Inequalities for dual quermassintegrals of the radial p th mean bodies

Abstract

Gardner and Zhang defined the notion of radial p th mean body ($p>-1$) in the Euclidean space ${\mathbb{R}}^n$. In this paper, we obtain inequalities for dual quermassintegrals of the radial p th mean bodies. Further, we establish dual quermassintegrals forms of the Zhang projection inequality and the Rogers-Shephard inequality, respectively. Finally, Shephard’s problem concerning the radial p th mean bodies is shown when $p>0$.

MSC:52A40, 52A20.

1 Introduction

Let ${\mathcal{K}}^n$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean space ${\mathbb{R}}^n$ for the set of convex bodies containing the origin in their interiors in ${\mathbb{R}}^n$ by ${\mathcal{K}}_o^n$. Let $S^{n-1}$ denote the unit sphere in ${\mathbb{R}}^n$, denote by $V(K)$ the n-dimensional volume of body K for the standard unit ball B in ${\mathbb{R}}^n$, define ${\omega }_n=V(B)$.

If K is a compact star-shaped (about the origin) in ${\mathbb{R}}^n$, its radial function, ${\rho }_K=\rho (K,\cdot )$, is defined by (see [1, 2])

$$\rho (K,u)=max\{\lambda \ge 0:\lambda u\in K\}$$

(1.1)

for all $u\in S^{n-1}$. If ${\rho }_K$ is positive and continuous, K will be called a star body (about the origin). Let ${\mathcal{S}}_o^n$ denote the set of star bodies (about the origin) in ${\mathbb{R}}^n$. Two star bodies K and L are said to be dilates (of one another) if ${\rho }_K(u)/{\rho }_L(u)$ is independent of $u\in S^{n-1}$.

The notion of radial p th mean body was given by Gardner and Zhang (see [3]). For $K\in {\mathcal{K}}^n$, the radial p th mean body $R_{p}K$ of K is defined for nonzero $p>-1$ by

$${\rho }_{R_{p}K}(u)={(\frac{1}{V(K)}{\int }_K{\rho }_K^p(x,u)dx)}^{\frac{1}{p}}$$

(1.2)

for each $u\in S^{n-1}$; define $R_{0}K$ by

$${\rho }_{R_{0}K}(u)=exp(\frac{1}{V(K)}{\int }_{K}log{\rho }_K(x,u)dx)$$

for each $u\in S^{n-1}$.

In [3], Gardner and Zhang showed the following.

Theorem 1.A If $K\in {\mathcal{K}}^n$, $-1<p<q$, then

$$DK\subseteq c_{n,q}{R}_{q}K\subseteq c_{n,p}{R}_{p}K\subseteq nV(K){\mathrm{\Pi }}^{\ast }K,$$

(1.3)

in each inclusion equality holds if and only if K is a simplex. Here

$$c_{n,p}={(nB(p+1,n))}^{-\frac{1}{p}}$$

(1.4)

for nonzero $p>-1$, $c_{n,0}={lim}_{p\to 0}{(nB(p+1,n))}^{-\frac{1}{p}}$, and DK and ${\mathrm{\Pi }}^{\ast }K$ denote the difference body and the polar of projection body, respectively.

From Theorem 1.A, Gardner and Zhang [3] again proved the Zhang projection inequality (also see [4]) and the Rogers-Shephard inequality (also see [5]).

Theorem 1.B (Zhang projection inequality)

If $K\in {\mathcal{K}}^n$, then

$$V\left({\mathrm{\Pi }}^{\ast }K\right)V{(K)}^{n-1}\ge \frac{1}{n^n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right),$$

(1.5)

with equality if and only if K is a simplex.

Theorem 1.C (Rogers-Shephard inequality)

If $K\in {\mathcal{K}}^n$, then

$$V(DK)\le \left(\genfrac{}{}{0ex}{}{2n}{n}\right)V(K),$$

(1.6)

with equality if and only if K is a simplex.

