The classical Brunn-Minkowski theory emerged at the turn of the 19th into the 20th century, when Minkowski began his study of the volume of the Minkowski sum of convex bodies. In the early 1960's, Firey (see e.g. Schneider [13]) introduced an $L_p$-extension of Minkowski's addition (now known as Firey-Minkowski $L_p$-addition) of convex bodies. In the middle of 1990s, it was shown in [9, 10], that a study of the volume of these Firey-Minkowski $L_p$-combinations leads to an embryonic $L_p$-Brunn-Minkowski theory. This theory was expanded rapidly (see e.g. [1-2, 4-6, 8-11, 14]).

The works of Haberl et al. [4-6] and the recent work of Ludwig and Reitzner [8], made it apparent that the time is ripe for the next step in the evolution of the Brunn-Minkowski theory towards the Orlicz-Brunn-Minkowski theory. Lutwak, Yang and Zhang recently introduced the notions of Orlicz projection bodies and Orlicz centroid bodies. It was shown in [11, 12] that a study of the Orlicz Petty projection inequality and Orlicz centroid inequality leads to the Orlicz Brunn-Minkowski theory which is a natural extension of the $L_p$-Brunn-Minkowski theory. Work of Haberl et al. [7] proved the even Orlicz Minkowski problem. Lutwak, Yang and Zhang (see [12]) established the Orlicz centroid inequality for convex bodies and conjectured that their inequality can be extended to star bodies. In [15], Zhu confirmed this conjecture. In [16], the reverse form of the Orlicz Busemann-Petty centroid inequalities was obtained in the two-dimensional case.

Let $\phi: \mathbb{R}\rightarrow [0,\infty)$ be an even strictly convex function such that $\phi(0) = 0$. The class of such a $\phi$ will be denoted by $\mathcal{C}$. Let $K$ be a convex body (i.e., a compact, convex set with non-empty interior) in $\mathbb{R}^n$ that contains the origin in its interior. Denote by $\mid K\mid$ the volume of $K$. The Orlicz centroid body $\Gamma_\phi K$ of $K$, as defined in [12], is the convex body whose support function at $x\in \mathbb{R}^n$ is given by

$\begin{equation} \label{eq:eps1.1} h_{\Gamma_\phi K}(x)=\inf\{\lambda>0:\frac{1}{\mid K \mid} \int_K \phi\left(\frac{x\cdot y}{\lambda}\right)dy \leq 1 \}, \end{equation}$

(1.1)

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

We say that a sequence $\{\phi_i\}$, where the $\phi_i\in\mathcal{C}$, is such that $\phi_i\rightarrow\phi_0\in\mathcal{C}$ provided

$\begin{equation} \mid\phi_i-\phi_0\mid_I:=\max\limits_{t\in I} \mid\phi_i(t)-\phi_0(t)\mid\rightarrow 0 \end{equation}$

for every compact interval $I\subset \mathbb{R}$.

We get the continuity of Orlicz centroid operator by the definition of the Orlicz centroid body as follows:

Theorem 1  Suppose $\phi_{i}\in{\mathcal{C}}$ and $K_{j}$ is a star body (about the origin) in $\mathbb{R}^n$. If $\phi_{i}\rightarrow\phi\in{\mathcal{C}}$ and $K_{j}\rightarrow{K}$, then $\Gamma_{{\phi}_{i}}K_{j}\rightarrow\Gamma_\phi{K}$.

Lutwak, Yang and Zhang also established the definition of the Orlicz projection body $\Pi_\phi K$ of $K$, whose support function is given by (see [11])

$\begin{equation} \label{eq:eps1.2} h_{\Pi_\phi K}(x)=\inf\{\lambda>0:\int_{S^{n-1}}\phi\left(\frac{x\cdot u}{\lambda h_K (u)}\right)h_K (u)dS(u)\leq n\mid K\mid \}. \end{equation}$

(1.2)

For $c>0$, we have

$\begin{equation} \label{eq:eps1.3} \Pi_\phi (cK)=\frac{1}{c} \Pi_\phi K. \end{equation}$

(1.3)

We get the continuity of Orlicz projection operator by the definition of the Orlicz projection body as follows:

Theorem 2  Suppose $\phi_{i}\in{\mathcal{C}}$ and $K_{j}$ is a convex body in $\mathbb{R}^n$ that contains the origin in its interior. If $\phi_{i}\rightarrow\phi\in \mathcal{{C}} $ and $K_{j}\rightarrow{K}$, then $\Pi_{\phi_{i}} K_{j}\rightarrow \Pi_{\phi} K $.

