© Wuhan University 2025

0 Introduction

The setting for this paper is $n$-dimensional Euclidean space ${\mathbb{R}}^n$. Denote by $B$ and $S^{n-\mathrm{1}}$ the unit ball and its boundary in ${\mathbb{R}}^n$, respectively. If $L$ is a star-shaped set with respect to the origin in ${\mathbb{R}}^n,$ then its radial function ${\rho }_L:S^{n-\mathrm{1}}\to [\mathrm{0},\mathrm{\infty })$ is defined, for $u\in S^{n-\mathrm{1}}$, by

${\rho }_L(u)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\lambda \ge \mathrm{0}:\lambda u\in L\right\}$(1)

And $L$ is called a star body about the origin if ${\rho }_L(\cdot )$ is continuous and positive. Denote by ${\mathcal{S}}_o^n$ the class of star bodies about the origin in ${\mathbb{R}}^n$ and by ${\mathcal{S}}_e^n$ the class of origin-symmetric star bodies in ${\mathbb{R}}^n$.

Intersection bodies were first named by Lutwak^[1] and have been intensively studied^[2-13]. With the advent of the 21st century, the research on intersection bodies has expanded in scope, encompassing a broader range of disciplines^[14-23]. Haberl and Ludwig^[24] defined a characterization of $L_p$ intersection bodies. Berck^[25] studied the convexity of the $L_p$-intersection bodies of origin-symmetric convex bodies. Wang and Li^[26] extended the general $L_p$ intersection bodies and settled its Busemann-Petty type problem. Intersection bodies have been identified as crucial in solving the Busemann-Petty type problem^{[2,8,11,13,21-22,26]}.

If $L$ is a star body, then the intersection body, $IL$, of $L$ is defined as follows. Its radial function in the direction $u\in S^{n-\mathrm{1}}$ equals the $(n-\mathrm{1})$-dimensional volume of $L$ section by $u^{\perp }$, i.e., for all $u\in S^{n-\mathrm{1}}$, $\rho (IL,u)=\mathrm{v}\mathrm{o}{\mathrm{l}}_{n-\mathrm{1}}\left(L\bigcap u^{\perp }\right),$$$ where $u^{\perp }$ denotes the hyperplane orthogonal to $u$.

Liu and Wang^[27] extended the $L_p$ intersection bodies to $L_p$ mixed intersection bodies. If $L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0}$) and $\mathrm{0}\le i\le n-\mathrm{1}$, then the $L_p$ mixed intersection body, $I_{p,i}L,$ of $L$ is an origin-symmetric star body, whose radial function is as follows,

$\rho (I_{p,i}L,u)=(\frac{\mathrm{1}}{n-p}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(v)|〈v,u〉{|}^{-p}{\mathrm{d}v)}^{\frac{\mathrm{1}}{p}},\forall u\in S^{n-\mathrm{1}}$(2)

In this paper, we first define the normalized $L_p$ mixed intersection body as follows. If $L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0}$) and $\mathrm{0}\le i\le n-\mathrm{1}$, then the normalized $L_p$ mixed intersection body, ${\overline{I}}_{p,i}L,$ of $L$ is an origin-symmetric star body, whose radial function is defined by

$\rho ({\overline{I}}_{p,i}L,u)=(\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(v)|〈v,u〉{|}^{-p}{\mathrm{d}v)}^{\frac{\mathrm{1}}{p}},\forall u\in S^{n-\mathrm{1}}.$(3)

From (2) and (3), we have

$I_{p,i}L={\tilde{W}}_i{(L)}^{-\frac{\mathrm{1}}{P}}{\overline{I}}_{p,i}L.$

One aim of this paper is to research the dual Brunn-Minkowski inequality for the normalized $L_p$ mixed intersection bodies in ${\mathbb{R}}^n$.