In this paper, we continuously research the radial p th mean body. First, we establish inequalities for dual quermassintegrals of the radial p th mean body $R_{p}K$ as follows.

Theorem 1.1 If $K\in {\mathcal{K}}^n$, $p>0$, real $i\ne n$, then there exists $x_0\in K$ such that for $i<n-p$ or $i>n$,

$${\tilde{W}}_i(R_{p}K)\le {\tilde{W}}_i(K-x_0);$$

(1.7)

for $n-p<i<n$,

$${\tilde{W}}_i(R_{p}K)\ge {\tilde{W}}_i(K-x_0).$$

(1.8)

In every inequality, equality holds if and only if $R_{p}K=K-x_0$. For $i=n-p$, (1.7) (or (1.8)) is identic. Here, ${\tilde{W}}_i(K)$ denotes the dual quermassintegrals of K which are given by (see [6])

$${\tilde{W}}_i(K)=\frac{1}{n}{\int }_{S^{n-1}}\rho {(K,u)}^{n-i}dS(u).$$

(1.9)

Obviously, let $i=0$ in (1.9), then

$${\tilde{W}}_0(K)=\frac{1}{n}{\int }_{S^{n-1}}\rho {(K,u)}^{n}dS(u)=V(K).$$

(1.10)

Let $i=0$ in Theorem 1.1 and notice that $V(K-x_0)=V(K)$, we easily get the following.

Corollary 1.1 If $K\in {\mathcal{K}}^n$, $p>0$, then for $p<n$,

$$V(R_{p}K)\le V(K);$$

for $p>n$,

$$V(R_{p}K)\ge V(K).$$

All with equality if and only if $R_{p}K=K$. For $p=n$, above inequalities are identic.

Note that Corollary 1.1 can be found in [7].

As an application of Theorem 1.1, we obtain the following dual quermassintegrals form of the Zhang projection inequality.

Theorem 1.2 If $K\in {\mathcal{K}}^n$, $p>0$, real $i\ne n$, then there exists $x_0\in K$ such that for $n-p\le i<n$,

$${\tilde{W}}_i\left({\mathrm{\Pi }}^{\ast }K\right)\ge {\left[\frac{c_{n,p}}{nV(K)}\right]}^{n-i}{\tilde{W}}_i(K-x_0),$$

(1.11)

with equality for $i=n-p$ if and only if K is a simplex, for $n-p<i<n$ if and only if K is a simplex and $R_{p}K=K-x_0$.

Note that the case of $p=n-i$ in (1.11) can be found in [8].

If p is a positive integer in Theorem 1.2, then by (1.4) we get that

$${(c_{n,p})}^{n-i}={\left(\genfrac{}{}{0ex}{}{n+p}{n}\right)}^{\frac{n-i}{p}}.$$

(1.12)

Hence, we have the following.

Corollary 1.2 If $K\in {\mathcal{K}}^n$, p is a positive integer, i is any real, if $n-p\le i<n$, then there exists $x_0\in K$ such that

$${\tilde{W}}_i\left({\mathrm{\Pi }}^{\ast }K\right)\ge {\left[\frac{1}{nV(K)}\right]}^{n-i}{\left(\genfrac{}{}{0ex}{}{n+p}{n}\right)}^{\frac{n-i}{p}}{\tilde{W}}_i(K-x_0),$$

with equality for $i=n-p$ if and only if K is a simplex, for $n-p<i<n$ if and only if K is a simplex and $R_{p}K=K-x_0$.

Let $i=0$ in Corollary 1.2, and together with (1.12) and (1.10), we have the following.