In this section we collect some basic well-known facts that we will use in the proofs of our main results. For references about the Brunn-Minkowski theory, see [3, 13].

Let $\rho(K, \cdot)=\rho_K: \mathbb{R}^n \backslash \{0\}\rightarrow [0,\infty)$ denote the radial function of the set $K\subset \mathbb{R}^n$, star-shaped about the origin; i.e. $\rho_K(x)=\max\{\lambda>0:\lambda x\in K\}$. If $\rho_K$ is strictly positive and continuous, then we call $K$ a star body and we denote the class of star bodies (about the origin) in $\mathbb{R}^n$ by $\mathcal{S}_0^n$. If $c>0$, then obviously for the dilate $cK=\{cx:x\in K\}$ we have

$\begin{equation} \label{eq:eps2.1} \rho_{cK}=c\rho_K. \end{equation}$

(2.1)

Let $h(K, \cdot)=h_K: \mathbb{R}^n \rightarrow \mathbb{R}$ denote the support function of the convex body $K$ in $\mathbb{R}^n$, i.e., $h_K(x)=\max\{x\cdot y: y\in K\}$, we have

$\begin{equation} \label{eq:eps2.2} h_{cK}(x)=ch_K(x) \quad \textrm{and} \quad h_K(cx)=ch_K(x). \end{equation}$

(2.2)

For $\phi\in \mathcal{C}$ define $\phi^\star\in\mathcal{C}$ by

$\begin{equation} \label{eq:eps2.3} \phi^\star (t)=\int_0 ^1 \phi(ts)ds^n, \end{equation}$

(2.3)

where $ds^n =ns^{n-1}ds$. Obviously, $\phi_i\rightarrow\phi_0 \in\mathcal{C}$ implies $\phi_i ^\star\rightarrow \phi_0^\star$.

It will be helpful to also use the alternate definition of Orlicz centroid body (see [12]):

$\begin{equation} \label{eq:eps2.4} h_{\Gamma_\phi K}(x)=\inf\{\lambda>0:\int_{\mathcal{S}^{n-1}}\phi^\star \left(\frac{1}{\lambda}(x\cdot u)\rho_K (u)\right)dV_K ^\ast (u)\leq 1\}, \end{equation}$

(2.4)

where $\phi^\star$ is defined by (2.3) and $dV_K ^\ast $ is the volume-normalized dual conical measure of $K$, defined by $ \mid K\mid dV_K^\ast =\frac{1}{n}\rho_K ^n dS, $ where $dS$ is Lebesgue measure on $S^{n-1}$ (i.e., $(n-1)$-dimensional Hausdorff measure). For $c>0$, an immediate consequence of definitions (2.4) and (2.1) is the fact that

$\begin{equation} \label{eq:eps2.5} \Gamma_\phi cK=c \Gamma_\phi K. \end{equation}$

(2.5)

Lemma 2.1 (see [12]) Suppose $ K \in S_{0}^{n}$ and $u_{0}\in S^{n-1}$. Then

$\begin{equation} \int_{S^{n-1}}\phi^{\star}\left(\frac{1}{\lambda_{0}}(u_{0}\cdot v)\rho_{K}(v)\right)dV_{K}^{\ast}(v)=1 \end{equation}$

if and only if $h_{\Gamma_{\phi}K}(u_{0})=\lambda_0.$

Associated with each $\phi\in\mathcal{C}$ is $c_\phi\in (0,\infty)$ defined by

$ c_\phi=\min\{c>0:\max\{\phi(c),\phi(-c)\}\leq 1\}. $

Throughout $B=\{x\in \mathbb{R}^n:\mid x \mid \leq 1\}$ will denote the unit ball centered at the origin, and $\omega_n= \mid B\mid$ will denote its $n$-dimensional volume. We shall make use of the trivial fact that for $u_0\in S^{n-1}$,

$ \omega_{n-1}=\int_{\mathcal{S}^{n-1}}(u_0 \cdot u)_+ d S(u)=\frac{1}{2}\int_{\mathcal{S}^{n-1}}\mid u_0 \cdot u \mid d S(u), $

where $(t)_+=\max\{t,0\}$ for $t\in \mathbb{R}$, and where $S$ denotes Lebesgue measure on $S^{n-1}$, i.e., $S$ is $(n-1)$-dimensional Hausdorff measure.