Theorem 1   If $K,L\in {\mathcal{S}}_o^n,$ and $\lambda ,\mu \ge \mathrm{0}$ (not both zero), $\mathrm{0}<p<\mathrm{1},$$\mathrm{0}\le i\le n-\mathrm{1}$, then

$V({\overline{I}}_{p,i}(\lambda \cdot K{+}_{-p}{\mu \cdot L))}^{\frac{p}{n}}\le \lambda V({\overline{I}}_{p,i}{K)}^{\frac{p}{n}}+\mu V({\overline{I}}_{p,i}{L)}^{\frac{p}{n}}$(4)

with the equality holds if $K$ is a dilation of $L$. If $p<\mathrm{0}$, the inequality (4) is reversed. Let ${+}_{-p}$ denotes the $L_{-p}$ harmonic Blaschke radial sum.

Studying the normalized $L_p$ Busemann-Petty problem is another aim of this paper.

Theorem 2   Let $L\in {\mathcal{S}}_o^n$ and $K$ be a normalized $L_p$ mixed intersection body, $\mathrm{0}<p<\mathrm{1}$ (or $p<\mathrm{0}$), and $\mathrm{0}\le i\le n-\mathrm{1}$. If ${\overline{I}}_{p,i}K\subseteq {\overline{I}}_{p,i}L$, then ${\tilde{W}}_i(K)\ge {\tilde{W}}_i(L)$, with the equality holds if and only if $K=L$.

1 Notation and Preliminaries

We refer to Schneider's works as a general source on the theory of convex (star) bodies ^[19].

Denote by $V(L)$ the volume of the compact set $L$ in ${\mathbb{R}}^n$ and by ${\tilde{W}}_{p,i}(L)$ the $L_p$ dual mixed quermassintegrals of the compact set $L$ in ${\mathbb{R}}^n$.

If $L\in {\mathcal{S}}_o^n$, then the support function, h_L, of L is defined by h_L(u)=max{〈x,u〉,x∈L},∀u∈S^n-1. Hence,

$h_L(-u)=h_{-L}(u),$(5)

where $-L=\left\{-x:x\in L\right\}$.

If $L\subset {\mathbb{R}}^n$ is a convex body with the origin in its interior, then the polar body, $L^{\mathrm{*}},$ of $L$ is defined by

$L^{\mathrm{*}}=\left\{x\in {\mathbb{R}}^n|〈x,y〉\le \mathrm{1},\forall y\in L\right\}.$

Obviously, for all $u\in S^{n-\mathrm{1}}$ (see Ref. [19]),

${\rho }_{L^{\mathrm{*}}}(u)=\frac{\mathrm{1}}{h_L(u)}.$(6)

For $L\in {\mathcal{S}}_o^n$, according to the definition of the radial function, then we have

${\rho }_L(-u)={\rho }_{-L}(u),\forall u\in S^{n-\mathrm{1}}$(7)

If $\frac{{\rho }_K(u)}{{\rho }_L(u)}$ is independent of $u\in S^{n-\mathrm{1}}$, then say the star body $K$ is a dilation of $L$. Let $K,L\in {\mathcal{S}}_o^n$, for all $u\in S^{n-\mathrm{1}}$,

$K\subseteq L\iff {\rho }_K(u)\le {\rho }_L(u).$(8)

If $s>\mathrm{0}$, then we have

${\rho }_{sL}(u)=s{\rho }_L(u),\forall u\in S^{n-\mathrm{1}}$(9)

If $\varphi \in SL(n),$ then we have

${\rho }_{\varphi L}(u)={\rho }_L({\varphi }^{-\mathrm{1}}u),\forall u\in S^{n-\mathrm{1}}.$(10)

Let $\tilde{\delta }$ denote the radial Hausdorff metric. If $K,L\in {\mathcal{S}}_o^n$, then $\tilde{\delta }(K,L)=\underset{u\in S^{n-\mathrm{1}}}{\mathrm{m}\mathrm{a}\mathrm{x}}|{\rho }_K(u)-{\rho }_L(u)|.$

A sequence $\{L_i\}$ of star bodies converges to $L$ if $\tilde{\delta }(L_i,L)\to \mathrm{0}$, as $i\to \mathrm{\infty }.$ Thus, $L_i\to L,$ as $i\to \mathrm{\infty }$ if and only if ${\rho }_{L_i}(\cdot )\to {\rho }_L(\cdot ),$ uniformly, as $i\to \mathrm{\infty }.$