Corollary 1.3 If $K\in {\mathcal{K}}^n$, $p\ge n$ and p is an integer, then

$$V\left({\mathrm{\Pi }}^{\ast }K\right)V{(K)}^{n-1}\ge \frac{1}{n^n}{\left(\genfrac{}{}{0ex}{}{n+p}{n}\right)}^{\frac{n}{p}},$$

(1.13)

with equality for $p=n$ if and only if K is a simplex, for $p>n$ if and only if K is a simplex and there exists $x_0\in K$ such that $R_{p}K=K-x_0$.

Compared to (1.13) and the Zhang projection inequality (1.5), inequality (1.13) may be regarded as a general Zhang projection inequality.

As another application of Theorem 1.1, we obtain the following dual quermassintegrals form of the Rogers-Shephard inequality.

Theorem 1.3 If $K\in {\mathcal{K}}^n$, $p>0$ and real $i\ne n$, if $i\le n-p$ or $i>n$, then there exists $x_0\in K$ such that

$${\tilde{W}}_i(DK)\le {(c_{n,p})}^{n-i}{\tilde{W}}_i(K-x_0),$$

(1.14)

with equality for $i=n-p$ if and only if K is a simplex, for $i<n-p$ or $i>n$ if and only if K is a simplex and $R_{p}K=K-x_0$.

Similarly, if p is a positive integer in Theorem 1.3, then by (1.12) we obtain the following.

Corollary 1.4 If $K\in {\mathcal{K}}^n$, p is a positive integer, i is any real, if $i\le n-p$ or $i>n$, then there exists $x_0\in K$ such that

$${\tilde{W}}_i(DK)\le {\left(\genfrac{}{}{0ex}{}{n+p}{n}\right)}^{\frac{n-i}{p}}{\tilde{W}}_i(K-x_0),$$

with equality for $i=n-p$ if and only if K is a simplex, for $i<n-p$ or $i>n$ if and only if K is a simplex and $R_{p}K=K-x_0$.

Taking $i=0$ in Corollary 1.4, and using (1.12) and (1.10), we get the following.

Corollary 1.5 If $K\in {\mathcal{K}}^n$, p is a positive integer and $p\le n$, then

$$V(DK)\le {\left(\genfrac{}{}{0ex}{}{n+p}{n}\right)}^{\frac{n}{p}}V(K),$$

(1.15)

with equality for $p=n$ if and only if K is a simplex, for $p<n$ if and only if K is a simplex and there exists $x_0\in K$ such that $R_{p}K=K-x_0$.

Compared to (1.15) and the Rogers-Shephard inequality (1.6), inequality (1.15) may be regarded as a general Rogers-Shephard inequality.

In addition, we also give the Shephard-type problem for the radial p th mean bodies in Section 4.

2 Preliminaries

2.1 Support function, difference body and projection body

If $K\in {\mathcal{K}}^n$, then its support function, $h_K=h(K,\cdot )$, is defined by (see [1, 2])

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

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

If $K\in {\mathcal{K}}_o^n$, the polar body of K, $K^{\ast }$, is defined by (see [1, 2])

$$K^{\ast }=\{x\in {\mathbb{R}}^n:x\cdot y\le 1,y\in K\}.$$

If $K\in {\mathcal{K}}^n$, the difference body, $DK=K+(-K)$, of K is defined by (see [1])

$$h(DK,u)=h(K,u)+h(K,-u)$$

for all $u\in S^{n-1}$.

For $K\in {\mathcal{K}}^n$, the projection body of K, ΠK, is a centered convex body whose support function is given by (see [1])

$$h(\mathrm{\Pi }K,u)=V_{n-1}(K|u^{\perp })$$

for all $u\in S^{n-1}$, where $V_{n-1}$ denotes $(n-1)$-dimensional volume, and $K|u^{\perp }$ denotes the image of orthogonal projection of K onto the codimensional 1 subspace orthogonal to u.