Lemma 2.2 (see [12]) If $K\in S_0 ^n $, then $ \frac{\omega_{n-1} r_{K}^{n+1}}{n c_{\phi ^\star}\mid K \mid} \leq h_{\Gamma_\phi K}(u)\leq \frac{R_{K}}{c_{\phi^{\star}}} $ for all $u\in S^{n-1}$, where the real numbers $R_{K}$ and $r_{K}$ are defined by

$R_{K}= \max\limits_{u\in S^{n-1}} \rho_{K}(u) \textrm{and} r_{K}= \min_{u\in S^{n-1}} \rho_{K}(u).$

It will be helpful to also use the alternate definition of Orlicz projection body (see [11]):

$ h_{\Pi_\phi K}(x)=\inf\{\lambda>0:\int_{\mathcal{S}^{n-1}}\phi\left(\frac{1}{\lambda}(x\cdot u)\rho_{K^\ast}(u)\right)dV_k (u)\leq 1\}, $

if $K\in \mathcal{K}_0 ^n$, then the polar body $K^\ast$ is defined by

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

it will be convenient to use the volume-normalized conical measure $V_K$ defined by

$ \mid K\mid dV_K=\frac{1}{n}h_K d S_K. $

Lemma 2.3 (see[11]) Suppose $ \phi \in \mathcal{C}$ and $ K \in \mathcal{K}_0 ^n$. If $x_0 \in \mathbb{R}^n \setminus \{0\}$, then

$ \int_{S^{n-1}}\phi\left(\frac{x_0\cdot u}{\lambda_0 h_K (u_0)}\right)dV_K (v)=1 $

if and only if $h_{\Pi_{\phi}K}(x_0)=\lambda_0.$

Lemma 2.4 (see[11]) If $\phi\in \mathcal{C}$ and $K\in\mathcal{K}_0 ^n$, then

$ \frac{1}{2n c_{\phi}R_K } \leq h_{\Pi_\phi K}(u)\leq \frac{1}{c_{\phi} r_K} $

for all $u\in S^{n-1}$, where the real numbers $R_{K}$ and $r_{K}$ are defined by

$R_{K}= \max\limits_{u\in S^{n-1}} h_{K}(u) \textrm{and} r_{K}= \min_{u\in S^{n-1}} h_{K}(u).$

Theorem 3.1 Suppose $\phi_{i}\in{\mathcal{C}}$ and $K_{j}\in{{S}_{0}}^{n}$. If $\phi_{i}\rightarrow\phi\in{\mathcal{C}}$ and $K_{j}\rightarrow{K}\in{{S}_{0}}^{n}$, then $\Gamma_{{\phi}_{i}}K_{j}\rightarrow\Gamma_\phi{K}$.

Proof (1) First, for fixed $j\in N^+ $ (the set of all the positive integer), suppose $K_{j}\in S_{0}^{n}$ and $u_{0}\in S^{n-1}$. We will show that $ h_{\Gamma_{\phi_{i}} K_{j}}(u_{0})\rightarrow h_{\Gamma_{\phi}K_{j}}(u_{0}). $ Let $ h_{\Gamma_{\phi_{i}} K_{j}}(u_{0})=\lambda_{i} $ and note that Lemma 2.2 gives

$\frac{\omega_{n-1} r_{K_{j}}^{n+1}}{nc_{\phi_{i} ^\star}\mid K_{j}\mid} \leq \lambda_{i}\leq \frac{R_{K_{j}}}{c_{\phi_{i}^{\star}}}.$

Since $\phi_{i}^\star\rightarrow \phi^\star \in \mathcal{C} $, we have $c_{\phi_{i}^\star}\rightarrow c_{\phi^\star} \in (0,\infty)$ and thus there exist $a,b$ such that $0<a\leq \lambda_{i}\leq b<\infty $ for all $i$.

