Let $ \mathcal{K}^n $ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean $ n $ -space $ \mathbb{R}^n $; Let $ \mathcal{K}_{o}^{n} $ denote the set of convex bodies containing the origin in their interiors in $ \mathbb{R}^{n} $; $ \mathcal{S}_{o}^{n} $ denotes the set of star bodies (about the origin) in $ \mathbb{R}^n $. We use $ |K| $ to denote the $ n $ -dimensional volume of a body $ K $. Besides, write $ S^{n-1} $ for the unit sphere in $ \mathbb{R}^{n} $.

For a compact set $ K\subset\mathbb{R}^{n} $ which is star-shaped with respect to the origin, we will use $ \rho_{K}=\rho(K,\cdot) :\mathbb{R}^{n}\backslash\{o\}\rightarrow[0,\infty) $ to denote its radial function. That is,

$\nonumber \rho_{K}(x)=\rho(K,x)=\max\{λ\geq0:λ x\in K\} \ \mbox{for all}\ x\in \mathbb{R}^{n}\backslash\{o\}.$

If $ \rho_{K} $ is continuous and positive, then $ K $ is called star body. In [1], Lutwak introduced the intersection body $ IK $ of each $ K\in \mathcal{S}_{o}^{n} $, which is the star body with radial function

$\nonumber \rho(IK,u)=v(K\cap u^{\bot}),\quad u\in S^{n-1},$

where $ v(\cdot) $ denotes $ (n-1) $ -dimensional volume and $ u^{\bot} $ is the hyperplane orthogonal to $ u $.

For $ 0<p<1 $, the $ L_{p} $ -intersection body $ I_{p}K $ of $ K\in\mathcal{S}_{o}^{n} $ is defined by (see, e.g., Cardner and Giannopoulos [2], Yuan [3])

$\label{1.1} \rho(I_{p}K,u)^{p}=\int_{K}|x\cdot u|^{-p}{\rm d}x,\quad u\in S^{n-1},$

(1.1)

where $ x\cdot u $ is the usual inner product of $ x,u\in\mathbb{R}^{n} $, and integration is with respect to Lebesgue measure.

Haberl and Ludwig [4] also gave the following definition of $ L_{p} $ -intersection body for convex polytopes and investigated its characterization. There is a different constant between the $ L_{p} $ -intersection body of (1.1) and the following $ L_{p} $ -intersection body in [5]. For $ u\in S^{n-1} $ and $ 0<p<1 $, the $ L_{p} $ -intersection body of $ K\in\mathcal{S}_{o}^{n} $ is defined by

$\rho(I_{p}K,u)^{p}=\frac{1}{\Gamma(1-p)}\int_{K}|x \cdot u|^{-p}{\rm d}x,$

where $ \Gamma $ denotes the Gamma function.

Intersection body [1] played an important role for the solution of the celebrated Busemann-Petty problem, and found applications in geometric tomography [6], affine isoperimetric inequalities [7, 8] and the geometry of Banach spaces [9]. For more details about intersection body we refer the reader to [10-14].

Progress towards an Orlicz Brunn-Minkowski theory was made by Lutwak, Yang and Zhang [15, 16]. This theory plays such a crucial role that it is undeniably applied to a number of areas of geometry. The more development of the Orlicz Brunn-Minkowski theory can be found in [17-31]. Recently, Ma et al. [32] considered convex function $ \varphi:\mathbb{R}\backslash\{o\}\rightarrow(0,\infty) $ such that $ \lim\limits_{t\rightarrow\infty}\varphi(t)=0 $, $ \lim\limits_{t\rightarrow0}\varphi(t)=\infty $ and $ \varphi(0) =\infty $. This means that $ \varphi $ must be increasing on $ (-\infty,0) $ and decreasing on $ (0,\infty) $. They assumed that $ \varphi $ is either strictly increasing on $ (-\infty,0) $ or strictly decreasing on $ (0,\infty) $, and denoted by $ \Phi $ the class of such $ \varphi $. Further, for $ K\in\mathcal{S}_{o}^{n} $, they gave the concept of Orlicz intersection body $ I_{\varphi}K $ as the star body whose radial function is given by

$\label{1.2} \rho_{I_{\varphi}K}^{-1}(x)=\sup\bigg\{λ>0: \frac{1}{|K|}\int_{K}\varphi\bigg(\frac{|x\cdot y|}{λ}\bigg){\rm d}y\leq1\bigg\},\ x\in\mathbb{R}^{n},$

(1.2)

when $ \varphi(t)=|t|^{-p} $, $ 0<p<1 $, then $ I_{\varphi}K=\frac{1}{|K|}I_{p}K. $

In this paper, we also consider the above convex function $ \varphi $ and assume that $ K,L $ are two star bodies (about the origin) in $ \mathbb{R}^{n} $ with volume $ |K| $ and $ |L| $. If $ \varphi\in\Phi $, then the Orlicz mixed intersection bodies $ I_{\varphi}(K,L) $ of $ K $, $ L $ as the star body whose radial function at $ x\in\mathbb{R}^{n} $ is given by

$\label{1.3} \rho_{I_{\varphi}(K,L)}^{-1}(x)=\sup\bigg\{λ>0: \frac{1}{|K||L|}\int_{K}\int_{L}\varphi(\frac{|x\cdot(y-z)|}{λ}){\rm d}y{\rm d}z\leq1\bigg\},$

(1.3)

when $ \varphi(t)=|t|^{-p} $ with $ 0<p<1 $, and $ K=L $ in (1.3), then $ I_{\varphi}(K,K)=I_{p}(K,K),$ where the radial function of $ I_{p}(K,K) $ is given by

$\nonumber \rho(I_{p}(K,K),x)^{p}=\frac{1}{|K|^{2}}\int_{K}\int_{K}|x\cdot (y-z)|^{-p}{\rm d}y{\rm d}z.$

Our main result deals with the affine isoperimetric inequality for Orlicz mixed intersection bodies. Here and in the following, for $ K\in \mathcal{K}^{n} $, we denote by $ B_{K} $ the $ n $ -ball with the same volume as $ K $ centered at the origin.

