_{-p}\left(L,Q\right)}{V(L)}\\ \ge V{\left(Q\right)}^{-\frac{p}{2n}}\left[\lambda V{\left(K\right)}^{\frac{p}{2n}}+\mu V{\left(L\right)}^{\frac{p}{2n}}\right]\end{array}$$(34)Taking $Q=\lambda \ast K{\widehat{+}}_p\mu \ast L$ and $\lambda \text{=}\mu \text{=}\frac{1}{2},K=L$ in (34), then $V({\widehat{\nabla }}_{p,C}L)\ge V(L)$ with equality if and only if L is an origin symmetric body.