To show that the bounded sequence $\{\lambda_{i}\}$ converges to $h_{\Gamma_{\phi}K_{j}}(u_{0})$, we show that every convergent subsequence of $\{\lambda_{i}\}$ converges to $h_{\Gamma_{\phi}K_{j}}(u_{0})$. Denote an arbitrary convergent subsequence of $\{\lambda_{i}\}$ by $\{\lambda_{i}\}$ as well, and suppose that for this subsequence we have $\lambda_{i}\rightarrow \lambda_\ast.$ Obviously, $ 0 < a\leq\lambda_\ast \leq b $. Since $ h_{\Gamma_{\phi_{i}} K_{j}} (u_{0})= \lambda_{i}$, Lemma 2.1 gives

$\begin{equation} 1=\int_{S^{n-1}}\phi_{i}^{\star}\left(\frac{u_{0}\cdot u}{\lambda_{i}}\rho_{K_{j}}(u)\right)dV_{K_{j}}^{\ast}(u). \end{equation}$

This, together with $\phi_{i}^{\star}\rightarrow \phi^{\star} \in \mathcal{C} $ and $\lambda_{i}\rightarrow \lambda_{\ast}$, gives

$\begin{equation} 1=\int_{S^{n-1}}\phi^{\star}\left(\frac{u_{0}\cdot u}{\lambda_{\ast}}\rho_{K_{j}}(u)\right)dV_{K_{j}}^{\ast}(u). \end{equation}$

By Lemma 2.1 this gives $ h_{\Gamma_{\phi}K_{j}}(u_{0})=\lambda_{\ast}.$ This shows that$\begin{equation} h_{{\Gamma_{\phi_{i}} K_{j}}} (u_{0})\rightarrow h_{{\Gamma_{\phi} K_{j}}} (u_{0}). \end{equation}$

Therefore, for any $ \varepsilon >0,$ there exists $N_{1}\in N^{+},$ for all $ i > N_{1}$, we have

$ \mid h_{{\Gamma_{\phi_{i}} K_{j}}}( u_{0})- h_{{\Gamma_{\phi} K_{j}}} (u_{0})\mid < \frac{\varepsilon}{2}. $

(2) Next, suppose $u_{0}\in S^{n-1}$, we will show that

$h_{\Gamma_{\phi} K_{j}}(u_{0})\rightarrow h_{\Gamma_{\phi}K }(u_{0}). $

Let $h_{\Gamma_{\phi} K_{j}}(u_{0})=\lambda_{j}, $ and note Lemma 2.2 gives

$\frac{\omega_{n-1} r_{K_{j}}^{n+1}}{nc_{\phi ^\star}\mid K_{j}\mid} \leq \lambda_{j}\leq \frac{R_{K_{j}}}{c_{\phi^{\star}}}.$

Since $K_{j}\rightarrow K \in S_{0}^{n} $, we have $r_{K_{j}}\rightarrow r_{K} > 0 $ and $R_{K_{j}}\rightarrow R_{K}<\infty $ and thus there exist $c,d$ such that $0<c\leq \lambda_{j}\leq d<\infty $, for all $j$. To show that the bounded sequence $\{\lambda_{j}\}$ converges to $h_{\Gamma_{\phi}K}(u_{0})$, we show that every convergent subsequence of $\{\lambda_{j}\}$ converges to $h_{\Gamma_{\phi}K}(u_{0})$. Denote an arbitrary convergent subsequence of $\{\lambda_{j}\}$ by $\{\lambda_{j}\}$ as well, and suppose that for this subsequence we have $\lambda_{j}\rightarrow \lambda_\diamond.$ Obviously, $c\leq\lambda_\diamond \leq b$. Let $\bar{K}_{j}=\lambda_{j}^{-1}K_{j}$. Since $\lambda_{j}^{-1}\rightarrow \lambda_\diamond^{-1}$ and $K_{j}\rightarrow K$, we have