For $K,L\in {\mathcal{S}}_o^n$, and $\lambda ,\mu \ge \mathrm{0}$ (not both zero), then the $L_p$ radial sum $\lambda \cdot K{+}_p\mu \cdot L(p\ne \mathrm{0})$, is defined by

${\rho }_{\lambda \cdot K{+}_p\mu \cdot L}^p(u)=\lambda {\rho }_K^p(u)+\mu {\rho }_L^p(u),\forall u\in S^{n-\mathrm{1}}$(11)

For $L\in {\mathcal{S}}_o^n$ and $\mathrm{0}\le i\le n-\mathrm{1}$, the dual quermassintegral ${\tilde{W}}_i(L)$ has the following integral representation:

${\tilde{W}}_i(L)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-i}(u)\mathrm{d}u.$(12)

When $i=\mathrm{0},{\tilde{W}}_{\mathrm{0}}(L)=V(L)$. For $K,L\in {\mathcal{S}}_o^n$,$\lambda ,\mu \ge \mathrm{0}$ (not both zero), and $\mathrm{0}\le i\le n-\mathrm{1}$. By using Minkowski integral inequality, if $\mathrm{0}<p\le n-i$, then we have

${\tilde{W}}_i(\lambda \cdot K{+}_p{\mu \cdot L)}^{\frac{p}{n-i}}\le \lambda {\tilde{W}}_i{(K)}^{\frac{p}{n-i}}+\mu {\tilde{W}}_i{(L)}^{\frac{p}{n-i}}.$(13)

If $p<\mathrm{0}$ or $p>n-i$, then the above inequality (13) is reversed. With the equality holds if and only if $K$ is a dilation of $L$.

For $K,L\in {\mathcal{S}}_o^n$ and $\mathrm{0}\le i\le n-\mathrm{1}$, the $L_p$ dual mixed quermassintegrals ${\tilde{W}}_{p,i}(K,L)$ is defined by

$\frac{n-i}{p}{\tilde{W}}_{p,i}(K,L)=\underset{\epsilon \to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\tilde{W}}_i(K{+}_p\epsilon \cdot L)-{\tilde{W}}_i(K)}{\epsilon }.$

The integral representation of the $L_p$ dual mixed quermassintegrals ${\tilde{W}}_{p,i}(K,L)$ is defined by

${\tilde{W}}_{p,i}(K,L)=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_K^{n-p-i}(u){\rho }_L^p(u)\mathrm{d}u.$(14)

When $K=L,{\tilde{W}}_{p,i}(L,L)={\tilde{W}}_i(L)$, by using Hölder inequality, the $L_p$ dual mixed Minkowski inequality is established in Ref. [28].

For $K,L\in {\mathcal{S}}_o^n$ and $\mathrm{0}\le i\le n-\mathrm{1}$, if $\mathrm{0}<p\le n-i$, then

${\tilde{W}}_{p,i}(K,L)\le {\tilde{W}}_i{(K)}^{\frac{n-p-i}{n-i}}{\tilde{W}}_i{(L)}^{\frac{p}{n-i}}.$(15)

If $p<\mathrm{0}$ or $p>n-i$, then the above inequality (15) is reversed. With the equality holds if and only if $K$ is a dilation of $L$.

Denote by $C(S^{n-\mathrm{1}})$ the class of real-valued, continuous functions on $S^{n-\mathrm{1}}$, by $C_e(S^{n-\mathrm{1}})$ the subset of $C(S^{n-\mathrm{1}})$ being the even functions, and by $C_e^{+}(S^{n-\mathrm{1}})$ the subset of $C_e(S^{n-\mathrm{1}})$ being the nonnegative functions. Assume $f,g\in C(S^{n-\mathrm{1}})$, then $〈f,g〉$ is defined by

$〈f,g〉=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}f(u)g(u)\mathrm{d}u.$(16)

Let $f\in C(S^{n-\mathrm{1}})$ and $\mathrm{0}<p<\mathrm{1}(\mathrm{o}\mathrm{r}p<\mathrm{0})$, then the $L_{-p}$ cosine transform, $C_{-p}f$, of $f$ is defined by (see Ref. [29])

$(C_{-p}f)(u)={\int }_{S^{n-\mathrm{1}}}|〈u,v〉{|}^{-p}f(v)\mathrm{d}v,\forall u\in S^{n-\mathrm{1}}$(17)