Theorem 1.1  If $ K,L\in \mathcal{K}_{o}^{n} $ and $ \varphi\in\Phi $, then

$\begin{eqnarray}|I^{*}_{\varphi}(K,L)|\geq |I^{*}_{\varphi}(B_{K},B_{L})|\nonumber\end{eqnarray}$

with equality if $ K $ and $ L $ are dilates of each other and have the same midpoints.

For a body $ K\in \mathcal{K}_{o}^{n} $, its support function, $ h_{K}=h(K,\cdot): S^{n-1}\rightarrow\mathbb{R} $ is defined by

$h_{K}(u)=\max\{u\cdot x:x\in K\},\quad u\in S^{n-1}.$

For $ c > 0 $, the support function of the convex body $ cK=\{cy:y\in K\} $ is

$\label{ma1.1}h_{cK}(x)=ch_{K}(x),\ \ \ x\in \mathbb{R}^{n}.$

Let $ {\rm GL}(n) $ denote the group of linear transformations. For $ T\in {\rm GL}(n) $, write $ T^{t} $ for the transpose of $ T $, $ T^{-1} $ for the inverse of $ T $ and $ T^{-t} $ for the inverse of the transpose of $ T $. The support function of the image $ T K=\{Ty:y\in K\} $ is given by

$\label{ma1.2}h_{TK}(x)=h_{K}(T^{t}x),\ \ \ x\in \mathbb{R}^{n}.$

According to the definition of radial function $ \rho_{K} $, for $ K\in \mathcal{S}_{o}^{n} $ and $ a>0 $, we easily get

$\label{2.1} \rho_{K}(a x)=a^{-1}\rho_{K}(x)\quad \mbox{and}\quad \rho_{a K}(x)=a\rho_{K}(x).$

(2.1)

For $ T\in {\rm GL}(n) $, the radial function of the image $ TK=\{Ty:y\in K\} $ of $ K\in \mathcal{S}_{o}^{n} $ is given by

$\label{2.2} \rho(TK,x)=\rho(K,T^{-1}x)\ \mbox{for all} \ x\in\mathbb{R}^{n}.$

(2.2)

The radial distance between $ K,L\in \mathcal{S}_{o}^{n} $ is defined by

$\tilde \delta (K,L) = \mathop {\sup }\limits_{u \in {S^{n - 1}}} {\mkern 1mu} |\rho (K,u) - \rho (L,u)|.$

For $ K\in \mathcal{S}_{o}^{n} $, define the real number $ R_{K} $ and $ r_{K} $ by

${R_K} = \mathop {\max }\limits_{x \in K} {\mkern 1mu} {\rho _K}(x),\quad {r_K} = \mathop {\min }\limits_{x \in K} {\mkern 1mu} {\rho _K}(x).$

(2.3)

For $ K\in \mathcal{K}_{o}^{n} $, then the polar body $ K^{*} $ of $ K $ is defined by

$K^{*}=\{x\in\mathbb{R}^{n}: x\cdot y\leq1,\forall y\in K\}.$

It is easily proved that $ (K^{*})^{*}=K $ if $ K\in\mathcal{K}_{o}^{n} $.

From the definitions of support and radial function, it follows obviously that for each $ K\in \mathcal{K}^{n}_{o} $, we easily get

$h_{K^{*}}(u)=\frac{1}{\rho_{K}(u)}\quad \mbox{and}\quad \rho_{K^{*}}(u)=\frac{1}{h_{K}(u)}\quad \mbox{for all}\ u\in S^{n-1}.$

In the following write $ x $, $ y $ for vectors in $ \mathbb{R}^{n} $ and $ x' $, $ y' $ for vectors in $ \mathbb{R}^{n-1} $. We will use $ (x',s) $, $ (y',s) $ for vectors in $ \mathbb{R}^{n}=\mathbb{R}^{n-1}×\mathbb{R} $. For a convex body $ K $ and a direction $ u\in S^{n-1} $, let $ K_{u} $ denote the image of the orthogonal projection of $ K $ onto $ u^{\bot} $, the subspace of $ \mathbb{R}^{n} $ orthogonal to $ u $. The undergraph and overgraph functions, $ \underline{l}_{u}(K,\cdot):K_{u}\rightarrow\mathbb{R} $ and $ \bar{l}_{u}(K,\cdot):K_{u}\rightarrow\mathbb{R} $, of $ K $ in the direction $ u $ are given by

$K=\{y'+tu : -\underline{l}_{u}(K,y')\leq t \leq \bar{l}_{u}(K,y') \ \mbox{for} \ y'\in K_{u}\}.$

Therefore the Steiner symmetral $ S_{u}K $ of $ K\in\mathcal{K}_{o}^{n} $ in the direction $ u $ is defined by the body whose orthogonal projection onto $ u^{\bot} $ is identical to that of $ K $ and whose undergraph and overgraph functions are

$\underline{l}_{u}(S_{u}K,y')=\bar{l}_{u}(S_{u}K,y')=\frac{1}{2}\underline{l}_{u}(K,y')+\frac{1}{2}\bar{l}_{u}(K,y').$

(2.4)

For more details about the Steiner symmetrization we can refer to[33].

The following lemma will be required.