$\begin{equation} \bar{K}_{j}\rightarrow\lambda_{\diamond}^{-1}K. \end{equation}$

Now (2.5), and the fact that $h_{\Gamma_{\phi} K_{j}}(u_{0})= \lambda_{j}$, shows that $h_{\Gamma_{\phi} \bar{K}_{j}}(u_{0})=1 $, i.e,

$\begin{equation} \int_{S^{n-1}}\phi^{\star} \left((u_{0}\cdot u)\rho_{\bar{K}_{j}}(u)\right)dV_{\bar{K}_{j}}^{\ast}(u)=1 \end{equation}$

for all $j$. But $\bar{K}_{j}\rightarrow\lambda_{\diamond}^{-1}K $ and the continuity of $\phi^{\star}$ now give

$\begin{equation} \int_{S^{n-1}}\phi^{\star} \left((u_{0}\cdot u)\rho_{\lambda_{\diamond}^{-1}K}(u)\right)dV_{\lambda_{\diamond}^{-1}K}^{\ast}(u)=1, \end{equation}$

which by Lemma 2.1 give $ h_{\Gamma_{\phi}\lambda_{\diamond}^{-1}K}(u_{0})=1.$ This (2.5) and (2.2) now give $ h_{\Gamma_{\phi}K}(u_{0})=\lambda_\diamond.$ This shows that $h_{\Gamma_{\phi}K_{j}}(u_{0})\rightarrow h_{\Gamma_{\phi}K}(u_{0})$.

Therefore, for any $ \varepsilon >0,$ there exists $ N_{2}\in N^{+}$ for all $ j > N_{2}$, we have

$\begin{equation} \mid h_{\Gamma_{\phi}K_{j}}(u_{0})- h_{\Gamma_{\phi}K}(u_{0})\mid < \frac{\varepsilon}{2}. \end{equation}$

(3) To sum up, for all $ \varepsilon >0,$ there exists $ N=\max\{N_{1},N_{2}\}\in N^{+}$ for all $ i, j > N $, we have

$\begin{eqnarray*} && \mid h_{{\Gamma_{\phi_{i}} K_{j}}}( u_{0})- h_{{\Gamma_{\phi} K}} (u_{0})\mid \\ &\leq& \mid h_{{\Gamma_{\phi_{i}} K_{j}}}( u_{0})- h_{{\Gamma_{\phi} K_{j}}} (u_{0})\mid + \mid h_{\Gamma_{\phi}K_{j}}(u_{0})- h_{\Gamma_{\phi}K}(u_{0}) \mid < \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon, \\ && h_{{\Gamma_{\phi_{i}} K_{j}}}( u_{0})\rightarrow h_{{\Gamma_{\phi} K}} (u_{0}). \end{eqnarray*}$

Hence $ \Gamma_{\phi_{i}}K_{j} \rightarrow \Gamma_{\phi} K. $

From the proof of Theorem 3.1, we can obtain the following two results that proved by Lutwak, Yang and Zhang (see [12]).

Corollary 3.2 Suppose $\phi\in{\mathcal{C}}$ and $K_{j}\in{{S}_{0}}^{n}$. If $K_{j}\rightarrow{K}\in{{S}_{0}}^{n}$, then $\Gamma_{{\phi}}K_{j}\rightarrow\Gamma_\phi{K}$.

Corollary 3.3 Suppose $\phi_{i}\in{\mathcal{C}}$ and $K\in{{S}_{0}}^{n}$. If $\phi_{i}\rightarrow\phi\in{\mathcal{C}}$, then $\Gamma_{{\phi}_{i}}K\rightarrow\Gamma_\phi{K}$.

Now, we prove Theorem 2 that is illustrated in Section 1, it is just the following theorem.

Theorem 3.4 Suppose $\phi_{i}\in{\mathcal{C}}$ and $K_{j}\in{\mathcal{K}_{0}}^{n}$. If $\phi_{i}\rightarrow\phi\in \mathcal{{C}} $ and $K_{j}\rightarrow{K}\in{\mathcal{{K}}_{0}}^{n}$, then $\Pi_{\phi_{i}} K_{j}\rightarrow \Pi_{\phi} K $.