2.2 Extended radial function

If K is compact star-shaped with respect to $x\in {\mathbb{R}}^n$, its radial function ${\rho }_K(x,\cdot )$ with respect to x is defined, for all $u\in S^{n-1}$ such that the line through x parallel to u intersects K, by (see [3])

$${\rho }_K(x,u)=max\{\lambda \ge 0:x+\lambda u\in K\}.$$

(2.1)

From (1.1) and (2.1), we easily know that

$${\rho }_K(x,u)={\rho }_{K-x}(u)$$

(2.2)

for all $u\in S^{n-1}$. We call ${\rho }_K(x,\cdot )$ the extended radial function of K with respect to x. If x is the origin o, then ${\rho }_K(x,u)={\rho }_K(u)$ for all $u\in S^{n-1}$.

From (2.2) and (1.2), obviously,

$${\rho }_{R_{p}K}(u)={(\frac{1}{V(K)}{\int }_K{\rho }_{K-x}^p(u)dx)}^{\frac{1}{p}}.$$

(2.3)

2.3 $L_p$-Dual mixed quermassintegrals

If $K,L\in {\mathcal{S}}_o^n$, $p>0$, $\lambda ,\mu \ge 0$ (not both zero), the $L_p$-radial combination, $\lambda \cdot K{\widetilde{+}}_p\mu \cdot L\in {\mathcal{S}}_o^n$, of K and L is defined by (see [9, 10])

$$\rho {(\lambda \cdot K{\widetilde{+}}_p\mu \cdot L,\cdot )}^p=\lambda \rho {(K,\cdot )}^p+\mu \rho {(L,\cdot )}^p.$$

(2.4)

Associated with (2.4) and (1.9), we define a class of $L_p$-dual mixed quermassintegrals as follows: For $K,L\in {\mathcal{S}}_o^n$, $p>0$, $\epsilon >0$ and real $i\ne n$, the $L_p$-dual mixed quermassintegrals, ${\tilde{W}}_{p,i}(K,L)$, of K and L are defined by

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

(2.5)

Let $i=0$ in definition (2.5), and together with (1.10), we write that ${\tilde{W}}_{p,0}(K,L)={\tilde{V}}_p(K,L)$, then

$$\frac{n}{p}{\tilde{V}}_p(K,L)=\underset{\epsilon \to 0^{+}}{lim}\frac{V(K{\widetilde{+}}_p\epsilon \cdot L)-V(K)}{\epsilon }.$$

Here ${\tilde{V}}_p(K,L)$ denotes a type of $L_p$-dual mixed volume of K and L which is defined in [9, 11] (for $p\ge 1$ also see [12]).

From definition (2.5), the integral representation of $L_p$-dual mixed quermassintegrals can be established as follows.

Theorem 2.1 If $K,L\in {\mathcal{S}}_o^n$, $p>0$, and real $i\ne n$, then

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

(2.6)

Proof From (2.4) and (2.5), for $i\ne n$, we have that

$$\begin{array}{c}\underset{\epsilon \to 0^{+}}{lim}\frac{{\tilde{W}}_i(K{\widehat{+}}_p\epsilon \cdot L)-{\tilde{W}}_i(K)}{\epsilon }\hfill \\ =\underset{\epsilon \to 0^{+}}{lim}\frac{1}{n}{\int }_{S^{n-1}}\frac{\rho {(K{\widehat{+}}_p\epsilon \cdot L,u)}^{n-i}-\rho {(K,u)}^{n-i}}{\epsilon }dS(u)\hfill \\ =\underset{\epsilon \to 0^{+}}{lim}\frac{1}{n}{\int }_{S^{n-1}}\frac{{[\rho {(K,u)}^p+\epsilon \rho {(L,u)}^p]}^{\frac{n-i}{p}}-\rho {(K,u)}^{n-i}}{\epsilon }dS(u).\hfill \end{array}$$