Lemma 2.1  (see [16], Lemma 1.2) Suppose $ K \in\mathcal{K}_{o}^{n} $ and $ u \in S^{n-1} $. For $ y'\in {\rm relint}K_{u} $, the overgraph and undergraph functions of $ K $ in direction $ u $ are given by

${\bar l_u}(K,y') = \mathop {\min }\limits_{x' \in {u^ \bot }} {\mkern 1mu} \{ {h_K}(x',1) - x' \cdot y'\} $

and

${{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{l}}}_{u}}(K,{y}')=\underset{{x}'\in {{u}^{\bot }}}{\mathop{\min }}\,\{{{h}_{K}}({x}',-1)-{x}'\cdot {y}'\}.$

From the definition of Orlicz mixed intersection bodies (1.3) and $ \varphi $ is strictly increasing on $ (-\infty,0) $ or strictly decreasing on $ (0,\infty) $, we have the function

$\nonumber λ\mapsto\int_{K}\int_{L}\varphi\bigg(\frac{|x\cdot (y-z)|}{λ}\bigg){\rm d}y{\rm d}z$

is strictly increasing on $ (0,\infty) $ and it is also continuous.Thus we have

Lemma 2.2  If $ K,L\in \mathcal{S}_{o}^{n} $, then for $ u_{0}\in S^{n-1} $, \begin{eqnarray}\frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|u_{0}\cdot(y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z=1\nonumber\end{eqnarray}if and only if $ \rho_{I_{\varphi}(K,L)}^{-1}(u_{0})=λ_{0}. $

Lemma 2.3  (see [34]) Let $ x=(x_{1},x_{2},\cdots,x_{n})\in \mathbb{R}^{n} $ and $ y=(y_{1},y_{2},\cdots,y_{n})\in \mathbb{R}^{n} $. If $ 0 <m_{1}\leq x_{k}\leq M_{1} $, $ 0<m_{2}\leq y_{k}\leq M_{2} $, $ k=1,\cdots,n $, then

$\bigg(\sum_{k=1}^{n}x_{k}^{2}\bigg)\bigg(\sum_{k=1}^{n}y_{k}^{2}\bigg)\leq \Bigg(\frac{\sqrt{\frac{M_{1}M_{2}}{m_{1}m_{2}}}+\sqrt{\frac{m_{1}m_{2}}{M_{1}M_{2}}}}{2}\Bigg)^{2}\bigg(\sum_{k=1}^{n}x_{k}y_{k}\bigg)^{2}.$

Lemma 2.3 implies that if $ x,y,z\in \mathbb{R}^{n} $, then there exists a constant $ c_{0}\in (0,1) $ such that

$\label{2.5} |x\cdot (y-z)|\geq c_{0}\| x \| \| y-z \|.$

(2.5)

If $ K $ and $ L $ are bodies in $ \mathbb{R}^{n} $, their multiplicative $ d_{ab}(K,L) $ is defined by (see [35]) $ d_{ab}(K,L)=\inf\{ab:a,b>0,K\subseteq bL,L\subseteq aK\} $.Whenever we write

$\label{2.6} \frac{1}{a}\|x\|\leq\|x\|_{K}\leq b\|x\|,\quad \frac{1}{b}\|y\|\leq \|y\|_{L}\leq a\|y\|.$

(2.6)

Denote by $ c_{\varphi}>0 $ the constant to meet $ c_{\varphi}=\min\{c>0: \max\{\varphi(c),\varphi(-c)\}\leq 1\} $ for each $ \varphi\in\Phi $. And denote distant $ d(K,L) $ of convex bodies $ K $, $ L $ by

$\label{2.4} d(K,L):=\max\{|x-y|:x\in K,y\in L\}.$

(2.7)

Lemma 2.4  If $ K,L\in \mathcal{S}_{o}^{n} $ and $ \varphi\in\Phi $, then for all $ u\in S^{n-1} $, there exists a positive constant $ b $ and $ c_{0}\in (0,1) $ such that

$\begin{eqnarray}\frac{c_{0}}{c_{\varphi}}\bigg|\frac{1}{bR_{K}}-\frac{b}{r_{L}}\bigg|\leq\rho_{I_{\varphi}(K,L)}^{-1}(u)\leq\frac{d(K,L)}{c_{\varphi}}.\nonumber\end{eqnarray}$

Proof  Let $ x_{0}\in S^{n-1} $ and $ \rho_{I_{\varphi}(K,L)}^{-1}(x_{0})=λ_{0} $. Then

$\label{2.8} \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|x_{0}\cdot (y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z=1.$

(2.8)

We first obtain the upper estimate. From the definition of $ c_{\varphi} $, either $ \varphi(c_{\varphi})=1 $ or $ \varphi(-c_{\varphi})=1 $. If $ \varphi(c_{\varphi})=1 $, by (2.7), we have

$\begin{align}\varphi(c_{\varphi})&=1=\frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|x_{0}\cdot(y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&\geq \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{d(K,L)}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&=\varphi\bigg(\frac{d(K,L)}{λ_{0}}\bigg).\nonumber\end{align}$

With the definition of $ \varphi $, $ \varphi $ is strictly increasing on $ (-\infty,0) $ or strictly decreasing on $ (0,\infty) $. So we can immediately obtain $ c_{\varphi}\leq\frac{d(K,L)}{λ_{0}} $, i.e., $ λ_{0}\leq\frac{d(K,L)}{c_{\varphi}} $.

On the other hand, we obtain the lower estimate. Note that $ \varphi(c_{\varphi})=1 $, $ x_{0}\in S^{n-1} $, from (2.5), (2.6) and (2.3), it follows that

$\begin{align}\varphi(c_{\varphi})&=1=\frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|x_{0}\cdot(y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&\leq \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|c_{0}\|y-z\||}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&\leq \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{c_{0}|\|y\|-\|z\||}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&\leq \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{c_{0}|\frac{1}{b} \|y\|_{K}-b\|z\|_{L}|}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&= \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{c_{0}}{λ_{0}}\bigg|\frac{1}{b\rho_{K}(y)}-\frac{b}{\rho_{L}(z)}\bigg|\bigg){\rm d}y{\rm d}z\nonumber\\&\leq \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{c_{0}}{λ_{0}}\bigg|\frac{1}{bR_{K}}-\frac{b}{r_{L}}\bigg|\bigg){\rm d}y{\rm d}z\nonumber\\&=\varphi\bigg(\frac{c_{0}}{λ_{0}}\bigg|\frac{1}{bR_{K}}-\frac{b}{r_{L}}\bigg|\bigg),\nonumber\end{align}$

where $ b>0 $ and $ c_{0}\in (0,1) $. Since $ \varphi $ is strictly decreasing on $ (0,\infty) $, we immediately get $ c_{\varphi}\geq\frac{c_{0}}{λ_{0}}\big|\frac{1}{bR_{K}}-\frac{b}{r_{L}}\big| $, i.e., $ λ_{0}\geq\frac{c_{0}}{c_{\varphi}}\big|\frac{1}{bR_{K}}-\frac{b}{r_{L}}\big| $.We complete the proof.