It is easy to verify that the linear transformation $C_{-p}:C(S^{n-\mathrm{1}})\to C(S^{n-\mathrm{1}})$ is self-adjoint (see Ref. [26]), i.e., if $f,g\in C(S^{n-\mathrm{1}})$, then

$〈C_{-p}f,g〉=〈f,C_{-p}g〉$(18)

By (3) and (1), then we have

${\rho }_{{\overline{I}}_{p,i}L}^p=\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(L)}{C}_{-p}{\rho }_L^{n-p-i}.$(19)

2 Main Results

Lemma 1   If $L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0}),\mathrm{0}\le i\le n-\mathrm{1}$ and $c>\mathrm{0},$ then ${\overline{I}}_{p,i}(cL)=\frac{\mathrm{1}}{c}{\overline{I}}_{p,i}L$.

Proof   From (3), and (9), we obtain that

$\begin{array}{c}\begin{array}{l}{\rho }_{{\overline{I}}_{p,i}cL}^p(u)=\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(cL)}{\int }_{S^{n-\mathrm{1}}}{\rho }_{cL}^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v=\frac{\mathrm{1}}{(n-p)c^{n-i}{\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{c}^{n-p-i}{\rho }_L^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v\\ =\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{c}^{-p}{\rho }_L^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v={\rho }_{\frac{\mathrm{1}}{c}{\overline{I}}_{p,i}L}^p(u).\end{array}\end{array}$

The harmonic Blaschke radial sum was first introduced by Lutwak^[30], and the $L_p$ analog was introduced by Wang and Zhang^[31]. Let $K,L\in {\mathcal{S}}_o^n$, and $\lambda ,\mu \ge \mathrm{0}$ (not both zero), $p\ne -n,$ the $L_p$ harmonic Blaschke radial sum, $\lambda \cdot K{+}_p\mu \cdot L,$ is defined by, for all $u\in S^{n-\mathrm{1}}$,

$\frac{{\rho }_{\lambda \cdot K{+}_p\mu \cdot L}^{n+p}(u)}{V(\lambda \cdot K{+}_p\mu \cdot L)}=\frac{\lambda {\rho }_K^{n+p}(u)}{V(K)}+\frac{\mu {\rho }_L^{n+p}(u)}{V(L)}.$(20)

Similarly, if $K,L\in {\mathcal{S}}_o^n,p\ne -n+i$, and $\lambda ,\mu \ge \mathrm{0}$ (not both zero), the generalized $L_p$ harmonic Blaschke radial sum, $\lambda \cdot K{+}_p\mu \cdot L$ can be stated as

$\frac{{\rho }_{\lambda \cdot K{+}_p\mu \cdot L}^{n-i+p}(u)}{{\tilde{W}}_i(\lambda \cdot K{+}_p\mu \cdot L)}=\frac{\lambda {\rho }_K^{n-i+p}(u)}{{\tilde{W}}_i(K)}+\frac{\mu {\rho }_L^{n-i+p}(u)}{{\tilde{W}}_i(L)},\forall u\in S^{n-\mathrm{1}}$(21)

When $p=\mathrm{1},i=\mathrm{0},\lambda \cdot K{+}_{\mathrm{1}}\mu \cdot L$ is just the harmonic Blaschke radial sum $\lambda \cdot K+\mu \cdot L.$

We demonstrate that the following $L_p$ dual mixed Brunn-Minkowski inequality is more general than Theorem 1.

Theorem 3   If $K,L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1},$$\mathrm{0}\le i\le n-\mathrm{1}$, and $\lambda ,\mu \ge \mathrm{0}$, then

${\tilde{W}}_i({\overline{I}}_{p,i}(\lambda \cdot K{+}_{-p}{\mu \cdot L))}^{\frac{p}{n-i}}\le \lambda {\tilde{W}}_i({\overline{I}}_{p,i}{K)}^{\frac{p}{n-i}}+\mu {\tilde{W}}_i({\overline{I}}_{p,i}{L)}^{\frac{p}{n-i}}$(22)

with the equality holds if $K$ is a dilation of L. If $p<\mathrm{0},$ then the above inequality is reversed.