By Hospital’s rule we see that

$$\begin{array}{c}\underset{\epsilon \to 0^{+}}{lim}\frac{{[\rho {(K,\cdot )}^p+\epsilon \rho {(L,\cdot )}^p]}^{\frac{n-i}{p}}-\rho {(K,\cdot )}^{n-i}}{\epsilon }\hfill \\ =\underset{\epsilon \to 0^{+}}{lim}{\rho }_K^{n-i}\frac{{[1+\epsilon {({\rho }_K/{\rho }_L)}^p]}^{\frac{n-i}{p}}-1}{\epsilon }\hfill \\ =\frac{n-i}{p}{\rho }_K^{n-p-i}{\rho }_L^p,\hfill \end{array}$$

thus we get formula (2.6) by definition (2.5). □

From (2.6), we easily know that

$${\tilde{W}}_{p,i}(K,K)=\frac{1}{n}{\int }_{S^{n-1}}\rho {(K,u)}^{n-i}dS(u)={\tilde{W}}_i(K),$$

(2.7)

$${\tilde{W}}_{p,n-p}(K,L)={\tilde{W}}_{n-p}(L).$$

(2.8)

The Minkowski inequality for the $L_p$-dual mixed quermassintegrals is given as follows.

Theorem 2.2 Let $K,L\in {\mathcal{S}}_o^n$, $p>0$, and real $i\ne n$, then for $i<n-p$,

$${\tilde{W}}_{p,i}(K,L)\le {\tilde{W}}_i{(K)}^{(n-p-i)/(n-i)}{\tilde{W}}_i{(L)}^{p/(n-i)};$$

(2.9)

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

$${\tilde{W}}_{p,i}(K,L)\ge {\tilde{W}}_i{(K)}^{(n-p-i)/(n-i)}{\tilde{W}}_i{(L)}^{p/(n-i)}.$$

(2.10)

In every inequality, equality holds if and only if K and L are dilates. For $i=n-p$, (2.9) (or (2.10)) is identic.

Proof For $i<n-p$, from (2.6) and together with the Hölder inequality (see [13]), we have that

$$\begin{array}{rl}{\tilde{W}}_{p,i}(K,L)& =\frac{1}{n}{\int }_{S^{n-1}}{\rho }_K^{n-p-i}(u){\rho }_L^p(u)dS(u)\\ \le {[\frac{1}{n}{\int }_{S^{n-1}}{[{\rho }_K^{n-p-i}(u)]}^{\frac{n-i}{n-p-i}}dS(u)]}^{\frac{n-p-i}{n-i}}{[\frac{1}{n}{\int }_{S^{n-1}}{[{\rho }_L^p(u)]}^{\frac{n-i}{p}}dS(u)]}^{\frac{p}{n-i}}\\ ={[\frac{1}{n}{\int }_{S^{n-1}}{\rho }_K^{n-i}(u)dS(u)]}^{\frac{n-p-i}{n-i}}{[\frac{1}{n}{\int }_{S^{n-1}}{\rho }_L^{n-i}(u)dS(u)]}^{\frac{p}{n-i}}\\ ={\tilde{W}}_i{(K)}^{\frac{n-p-i}{n-i}}{\tilde{W}}_i{(L)}^{\frac{p}{n-i}},\end{array}$$

which gives inequality (2.9) when $i<n-p$. According to the condition that equality holds for the Hölder inequality, we know that the equality holds in inequality (2.9) if and only if K and L are dilates.

Similarly, we can prove for $n-p<i<n$ or $i>n$, inequality (2.10) is true.

For $i=n-p$, by (2.8) and (2.3) then

$${\tilde{W}}_{p,i}{(K,L)}^{n-i}={\tilde{W}}_{p,n-p}{(K,L)}^p={\tilde{W}}_{n-p}{(L)}^p$$

and

$${\tilde{W}}_i{(K)}^{n-p-i}{\tilde{W}}_i{(L)}^p={\tilde{W}}_{n-p}{(K)}^{n-p-i}{\tilde{W}}_{n-p}{(L)}^p={\tilde{W}}_{n-p}{(L)}^p,$$

thus (2.9) (or (2.10)) is identic when $i=n-p$. □

3 Proofs of the theorems

The proofs of the theorems require the following lemma.