The following shows that the Orlicz mixed intersection bodies operator $ I_{\varphi}: \mathcal{S}_{o}^{n}\rightarrow\mathcal{S}_{o}^{n} $ is continuous.

Lemma 2.5  Suppose $ \varphi\in\Phi $. If $ K_{i},L_{i}\in\mathcal{S}_{o}^{n} $ and $ K_{i}\rightarrow K\in\mathcal{S}_{o}^{n} $, $ L_{i}\rightarrow L\in \mathcal{S}_{o}^{n} $, then $ I_{\varphi}(K_{i},L_{i})\rightarrow I_{\varphi}(K,L) $.

Proof  For $ \varphi\in\Phi $ and $ \varphi $ is convex, continuous, and either strictly increasing on $ (-\infty,0) $ or strictly decreasing on $ (0,\infty) $, then for $ u_{0}\in S^{n-1} $, we will show that

$\nonumber \rho^{-1}_{I_{\varphi}(K_{i},L_{i})}(u_{0})\rightarrow \rho^{-1}_{I_{\varphi}(K,L)}(u_{0}).$

Suppose $ \rho_{I_{\varphi}(K_{i},L_{i})}^{-1}(u_{0})=λ_{i} $, by Lemma 2.2, we have

$\frac{1}{|K_{i}||L_{i}|}\int_{K_{i}}\int_{L_{i}}\varphi\bigg(\frac{|u_{0}\cdot (y-z)|}{λ_{i}}\bigg){\rm d}y{\rm d}z=1.$

(2.9)

From Lemma 2.4, there exists a positive constant $ b $ and $ c_{0}\in(0,1) $ such that

$\begin{eqnarray}\frac{c_{0}}{c_{\varphi}}\bigg|\frac{1}{bR_{K_{i}}}-\frac{b}{r_{L_{i}}}\bigg|\leq λ_{i}\leq \frac{d(K_{i},L_{i})}{c_{\varphi}}.\nonumber\end{eqnarray}$

Since $ K_{i}\rightarrow K\in \mathcal{S}_{o}^{n} $, $ L_{i}\rightarrow L\in \mathcal{S}_{o}^{n} $, we have $ d(K_{i},L_{i})\rightarrow d(K,L)>0 $, $ R_{K_{i}}\rightarrow R_{K}>0 $, $ r_{L_{i}}\rightarrow r_{L}>0 $ and there exist $ \alpha $, $ \beta $ such that $ 0< \alpha\leq λ_{i}\leq \beta< \infty $ for all $ i $. Let $ \{λ_{*}\} $ be a convergent subsequence of $ \{λ_{i}\} $, and suppose that $ λ_{*}\rightarrowλ_{0} $, together with the continuous of $ \varphi $ and (2.9), we have

$\nonumber \frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|u_{0}\cdot (y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z=1,$

which by Lemma 2.2 yields $ \rho^{-1}_{I_{\varphi}(K,L)}(u_{0})=λ_{0} $. This shows that

$\rho^{-1}_{I_{\varphi}(K_{i},L_{i})}(u_{0})\rightarrow \rho^{-1}_{I_{\varphi}(K,L)}(u_{0}).$

But for radial function on $ S^{n-1} $ pointwise and uniform convergence are equivalent (see Schneider [8], p.54). Thus the pointwise convergence $ \rho^{-1}_{I_{\varphi}(K_{i},L_{i})}\rightarrow \rho^{-1}_{I_{\varphi}(K,L)} $ on $ S^{n-1} $ completes the proof.

We next show that the Orlicz mixed intersection bodies operator is also continuous in $ \varphi $.

Lemma 2.6  For $ K,L\in \mathcal{S}_{o}^{n} $ and $ \varphi_{i}\rightarrow \varphi\in \Phi $, then $ I_{\varphi_{i}}(K,L)\rightarrow I_{\varphi}(K,L) $.

Proof  Let $ K,L\in \mathcal{S}_{o}^{n} $ and $ u_{0}\in S^{n-1} $. We will show that $ \rho^{-1}_{I_{\varphi_{i}}(K,L)}\rightarrow \rho^{-1}_{I_{\varphi}(K,L)}. $ For $ \varphi\in\Phi $ with $ \varphi $ is convex, continuous, and either strictly increasing on $ (-\infty,0) $ or strictly decreasing on $ (0,\infty) $, and let $ \rho^{-1}_{I_{\varphi_{i}}(K,L)}(u_{0})=λ_{i} $, i.e.,

$\label{2.9} \frac{1}{|K||L|}\int_{K}\int_{L}\varphi_{i} \bigg(\frac{|u_{0}\cdot (y-z)|}{λ_{i}}\bigg){\rm d}y{\rm d}z=1,$

(2.10)

then together with Lemma 2.4 we have that there exists a positive constant $ b $ and $ c_{0}\in (0,1) $ such that

$\begin{eqnarray}\frac{c_{0}}{c_{\varphi_{i}}}\bigg|\frac{1}{bR_{K}}-\frac{b}{r_{L}}\bigg|\leqλ_{i}\leq \frac{d(K,L)}{c_{\varphi_{i}}}.\nonumber\end{eqnarray}$