Proof   From (3) and (21), for all $u\in S^{n-\mathrm{1}}$, we have

$\begin{array}{l}{\rho }_{{\overline{I}}_{p,i}(\lambda \cdot K{+}_{-p}\mu \cdot L}^p(u)=\frac{\mathrm{1}}{n-p}{\int }_{S^{n-\mathrm{1}}}\frac{{\rho }_{\lambda \cdot K{+}_{-p}\mu \cdot L}^{n-p-i}(v)}{{\tilde{W}}_i(\lambda \cdot K{+}_{-p}\mu \cdot L)}|〈v,u〉{|}^{-p}\mathrm{d}v=\frac{\mathrm{1}}{n-p}{\int }_{S^{n-\mathrm{1}}}\left(\lambda \frac{{\rho }_K^{n-p-i}(v)}{{\tilde{W}}_i(K)}+\mu \frac{{\rho }_L^{n-p-i}(v)}{{\tilde{W}}_i(L)}\right)|〈v,u〉{|}^{-p}\mathrm{d}v\\ =\lambda {\rho }_{{\overline{I}}_{p,i}K}^p(u)+\mu {\rho }_{{\overline{I}}_{p,i}L}^p(u).\end{array}$(23)

If $\mathrm{0}<\mathit{p}<\mathrm{1}$, then $\frac{\mathit{n}-\mathit{i}}{\mathit{p}}>\mathrm{1}$. By (12), (23), and Minkowski integral inequality, it follows that

$\begin{array}{l}{\tilde{W}}_i({\overline{I}}_{p,i}(\lambda \cdot K{+}_{-p}{\mu \cdot L))}^{\frac{p}{n-i}}={\left[\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_{{\overline{I}}_{p,i}\lambda \cdot K{+}_{-p}\mu \cdot L}^{n-i}(u)\mathrm{d}u\right]}^{\frac{p}{n-i}}={\left[\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}(\lambda {\rho }_{{\overline{I}}_{p,i}K}^p(u)+\mu {\rho }_{{\overline{I}}_{p,i}L}^p{(u))}^{\frac{n-i}{p}}\mathrm{d}u\right]}^{\frac{p}{n-i}}\\ \le \lambda {\left[\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_{{\overline{I}}_{p,i}K}^{n-i}(u)\mathrm{d}u\right]}^{\frac{p}{n-i}}+\mu {\left[\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_{{\overline{I}}_{p,i}L}^{n-i}(u)\mathrm{d}u\right]}^{\frac{p}{n-i}}=\lambda {\tilde{W}}_i({\overline{I}}_{p,i}{(K))}^{\frac{p}{n-i}}+\mu {\tilde{W}}_i({\overline{I}}_{p,i}{(L))}^{\frac{p}{n-i}}.\end{array}$(24)

If $K$ and $L$ are dilations of each other, then there exists a constant c, such that $K=cL$. By Lemma 1, then we have ${\rho }_{{\overline{I}}_{p,i}K}(u)={\rho }_{{\overline{I}}_{p,i}cL}(u)={\rho }_{\frac{\mathrm{1}}{c}{\overline{I}}_{p,i}L}(u)$ for all $u\in S^{n-\mathrm{1}}$. This means that ${\overline{I}}_{p,i}K$ and ${\overline{I}}_{p,i}L$ are dilations of each another. From the equality condition of Minkowski integral inequality, the equality in (24) holds if $K$ is a dilation of L.

When $p<\mathrm{0},$ we get $\frac{n-i}{p}<\mathrm{0}$ and the inequality in (22) is reversed.

When $i=\mathrm{0},$ Theorem 3 is Theorem 1. For brevity, let ${\widehat{▽}}_{-p}L:=\frac{\mathrm{1}}{\mathrm{2}}\cdot L{+}_{-p}\frac{\mathrm{1}}{\mathrm{2}}\cdot (-L)$.

Lemma 2   If $L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0}),$ and $\mathrm{0}\le i\le n-\mathrm{1}$, then

${\tilde{W}}_i({\widehat{▽}}_{-p}L)\ge {\tilde{W}}_i(L),$(25)

with the equality holds if and only if $L$ is origin-symmetric.