Lemma 3.1 If $K\in {\mathcal{K}}^n$, $p>0$, and real $i\ne n$, then for any $Q\in {\mathcal{S}}_o^n$,

$${\tilde{W}}_{p,i}(Q,R_{p}K)=\frac{1}{V(K)}{\int }_K{\tilde{W}}_{p,i}(Q,K-x)dx.$$

(3.1)

Proof Using (2.6) and (2.3), then for any $Q\in {\mathcal{S}}_o^n$, we have that

$$\begin{array}{rl}{\tilde{W}}_{p,i}(Q,R_{p}K)& =\frac{1}{n}{\int }_{S^{n-1}}\rho {(Q,u)}^{n-p-i}\rho {(R_{p}K,u)}^{p}dS(u)\\ =\frac{1}{nV(K)}{\int }_{S^{n-1}}{\int }_K\rho {(Q,u)}^{n-p-i}\rho {(K-x,u)}^{p}dxdS(u)\\ =\frac{1}{V(K)}{\int }_K{\tilde{W}}_{p,i}(Q,K-x)dx.\end{array}$$

 □

Proof of Theorem 1.1 For $i<n-p$, let $Q=R_{p}K$ in (3.1), this together with (2.6), (2.7) and (2.9) gives

$$\begin{array}{rl}{\tilde{W}}_i(R_{p}K)& =\frac{1}{V(K)}{\int }_K{\tilde{W}}_{p,i}(R_{p}K,K-x)dx\\ \le \frac{1}{V(K)}{\int }_K{\tilde{W}}_i{(R_{p}K)}^{\frac{n-p-i}{n-i}}{\tilde{W}}_i{(K-x)}^{\frac{p}{n-i}}dx\\ =\frac{1}{V(K)}{\tilde{W}}_i{(R_{p}K)}^{\frac{n-p-i}{n-i}}{\int }_K{\tilde{W}}_i{(K-x)}^{\frac{p}{n-i}}dx,\end{array}$$

i.e.,

$${\tilde{W}}_i{(R_{p}K)}^{\frac{p}{n-i}}\le \frac{1}{V(K)}{\int }_K{\tilde{W}}_i{(K-x)}^{\frac{p}{n-i}}dx.$$

Therefore, according to the integral mean value theorem, there exists $x_0\in K$ such that

$${\tilde{W}}_i{(R_{p}K)}^{\frac{p}{n-i}}\le \frac{1}{V(K)}{\tilde{W}}_i{(K-x_0)}^{\frac{p}{n-i}}{\int }_{K}dx={\tilde{W}}_i{(K-x_0)}^{\frac{p}{n-i}}.$$

Since $p>0$ and $i<n-p$, thus we get inequality (1.7). According to the condition that equality holds in inequality (2.9), we see that with equality in (1.7) if and only if $R_{p}K$ and $K-x_0$ are dilates. This combined with (1.7), we know that equality holds in (1.7) if and only if $R_{p}K=K-x_0$.

Similarly, for $n-p<i<n$ or $i>n$, from inequality (2.10) and equality (3.1), then

$${\tilde{W}}_i{(R_{p}K)}^{\frac{p}{n-i}}\ge {\tilde{W}}_i{(K-x_0)}^{\frac{p}{n-i}}.$$

Hence, we have that for $i>n$ and $p>0$,

$${\tilde{W}}_i(R_{p}K)\le {\tilde{W}}_i(K-x_0);$$

for $n-p<i<n$ and $p>0$,

$${\tilde{W}}_i(R_{p}K)\ge {\tilde{W}}_i(K-x_0).$$

From this, we get inequality (1.7) and inequality (1.8), respectively, and equality holds in the above inequalities if and only if $R_{p}K=K-x_0$.