Since $ \varphi_{i}\rightarrow \varphi\in \Phi $, we have $ c_{\varphi_{i}}\rightarrow c_{\varphi}>0 $ and thus there exist $ \alpha $, $ \beta $ such that $ 0< \alpha\leq λ_{i}\leq \beta<\infty $ for all $ i $. Denote by $ \{λ_{*}\} $ the arbitrary convergent subsequence of $ \{λ_{i}\} $, and suppose that $ λ_{*}\rightarrow λ_{0} $, together with (2.10), we immediately have

$\nonumber \frac{1}{|K||L|}\int_{K}\int_{L}\varphi \bigg(\frac{|u_{0}\cdot (y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z=1,$

which by Lemma 2.2 yields $ \rho^{-1}_{I_{\varphi}(K,L)}(u_{0})=λ_{0} $. This shows that $ \rho^{-1}_{I_{\varphi_{i}}(K,L)}(u_{0})\rightarrow\rho^{-1}_{I_{\varphi}(K,L)}(u_{0}) $. Since the radial function $ \rho_{I_{\varphi_{i}}(K,L)}\rightarrow \rho_{I_{\varphi}(K,L)}(u_{0}) $ pointwise on $ S^{n-1} $ they converge uniformly and hence $ I_{\varphi_{i}}(K,L)\rightarrow I_{\varphi}(K,L). $

Lemma 2.7  Suppose $ \varphi\in \Phi $. For $ K,L\in\mathcal{S}_{o}^{n} $ and a linear transformation $ T\in {\rm GL}(n) $, then $ I_{\varphi}(TK,TL)= T^{-t}(I_{\varphi}(K,L)). $

Proof  Suppose $ x_{0}\in\mathbb{R}^{n} $ and

$\label{2.10} \rho^{-1}(I_{\varphi}(TK,TL),x_{0})=λ_{0}.$

(2.11)

Let $ s=Ty $, $ t=Tz $, then $ |TK|=|{\rm det}T||K| $, $ |TL|=|{\rm det}T||L| $, $ {\rm d}s=|{\rm det}T|{\rm d}y $, $ {\rm d}t=|{\rm det}T|{\rm d}z $, where $ |{\rm det}T| $ is the absolute value of the determinant of $ T $. From Lemma 2.2, we have

$\begin{align}1&=\frac{1}{|TK||TL|}\int_{TK}\int_{TL}\varphi\bigg(\frac{|x_{0}\cdot(s-t)|}{λ_{0}}\bigg){\rm d}s{\rm d}t\nonumber\\&=\frac{1}{|{\rm det}T|^{2}|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|x_{0}\cdot(Ty-Tz)|}{λ_{0}}\bigg)|{\rm det}T|^{2}{\rm d}y{\rm d}z\nonumber\\&=\frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|(x_{0}\cdot Ty)-(x_{0}\cdot Tz)|}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&=\frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|(T^{t}x_{0}\cdot y)-(T^{t}x_{0}\cdot z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z\nonumber\\&=\frac{1}{|K||L|}\int_{K}\int_{L}\varphi\bigg(\frac{|T^{t}x_{0}\cdot(y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z,\nonumber\end{align}$

i.e., $ \rho^{-1}_{I_{\varphi}(K,L)}(T^{t}x_{0})=λ_{0}. $ By (2.2), we have $ λ_{0}=\rho^{-1}_{I_{\varphi}(K,L)}(T^{t}x_{0})=\rho^{-1}_{T^{-t}(I_{\varphi}(K,L))}(x_{0}). $ Combining with equality (2.11), we immediately obtain

$\nonumber \rho^{-1}(I_{\varphi}(TK,TL),x_{0})= \rho^{-1}(T^{-t}(I_{\varphi}(K,L)),x_{0}).$

That is, $ I_{\varphi}(TK,TL)= T^{-t}(I_{\varphi}(K,L)). $

Lemma 3.1  Suppose $ \varphi\in \Phi $ is strictly convex and $ K,L\in \mathcal{K}_{o}^{n} $. If $ u\in S^{n-1} $ and $ x_{1}',x_{2}'\in u^{\bot} $, then

$\label{3.1}\rho^{-1}_{I_{\varphi}(S_{u}K,S_{u}L)}(\frac{x_{1}'+x_{2}'}{2},1) \leq \frac{1}{2} \rho^{-1}_{I_{\varphi}(K,L)}(x_{1}',1) +\frac{1}{2} \rho^{-1}_{I_{\varphi}(K,L)}(x_{2}',-1) $

(3.1)

and

$\label{3.2}\rho^{-1}_{I_{\varphi}(S_{u}K,S_{u}L)}(\frac{x_{1}'+x_{2}'}{2},-1) \leq \frac{1}{2} \rho^{-1}_{I_{\varphi}(K,L)}(x_{1}',1) +\frac{1}{2} \rho^{-1}_{I_{\varphi}(K,L)}(x_{2}',-1) .$

(3.2)

Equality in either inequality holds if $ \rho_{I_{\varphi}(K,L)}(x_{1}',1) =\rho_{I_{\varphi}(K,L)}(x_{2}',-1) $ with $ K $ and $ L $ are dilates having the same midpoints.

Proof  We only prove (3.1). Inequality (3.2) can be established in the same way.

For each $ z'\in K_{u} $, $ y'\in L_{u} $, let $ \sigma_{z'}=\sigma_{z'}(u)=|K\cap (z'+\mathbb{R}u)| $ and $ \sigma_{y'}=\sigma_{y'}(u)=|L\cap (y'+\mathbb{R}u)| $ be the lengths of the chords $ K\cap (z'+\mathbb{R}u) $, $ L\cap(y'+\mathbb{R}u) $.