Proof (1) First, for fixed $j\in N^+$, suppose $K_{j}\in \mathcal{K}_{0}^{n}$ and $u_{0}\in S^{n-1}$. We will show that $h_{\Pi_{\phi_{i}} K_{j}}(u_{0})\rightarrow h_{\Pi_{\phi}K_{j}}(u_{0}). $ Let $h_{\Pi_{\phi_{i}} K_{j}}(u_{0})=\lambda_{i}, $ and note that Lemma 2.4 gives

$\begin{equation} \frac{1}{2n c_{\phi_i} R_{K_j}} \leq \lambda_{i} \leq \frac{1}{c_{\phi_i} r_{K_j}}. \end{equation}$

Since $\phi_{i}\rightarrow \phi \in \mathcal{C} $, we have $c_{\phi_i}\rightarrow c_{\phi} \in (0,\infty)$ and thus there exist $a,b$ such that $0<a\leq \lambda_{i}\leq b<\infty $ for all $i$. To show that the bounded sequence $\{\lambda_{i}\}$ converges to $h_{\Pi_{\phi}K_{j}}(u_{0})$, we show that every convergent subsequence of $\{\lambda_{i}\}$ converges to $h_{\Pi_{\phi} K_{j}}(u_{0})$. Denote an arbitrary convergent subsequence of $\{\lambda_i\}$ by $\{\lambda_i\}$ as well, and suppose that for this subsequence we have $\lambda_{i}\rightarrow \lambda_\ast.$ Obviously, $ 0 < a \leq \lambda_\ast \leq b $. Since $ h_{\Pi_{\phi_i} K_j} (u_{0})= \lambda_{i}$, Lemma 2.3 gives

$\begin{equation} 1=\int_{S^{n-1}}\phi_{i}\left(\frac{u_{0}\cdot u}{\lambda_i h_{K_j}(u)}\right)dV_{K_j}(u). \end{equation}$

This, together with the facts that $\phi_{i}\rightarrow \phi \in \mathcal{C} $ and $\lambda_{i}\rightarrow \lambda_{\ast}\in (0,\infty)$, gives

$\begin{equation} 1=\int_{S^{n-1}}\phi\left(\frac{u_{0}\cdot u}{\lambda_{\ast}h_{K_j}(u)}\right)dV_{K_j}(u). \end{equation}$

When combined with Lemma 2.3, this gives the desired $h_{\Pi_{\phi}K_{j}}(u_{0})=\lambda_{\ast},$ and completes the argument showing that$\begin{equation} h_{\Pi_{\phi_i} K_j}(u_{0})\rightarrow h_{\Gamma_{\phi} K_j}(u_{0}). \end{equation}$

Therefore, for all $\varepsilon >0,$ there exists $ N_{1}\in N^{+},$ when $ i > N_{1}$, we have

$\begin{equation} \mid h_{\Pi_{\phi_i} K_j}( u_{0})- h_{\Pi_{\phi} K_j}(u_{0})\mid < \frac{\varepsilon}{2}. \end{equation}$

(2) Next, suppose $u_{0}\in S^{n-1}$, we will show that $ h_{\Pi_{\phi} K_j}(u_{0})\rightarrow h_{\Pi_{\phi}K }(u_{0}). $ Let

$ h_{\Pi_{\phi} K_j}(u_{0})=\lambda_{j}, $

and note Lemma 2.4 gives $\frac{1}{2nc_{\phi}R_{K_j}} \leq \lambda_{j}\leq \frac{1}{c_{\phi}r_{K_j}}.$ Since $K_{j}\rightarrow K \in \mathcal{K}_{0}^{n} $, we have $r_{K_{j}}\rightarrow r_{K} > 0 $ and $R_{K_{j}}\rightarrow R_{K}<\infty $ and thus there exist $c,d$ such that $0<c\leq \lambda_{j}\leq d<\infty $ for all $j$. To show that the bounded sequence $\{\lambda_j\}$ converges to $h_{\Pi_{\phi}K}(u_{0})$, we show that every convergent subsequence of $\{\lambda_{j}\}$ converges to $h_{\Pi_{\phi}K}(u_{0})$. Denote an arbitrary convergent subsequence of $\{\lambda_j\}$ by $\{\lambda_j\}$ as well, and suppose that for this subsequence we have $\lambda_{j}\rightarrow \lambda_\diamond.$ Obviously, $0< c\leq\lambda_\diamond \leq b $. Let $\bar{K}_{j}=\lambda_j K_j$. Since $\lambda_j \rightarrow \lambda_\diamond $ and $K_{j}\rightarrow K$, we have