Proof   By (21), one can obtain

$\frac{{\rho }_{{\widehat{▽}}_{-p}L}^{n-p-i}(u)}{{\tilde{W}}_i({\widehat{▽}}_{-p}L)}=\frac{\mathrm{1}}{\mathrm{2}}\frac{{\rho }_L^{n-p-i}(u)}{{\tilde{W}}_i(L)}+\frac{\mathrm{1}}{\mathrm{2}}\frac{{\rho }_{-L}^{n-p-i}(u)}{{\tilde{W}}_i(-L)}.$(26)

Equivalently,

${\rho }_{{\widehat{▽}}_{-p}L(u)}={\left[\frac{{\tilde{W}}_i({\widehat{▽}}_{-p}L)}{{\tilde{W}}_i(L)}\left(\frac{\mathrm{1}}{\mathrm{2}}{\rho }_L^{n-p-i}(u)+\frac{\mathrm{1}}{\mathrm{2}}{\rho }_{-L}^{n-p-i}(u)\right)\right]}^{\frac{\mathrm{1}}{n-p-i}}.$

Since $\mathrm{0}<p<\mathrm{1}$ and $\mathrm{0}\le i\le n-\mathrm{1}$, it follows from (12), (26), and Minkowski integral inequality that

$\begin{array}{l}{\tilde{W}}_i({\widehat{▽}}_{-p}{L)}^{\frac{n-p-i}{n-i}}={\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_{{\widehat{▽}}_{-p}L}^{n-i}(u)\mathrm{d}u\right)}^{\frac{n-p-i}{n-i}}={\left\{\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\left[\frac{{\tilde{W}}_i({\widehat{▽}}_{-p}L)}{{\tilde{W}}_i(L)}\left(\frac{\mathrm{1}}{\mathrm{2}}{\rho }_L^{n-p-i}(u)+\frac{\mathrm{1}}{\mathrm{2}}{\rho }_{-L}^{n-p-i}(u)\right)\right]}^{\frac{n-i}{n-p-i}}\mathrm{d}u\right\}}^{\frac{n-p-i}{n-i}}\\ \le \frac{{\tilde{W}}_i({\widehat{▽}}_{-p}L)}{\mathrm{2}{\tilde{W}}_i(L)}\left[{\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-i}(u)\mathrm{d}u)\right)}^{\frac{n-p-i}{n-i}}+{\left(\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}{\rho }_{-L}^{n-i}(u)\mathrm{d}u)\right)}^{\frac{n-p-i}{n-i}}\right]={\tilde{W}}_i({\widehat{▽}}_{-p}L){\tilde{W}}_i{(L)}^{-\frac{p}{n-i}}.\end{array}$

Thus, ${\tilde{W}}_i({\widehat{▽}}_{-p}L)\ge {\tilde{W}}_i(L).$

From the equality condition of Minkowski integral inequality, we see that equality holds if and only if $L$ and $-L$ are dilations of each other, which means that $L$ is the origin-symmetric star body.

Trivially, we can obtain the same result if $p<\mathrm{0}$.

Lemma 3   If $L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0})$ and $\mathrm{0}\le i\le n-\mathrm{1}$, then ${\overline{I}}_{p,i}({\widehat{▽}}_{-p}L)={\overline{I}}_{p,i}L.$

Proof   From (3), (7), and (26), we have that, for all $u\in S^{n-\mathrm{1}}$,

$\begin{array}{l}{\rho }_{{\overline{I}}_{p,i}({\widehat{▽}}_{-p}L)}^p(u)=\frac{\mathrm{1}}{(n-p){\tilde{W}}_i({\widehat{▽}}_{-p}L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_{{\widehat{▽}}_{-p}L}^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v\\ =\frac{\mathrm{1}}{\mathrm{2}(n-p){\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v+\frac{\mathrm{1}}{\mathrm{2}(n-p){\tilde{W}}_i(-L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_{-L}^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v\\ =\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(v)|〈v,u〉{|}^{-p}\mathrm{d}v={\rho }_{{\overline{I}}_{p,i}L}^p(u).\end{array}$

Lemma 4   If $K\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0})$and $\mathrm{0}\le i\le n-\mathrm{1}$, for all $L\in {\mathcal{S}}_e^n,$then $\frac{{\tilde{W}}_{p,i}({\widehat{▽}}_{-p}K,L)}{{\tilde{W}}_i({\widehat{▽}}_{-p}K)}=\frac{{\tilde{W}}_{p,i}(K,L)}{{\tilde{W}}_i(K)}$.