Define $ m_{z'}=m_{z'}(u) $ by $ m_{z'}(u)=\frac{1}{2}\underline{l}_{u}(K,z')+\frac{1}{2}\bar{l}_{u}(K,z') $ such that $ z'+m_{z'}u $ is the midpoint of the chord $ K\cap (z'+\mathbb{R}u) $. And define $ m_{y'}=m_{y'}(u) $ by $ m_{y'}(u)=\frac{1}{2}\underline{l}_{u}(L,y')+\frac{1}{2}\bar{l}_{u}(L,y') $ such that $ y'+m_{y'}u $ is the midpoint of the chord $ L\cap (y'+\mathbb{R}u) $. Note that the midpoints of the chords of $ K $ in the direction $ u $ lie in a subspace if and only if there exists a $ \mu\in K_{u} $ such that $ \mu\cdot z'=m_{z'} \ \mbox{for all} \ z'\in K_{u}. $ Similarly, the midpoints of the chords of $ L $ in the direction $ u $ lie in a subspace if and only if there exists a $ \mu\in L_{u} $ such that $ \mu\cdot y'=m_{y'} \ \mbox{for all} \ y'\in L_{u}. $ If $ λ_{1}> 0 $, then we have

$\begin{align}&\int_{K} \int_{L} \varphi\bigg(\frac{|(x_{1}',1) \cdot(y-z)|}{λ_{1}}\bigg){\rm d}y{\rm d}z\nonumber\\&=\int_{K} \int_{L} \varphi\bigg(\frac{|(x_{1}',1) \cdot((y',s)-(z',t))|}{λ_{1}}\bigg){\rm d}(y',s){\rm d}(z',t)\nonumber\\&=\int_{K_{u}}\int_{m_{z'}-\frac{\sigma_{z'}}{2}}^{m_{z'}+\frac{\sigma_{z'}}{2}}\int_{L_{u}} \int_{m_{y'}-\frac{\sigma_{y'}}{2}}^{m_{y'}+\frac{\sigma_{y'}}{2}}\varphi\bigg(\frac{|x_{1}'\cdot(y'-z')+(s-t)|}{λ_{1}}\bigg){\rm d}y'{\rm d}s{\rm d}z'{\rm d}t\nonumber\\&=\int_{K_{u}}\int_{-\frac{\sigma_{z'}}{2}}^{\frac{\sigma_{z'}}{2}} \int_{L_{u}}\int_{-\frac{\sigma_{y'}}{2}}^{\frac{\sigma_{y'}}{2}}\varphi\bigg(\frac{|x_{1}'\cdot(y'-z')+m_{y'}+s_{1}-m_{z'}-t_{1}|}{λ_{1}}\bigg){\rm d}y'{\rm d}s_{1}{\rm d}z'{\rm d}t_{1}\nonumber\\\label{3.3}&=\int_{S_{u}K} \int_{S_{u}L}\varphi\bigg(\frac{|x_{1}'\cdot(y'-z')+(s_{1}-t_{1})+(m_{y'}-m_{z'})|}{λ_{1}}\bigg){\rm d}(y',s_{1}){\rm d}(z',t_{1})\end{align}$

(3.3)

by making the change of variables $ s=m_{y'}+s_{1} $ and $ t=m_{z'}+t_{1} $.

On the other hand, for $ λ_{2}>0 $, we have

$\begin{align}&\int_{K} \int_{L} \varphi\bigg(\frac{(|x_{2}',-1) \cdot(y-z)|}{λ_{2}}\bigg){\rm d}y{\rm d}z\nonumber\\=&\int_{K} \int_{L} \varphi\bigg(\frac{(|x_{2}',-1) \cdot((y',s)-(z',t))|}{λ_{2}}\bigg){\rm d}(y',s){\rm d}(z',t)\nonumber\\=&\int_{K_{u}}\int_{m_{z'}-\frac{\sigma_{z'}}{2}}^{m_{z'}+\frac{\sigma_{z'}}{2}}\int_{L_{u}} \int_{m_{y'}-\frac{\sigma_{y'}}{2}}^{m_{y'}+\frac{\sigma_{y'}}{2}}\varphi\bigg(\frac{|x_{2}'\cdot(y'-z')-(s-t)|}{λ_{2}}\bigg){\rm d}y'{\rm d}s{\rm d}z'{\rm d}t\nonumber\\=&\int_{K_{u}}\int_{-\frac{\sigma_{z'}}{2}}^{\frac{\sigma_{z'}}{2}} \int_{L_{u}}\int_{-\frac{\sigma_{y'}}{2}}^{\frac{\sigma_{y'}}{2}}\varphi\bigg(\frac{|x_{2}'\cdot(y'-z')-m_{y'}+s_{1}+m_{z'}-t_{1}|}{λ_{2}}\bigg){\rm d}y'{\rm d}s_{1}{\rm d}z'{\rm d}t_{1}\nonumber\\\label{3.4}=&\int_{S_{u}K} \int_{S_{u}L}\varphi\bigg(\frac{|x_{2}'\cdot(y'-z')+(s_{1}-t_{1})-(m_{y'}-m_{z'})|}{λ_{2}}\bigg){\rm d}(y',s_{1}){\rm d}(z',t_{1})\end{align}$

(3.4)

by making the change of variables $ s=m_{y'}-s_{1} $ and $ t=m_{z'}-t_{1} $.

Let

$\nonumber x_{0}'=\frac{1}{2}x_{1}'+\frac{1}{2}x_{2}' \quad \mbox{and} \quad λ_{0}=\frac{1}{2}λ_{1}+\frac{1}{2}λ_{2}.$

From the convexity of $ \varphi $, it follows that

$\begin{align}\label{3.5}2\varphi\bigg(\frac{|x_{0}'\cdot(y'-z')+(s_{1}-t_{1})|}{λ_{0}}\bigg)\nonumber\leq& \frac{λ_{1}}{λ_{0}}\varphi\bigg(\frac{|x_{1}'\cdot(y'-z')+(s_{1}-t_{1})+(m_{y'}-m_{z'})|}{λ_{1}}\bigg)\nonumber\\&+\frac{λ_{2}}{λ_{0}}\varphi\bigg(\frac{|x_{2}'\cdot(y'-z')+(s_{1}-t_{1})-(m_{y'}-m_{z'})|}{λ_{2}}\bigg).\end{align}$