For $i=n-p$, by (2.8) and (3.1) we see that (1.7) (or (1.8)) is identic. □

Proof of Theorem 1.2 From (1.3), we have that $c_{n,p}{R}_{p}K\subseteq nV(K){\mathrm{\Pi }}^{\ast }K$ for $p>-1$, then

$${(c_{n,p})}^{n-i}{\tilde{W}}_i(R_{p}K)\le {(nV(K))}^{n-i}{\tilde{W}}_i\left({\mathrm{\Pi }}^{\ast }K\right),$$

(3.2)

with equality if and only if K is a simplex. Hence, together with (1.8), then for $n-p<i<n$ and $p>0$, we obtain that

$${\tilde{W}}_i\left({\mathrm{\Pi }}^{\ast }K\right)\ge {\left[\frac{c_{n,p}}{nV(K)}\right]}^{n-i}{\tilde{W}}_i(K-x_0),$$

which is desired (1.11).

Associated with the cases of equality holding in (3.2) and (1.8), we see that equality holds in (1.11) for $i=n-p$ if and only if K is a simplex, for $n-p<i<n$ if and only if K is a simplex and $R_{p}K=K-x_0$. □

Proof of Theorem 1.3 From (1.3), we know that $DK\subseteq c_{n,p}{R}_{p}K$ for $p>-1$, thus

$${\tilde{W}}_i(DK)\le {(c_{n,p})}^{n-i}{\tilde{W}}_i(R_{p}K),$$

(3.3)

with equality if and only if K is a simplex. Hence, together with (1.7), then for $p>0$, $i<n-p$ or $i>n$, we get that

$${\tilde{W}}_i(DK)\le {(c_{n,p})}^{n-i}{\tilde{W}}_i(K-x_0),$$

this is just (1.14).

Combining with the cases of equality holding in (3.3) and (1.7), we see that equality holds in (1.14) for $i=n-p$ if and only if K is a simplex, for $i<n-p$ or $i>n$ if and only if K is a simplex and $R_{p}K=K-x_0$. □

4 Shephard-type problem

In this section, we research the Shephard-type problem for the radial p th mean bodies. Recall that Zhou and Wang in [7] gave the Shephard-type problem for the radial p th mean bodies as follows.

Theorem 4.A Let $K,L\in {\mathcal{K}}^n$, $p>0$, if $R_{p}K\subseteq R_{p}L$, then

$$V(K)\le V(L),$$

with equality if and only if $R_{p}K=R_{p}L$ and K is a translation of L.

Here, we obtain a stronger result for the Shephard-type problem of the radial p th mean bodies. Our result is the following theorem.

Theorem 4.1 Let $K,L\in {\mathcal{K}}^n$, $p>0$, if $R_{p}K\subseteq R_{p}L$, then there exist $x_0\in K$ and $y_0\in L$ such that

$$K-x_0\subseteq L-y_0,$$

(4.1)

with equality if and only if $R_{p}K=R_{p}L$ and $K-x_0=L-y_0$.

Proof Since $R_{p}K\subseteq R_{p}L$ for $p>0$, thus ${\rho }_{R_{p}K}^p(u)\le {\rho }_{R_{p}L}^p(u)$ for all $u\in S^{n-1}$, i.e.,

$$\frac{1}{V(K)}{\int }_K{\rho }_{K-x}^p(u)dx\le \frac{1}{V(L)}{\int }_L{\rho }_{L-y}^p(u)dy.$$

Therefore, by the integral mean value theorem, there exist $x_0\in K$ and $y_0\in L$ such that

$${\rho }_{K-x_0}^p(u)\frac{1}{V(K)}{\int }_{K}dx\le {\rho }_{L-y_0}^p(u)\frac{1}{V(L)}{\int }_{L}dy,$$

thus

$${\rho }_{K-x_0}^p(u)\le {\rho }_{L-y_0}^p(u)$$

for all $u\in S^{n-1}$. This yields (4.1). □