$\begin{equation} \bar{K}_{j}\rightarrow\lambda_{\diamond}K. \end{equation}$

The fact that $h_{\Pi_{\phi} K_{j}}(u_{0})= \lambda_{j}$, together with (2.2) and (1.3), shows that $h_{\Pi_{\phi} \bar{K}_{j}}(u_{0})=1 $, i.e.,

$\begin{equation} \int_{S^{n-1}}\phi\left(\frac{u_{0}\cdot u}{h_{\bar{K}_j}(u)}\right)dV_{\bar{K}_j}(u)=1 \end{equation}$

for all $j$. But $\bar{K}_{j}\rightarrow\lambda_{\diamond} K $ implies that the functions $h_{\bar{K}_j}\rightarrow h_{\lambda_{\diamond} K} $, uniformly, and the measures $S_{\bar{K}_j}\rightarrow S_{\lambda_{\diamond} K}$, weakly. This in turn implies that the measures $V_{\bar{K}_j}\rightarrow V_{\lambda_{\diamond} K}$, weakly, and hence using the continuity of $\phi$ we have

$\begin{equation} \int_{S^{n-1}}\phi\left(\frac{u_{0}\cdot u}{h_{\lambda_{\diamond}K}(u)}\right)dV_{\lambda_{\diamond}K}(u)=1, \end{equation}$

which by Lemma 2.3 give $h_{\Pi_{\phi}\lambda_{\diamond}K}(u_{0})=1.$ This, together with (2.2) and (1.3), yields the desired $ h_{\Pi_{\phi}K}(u_{0})=\lambda_\diamond,$ and shows that $h_{\Pi_{\phi}K_j}(u_{0})\rightarrow h_{\Pi_{\phi}K}(u_{0})$.

Therefore, for any $\varepsilon >0,$ there exists $N_{2}\in N^{+},$ for all $j > N_{2}$, we have

$\begin{equation} \mid h_{\Pi_{\phi}K_{j}}(u_{0})- h_{\Pi_{\phi}K}(u_{0})\mid < \frac{\varepsilon}{2}. \end{equation}$

(3) To sum up, for all $ \varepsilon >0,$ there exists $ N=\max\{N_{1},N_{2}\}\in N^{+},$ for all $ i, j > N $, we have

$\begin{eqnarray*} && \mid h_{\Pi_{\phi_i} K_j}( u_{0})- h_{\Pi_{\phi} K} (u_{0}) \mid \\ &\leq& \mid h_{\Pi_{\phi_i} K_j}( u_{0})- h_{\Pi_{\phi} K_j} (u_{0})\mid + \mid h_{\Pi_{\phi}K_j}(u_{0})- h_{\Pi_{\phi}K}(u_{0}) \mid < \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon, \\ && h_{\Pi_{\phi_i} K_j}( u_{0})\rightarrow h_{{\Pi_{\phi} K}} (u_{0}), \end{eqnarray*}$

Hence $ \Pi_{\phi_i} K_j \rightarrow \Pi{_{\phi} K }. $

From the proof of Theorem 3.4, we can obtain the following results that were proved by Lutwak, Yang and Zhang (see [11]).

Corollary 3.5  Suppose $\phi\in{\mathcal{C}}$ and $K_{j}\in{\mathcal{K}_{0}}^{n}$. If $K_{j}\rightarrow{K}\in{\mathcal{{K}}_{0}}^{n}$, then $\Pi_{\phi} K_{j}\rightarrow \Pi_{\phi} K $.

Corollary 3.6 Suppose $\phi_{i}\in{\mathcal{C}}$ and $K\in{\mathcal{K}_{0}}^{n}$. If $\phi_{i}\rightarrow\phi\in \mathcal{{C}}$, then $\Pi_{\phi_{i}} K\rightarrow \Pi_{\phi} K $.