(3.5)

By (3.3), (3.4) and (3.5), it yields that

$\begin{align}\label{3.6}&\frac{λ_{1}}{λ_{0}} \int_{K} \int_{L}\varphi\bigg(\frac{|(x_{1}',1) \cdot(y-z)|}{λ_{1}}\bigg){\rm d}y{\rm d}z+\frac{λ_{2}}{λ_{0}} \int_{K} \int_{L} \varphi\bigg(\frac{|(x_{2}',-1) \cdot(y-z)|}{λ_{2}}\bigg){\rm d}y{\rm d}z\nonumber\\=&\frac{λ_{1}}{λ_{0}} \int_{S_{u}K} \int_{S_{u}L} \varphi\bigg(\frac{|x_{1}'\cdot(y'-z')+(s_{1}-t_{1})+(m_{y'}-m_{z'})|}{λ_{1}}\bigg){\rm d}(y',s_{1}){\rm d}(z',t_{1})\nonumber\\&+\frac{λ_{2}}{λ_{0}} \int_{S_{u}K} \int_{S_{u}L} \varphi\bigg(\frac{|x_{2}'\cdot(y'-z')+(s_{1}-t_{1})-(m_{y'}-m_{z'})|}{λ_{2}}\bigg){\rm d}(y',s_{1}){\rm d}(z',t_{1})\nonumber\\\geq& 2\int_{S_{u}K} \int_{S_{u}L} \varphi\bigg(\frac{|x_{0}'\cdot(y'-z')+(s_{1}-t_{1})|}{λ_{0}}\bigg){\rm d}(y',s_{1}){\rm d}(z',t_{1})\nonumber\\=&2\int_{S_{u}K} \int_{S_{u}L}\varphi\bigg(\frac{|(x_{0}',1) \cdot(y-z)|}{λ_{0}}\bigg){\rm d}y{\rm d}z.\end{align}$

(3.6)

Let

$\nonumber \rho^{-1}_{I_{\varphi}(K,L)}(x_{1}',1) =λ_{1}\quad \mbox{and} \quad \rho^{-1}_{I_{\varphi}(K,L)}(x_{2}',-1) =λ_{2}.$

From Lemma 2.2, we get

$\nonumber \frac{1}{|K||L|}\int_{K}\int_{L}\varphi \bigg(\frac{|(x_{1}',1) \cdot (y-z)|}{λ_{1}}\bigg){\rm d}y{\rm d}z=1$

and

$\nonumber \frac{1}{|K||L|}\int_{K}\int_{L}\varphi \bigg(\frac{|(x_{2}',-1) \cdot (y-z)|}{λ_{2}}\bigg){\rm d}y{\rm d}z=1.$

Combining with the fact that $ |K|=|S_{u}K| $, $ |L|=|S_{u}L| $ and (3.6), we get

$\nonumber \frac{1}{|S_{u}K||S_{u}L|}\int_{S_{u}K}\int_{S_{u}L}\varphi \bigg(\frac{(x_{0}',1) \cdot (y-z)}{λ_{0}}\bigg){\rm d}y{\rm d}z \leq 1.$

In light of the continuity of $ \varphi $ and (1.3) yields

$\nonumber \rho^{-1}_{I_{\varphi}}(S_{u}K,S_{u}L)(\frac{x_{1}'+x_{2}'}{2},1) \leq \frac{1}{2}λ_{1}+\frac{1}{2}λ_{2}$

with the equality requiring equality in (3.5) for all $ z'\in K_{u} $, $ y'\in L_{u} $, and $ s_{1} \in [-\frac{\sigma_{y'}}{2},\frac{\sigma_{y'}}{2}] $, $ t_{1} \in [-\frac{\sigma_{z'}}{2},\frac{\sigma_{z'}}{2}] $.

Since $ \varphi $ is strictly convex, this means that we must have $ \varphi $ can not be linear in a neighborhood of the origin given by

$\label{3.7}\frac{x_{1}'\cdot(y'-z')+(s_{1}-t_{1})+(m_{y'}-m_{z'})}{λ_{1}}=\frac{x_{2}'\cdot(y'-z')+(s_{1}-t_{1})-(m_{y'}-m_{z'})}{λ_{2}}$

(3.7)

for all $ s_{1} \in (-\frac{\sigma_{y'}}{2},\frac{\sigma_{y'}}{2}) $, $ t_{1} \in (-\frac{\sigma_{z'}}{2},\frac{\sigma_{z'}}{2}) $. Choosing $ λ_{1}=\rho^{-1}_{I_{\varphi}(K,L)}(x_{1}',1) $ and $ λ_{2}=\rho^{-1}_{I_{\varphi}(K,L)}(x_{2}',-1) $, equation (3.7) immediately yields

$\rho^{-1}_{I_{\varphi}(K,L)}(x_{1}',1) =λ_{1}=λ_{2}=\rho^{-1}_{I_{\varphi}(K,L)}(x_{2}',-1) $

and

$(x_{2}'-x_{1}')\cdot y'=2m_{y'},\quad (x_{2}'-x_{1}')\cdot z'=2m_{z'}$

for all $ y'\in L_{u} $ and $ z'\in K_{u} $.

But this means that the midpoints $ \{(y',m_{y'}): y'\in L_{u} \} $ and $ \{(z',m_{z'}): z'\in K_{u} \} $ of the chords of $ L $, $ K $ parallel to $ u $ lie in the subspaces

$\{(y',\frac{x_{2}'-x_{1}'}{2} \cdot y'): y'\in L_{u} \} \quad \mbox{and} \quad \{(z',\frac{x_{2}'-x_{1}'}{2} \cdot z'): z'\in K_{u} \}$

of $ \mathbb{R}^{n} $, respectively.As we can observe the equality holds if $ \rho_{I_{\varphi}(K,L)}(x_{1}',1) =\rho_{I_{\varphi}(K,L)}(x_{2}',-1) $ with $ K $ and $ L $ are dilates having the same midpoints.