Proof   By (7), (14), and (26), we see that

$\begin{array}{l}\frac{{\tilde{W}}_{p,i}({\widehat{▽}}_{-p}K,L)}{{\tilde{W}}_i({\widehat{▽}}_{-p}K)}=\frac{\mathrm{1}}{n{\tilde{W}}_i({\widehat{▽}}_{-p}K)}{\int }_{S^{n-\mathrm{1}}}{\rho }_{{\widehat{▽}}_{-p}K}^{n-p-i}(u){\rho }_L^p(u)\mathrm{d}u=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}\frac{{\rho }_K^{n-p-i}(u)}{\mathrm{2}{\tilde{W}}_i(K)}{\rho }_L^p(u)\mathrm{d}u+\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}\frac{{\rho }_{-K}^{n-p-i}(u)}{\mathrm{2}{\tilde{W}}_i(-K)}{\rho }_L^p(u)\mathrm{d}u\\ =\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}\frac{{\rho }_K^{n-p-i}(u)}{\mathrm{2}{\tilde{W}}_i(K)}{\rho }_L^p(u)\mathrm{d}u+\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}\frac{{\rho }_K^{n-p-i}(u)}{\mathrm{2}{\tilde{W}}_i(K)}{\rho }_L^p(-u)\mathrm{d}u=\frac{\mathrm{1}}{n}{\int }_{S^{n-\mathrm{1}}}\frac{{\rho }_K^{n-p-i}(u)}{{\tilde{W}}_i(K)}{\rho }_L^p(u)\mathrm{d}u=\frac{{\tilde{W}}_{p,i}(K,L)}{{\tilde{W}}_i(K)}.\end{array}$

The following result is necessary to prove Theorem 2.

Theorem 4   If $K,L\in {\mathcal{S}}_o^n,\mathrm{0}<p<\mathrm{1}$(or $p<\mathrm{0})$ and $\mathrm{0}\le i\le n-\mathrm{1}$, then $\frac{{\tilde{W}}_{p,i}(K,{\overline{I}}_{p,i}L)}{{\tilde{W}}_i(K)}=\frac{{\tilde{W}}_{p,i}(L,{\overline{I}}_{p,i}K)}{{\tilde{W}}_i(L)}$.

Proof   From (3), (14), and Fubini's theorem, we obtain that

$\begin{array}{l}\frac{{\tilde{W}}_{p,i}(K,{\overline{I}}_{p,i}L)}{{\tilde{W}}_i(K)}=\frac{\mathrm{1}}{n{\tilde{W}}_i(K)}{\int }_{S^{n-\mathrm{1}}}{\rho }_K^{n-p-i}(u){\rho }_{{\overline{I}}_{p,i}L}^p(u)\mathrm{d}u=\frac{\mathrm{1}}{n{\tilde{W}}_i(K)}{\int }_{S^{n-\mathrm{1}}}{\rho }_K^{n-p-i}(u)\left(\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(v)|u\cdot v{|}^{-p}\mathrm{d}v\right)\mathrm{d}u\\ =\frac{\mathrm{1}}{n{\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(u)\left(\frac{\mathrm{1}}{(n-p){\tilde{W}}_i(K)}{\int }_{S^{n-\mathrm{1}}}{\rho }_K^{n-p-i}(v)|u\cdot v{|}^{-p}\mathrm{d}u\right)\mathrm{d}v=\frac{\mathrm{1}}{n{\tilde{W}}_i(L)}{\int }_{S^{n-\mathrm{1}}}{\rho }_L^{n-p-i}(v){\rho }_{{\overline{I}}_{p,i}K}^n(v)\mathrm{d}v=\frac{{\tilde{W}}_{p,i}(L,{\overline{I}}_{p,i}K)}{{\tilde{W}}_i(L)}.\end{array}$