Lemma 3.2  Suppose $ K,L\in \mathcal{K}_{o}^{n} $, $ \varphi\in\Phi $. If $ u\in S^{n-1} $, then

$\nonumber I^{*}_{\varphi}(S_{u}K,S_{u}L)\subseteq S_{u}(I^{*}_{\varphi}(K,L)).$

Proof  Let $ y'\in {\rm relint}(I^{*}_{\varphi}(K,L)_{u}) $. According to Lemma 2.1, there exist $ x_{1}'=x_{1}'(y') $ and $ x_{2}'=x_{2}'(y') $ in $ u^{\bot} $ such that

$\label{3.8} \bar{l}_{u}(I^{*}_{\varphi}(K,L),y')=h_{I^{*}_{\varphi}(K,L)}(x_{1}',1) -x_{1}'\cdot y'$

(3.8)

and

$\label{3.9} \underline{l}_{u}(I^{*}_{\varphi}(K,L),y')=h_{I^{*}_{\varphi}(K,L)}(x_{2}',-1) -x_{2}'\cdot y'.$

(3.9)

From (2.4), (3.8), (3.9) and Lemma 3.1, it follows that

$\begin{align} \bar{l}_{u}(S_{u}I^{*}_{\varphi}(K,L),y')&=\frac{1}{2}\bar{l}_{u}(I^{*}_{\varphi}(K,L),y')+\frac{1}{2}\underline{l}_{u}(I^{*}_{\varphi}(K,L),y')\nonumber\\ &=\frac{1}{2}(h_{I^{*}_{\varphi}(K,L)}(x_{1}',1) -x_{1}'\cdot y')+\frac{1}{2}(h_{I^{*}_{\varphi}(K,L)}(x_{2}',-1) -x_{2}'\cdot y')\nonumber\\ &=\frac{1}{2}h_{I^{*}_{\varphi}(K,L)}(x_{1}',1) +\frac{1}{2}h_{I^{*}_{\varphi}(K,L)}(x_{2}',-1) -\big(\frac{x_{1}'+x_{2}'}{2}\big)\cdot y'\nonumber\\ &\geq h_{I^{*}_{\varphi}(S_{u}K,S_{u}L)}(\frac{x_{1}'+x_{2}'}{2},1) -\big(\frac{x_{1}'+x_{2}'}{2}\big)\cdot y'\nonumber\\ &\geq \min_{x'\in u^{\bot}}\{h_{I^{*}_{\varphi}(S_{u}K,S_{u}L)}(x',1) -x'\cdot y' \}\nonumber\\ \label{3.10} &=\bar{l}_{u}(I^{*}_{\varphi}(S_{u}K,S_{u}L),y')\end{align}$

(3.10)

and

$\begin{align} \underline{l}_{u}(S_{u}I^{*}_{\varphi}(K,L),y')&=\frac{1}{2}\bar{l}_{u}(I^{*}_{\varphi}(K,L),y')+\frac{1}{2}\underline{l}_{u}(I^{*}_{\varphi}(K,L),y')\nonumber\\ &=\frac{1}{2}(h_{I^{*}_{\varphi}(K,L)}(x_{1}',1) -x_{1}'\cdot y')+\frac{1}{2}(h_{I^{*}_{\varphi}(K,L)}(x_{2}',-1) -x_{2}'\cdot y')\nonumber\\ &=\frac{1}{2}h_{I^{*}_{\varphi}(K,L)}(x_{1}',1) +\frac{1}{2}h_{I^{*}_{\varphi}(K,L)}(x_{2}',-1) -\big(\frac{x_{1}'+x_{2}')}{2}\big)\cdot y'\nonumber\\ &\geq h_{I^{*}_{\varphi}(S_{u}K,S_{u}L)}(\frac{x_{1}'+x_{2}'}{2},-1) -\big(\frac{x_{1}'+x_{2}'}{2}\big)\cdot y'\nonumber\\ &\geq \min_{x'\in u^{\bot}}\{h_{I^{*}_{\varphi}(S_{u}K,S_{u}L)}(x',-1) -x'\cdot y' \}\nonumber\\ \label{3.11} &=\underline{l}_{u}(I^{*}_{\varphi}(S_{u}K,S_{u}L),y'),\end{align}$

(3.11)

(3.10) and (3.11) give the inclusion.

Proof of Theorem 1.1  Combining with the Steiner symmetrization argument, there is a sequence of direction $ \{u_{i}\} $, such that the sequences $ \{K_{i}\} $ and $ \{L_{i}\} $ converge to $ B_{K} $ and $ B_{L} $, respectively, where the sequences $ \{K_{i}\} $ and $ \{L_{i}\} $ are defined by

$\nonumber K_{i}=S_{u_{i}}\cdots S_{u_{1}}K \quad \mbox{and} \quad L_{i}=S_{u_{i}}\cdots S_{u_{1}}L$

with $ |K|=|K_{i}| $ and $ |L|=|L_{i}| $. Thus $ |K|=|B_{K}| $ and $ |L|=|B_{L}| $.

Since the Steiner symmetrization keeps the volume, by Lemma 3.2, we have

$\begin{align}\nonumber |I^{*}_{\varphi}(K_{i},L_{i})|= |I^{*}_{\varphi}(S_{u_{i}}K_{i-1},S_{u_{i}}L_{i-1})| \leq |I^{*}_{\varphi}(K_{i-1},L_{i-1})| \leq \cdots \leq |I^{*}_{\varphi}(K,L)|,\end{align}$

when $ i\rightarrow\infty $, we have $ |I^{*}_{\varphi}(B_{K},B_{L})|\leq |I^{*}_{\varphi}(K,L)|. $

According to the equality condition of Lemma 3.1, above equality holds if $ K $ and $ L $ are dilates of each other and have the same midpoints.