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# Some inequalities related to $(i,j)$-type ${L}_{p}$-mixed affine surface area and ${L}_{p}$-mixed curvature image

- Tong-Yi Ma
^{1}Email author

**2013**:470

https://doi.org/10.1186/1029-242X-2013-470

© Ma; licensee Springer. 2013

**Received:**29 March 2013**Accepted:**5 September 2013**Published:**7 November 2013

## Abstract

In this article, we introduce two concepts: the $(i,0)$-type ${L}_{p}$-mixed affine surface area and $(i,j)$-type ${L}_{p}$-mixed affine surface area in the set of convex bodies such that ${L}_{p}$-affine surface area by Lutwak *et al.* is proposed in its special cases. Besides, applying these concepts, we establish the extension results of the well-known ${L}_{p}$-Petty affine projection inequality, ${L}_{p}$-Busemann centroid inequality and its dual inequality.

**MSC:**52A40, 52A20.

## Keywords

- convex body
- $(i,0)$-type ${L}_{p}$-mixed affine surface area
- $(i,j)$-type ${L}_{p}$-mixed affine surface area
- ${L}_{p}$-mixed curvature image

## 1 Introduction

Let ${\mathcal{K}}^{n}$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space ${\mathbb{R}}^{n}$. For the set of convex bodies, which contain the origin in their interiors, and the set of origin-symmetric convex bodies in ${\mathcal{K}}^{n}$, we write ${\mathcal{K}}_{o}^{n}$ and ${\mathcal{K}}_{e}^{n}$, respectively. Let ${S}_{o}^{n}$ denote the set of star bodies (about the origin) in ${\mathbb{R}}^{n}$, and let ${S}_{e}^{n}$ denote the set of origin-symmetric star bodies in ${\mathcal{S}}_{o}^{n}$. Let ${S}^{n-1}$ denote the unit sphere in ${\mathbb{R}}^{n}$, and let $V(K)$ denote the *n*-dimensional volume of body *K*. If *K* is the standard unit ball *B* in ${\mathbb{R}}^{n}$, then it is denoted by ${\omega}_{n}=V(B)$. Note that ${\omega}_{n}=\frac{{\pi}^{n/2}}{\mathrm{\Gamma}(1+n/2)}$ defines ${\omega}_{n}$ for all non-negative real *n* (not just the positive integers).

*i*th curvature function ${f}_{i}(K,\cdot ):{S}^{n-1}\to [0,\mathrm{\infty})$ if and only if ${S}_{i}(K,\cdot )$ is absolutely continuous with respect to

*S*and has the Radon-Nikodym derivative (see [1])

Let ${\mathcal{F}}_{i}^{n}$ denote the subset of all bodies ${\mathcal{K}}^{n}$ which have a positive continuous *i* th curvature function. Let ${\mathcal{F}}_{i,o}^{n}$, ${\mathcal{F}}_{i,e}^{n}$ denote the set of all bodies in ${\mathcal{K}}_{o}^{n}$, ${\mathcal{K}}_{e}^{n}$, respectively, and both of them have a positive continuous *i* th curvature function.

*S*, and it has the Radon-Nikodym derivative (see [2])

*K*by (see [3–5])

*K*is an origin-symmetric convex body whose support function is given by (see [6])

where ${c}_{n,p}=\frac{{\omega}_{n+p}}{{\omega}_{2}{\omega}_{n}{\omega}_{p-1}}$. When $p=1$, (1.4) is the notion of projection body (see [7]).

*E*is an ellipsoid which is centered at the origin, then (see [[8], p.105])

The well-known ${L}_{p}$-Petty affine projection inequality is expressed as follows (see [8–10]).

**Theorem A** (${L}_{p}$-Petty affine projection inequality)

*If*$K,L\in {\mathcal{F}}_{o}^{n}$, $p\ge 1$,

*then*

*with equality if and only if* *K* *is an ellipsoid which is centered at the origin*.

*K*is the origin-symmetric convex body whose support function is given by (see [6, 11])

If *E* is an ellipsoid which is centered at the origin, then ${\mathrm{\Gamma}}_{p}E=E$. In particular, ${\mathrm{\Gamma}}_{p}B=B$.

The well-known ${L}_{p}$-Busemann-Petty centroid inequality is as follows (see [6]).

**Theorem B** (${L}_{p}$-Busemann-Petty centroid inequality)

*If*$K\in {S}_{o}^{n}$

*and*$p\ge 1$,

*then*

*with the equality if and only if* *K* *is an ellipsoid which is centered at the origin*.

*et al.*introduced the concept of dual ${L}_{p}$-centroid bodies (see [12]). We give the concept of the unusual normalization of dual ${L}_{p}$-centroid bodies such that ${\mathrm{\Gamma}}_{-p}B=B$: Let $K\in {\mathcal{K}}_{o}^{n}$ and real $p>0$, then radial function of dual ${L}_{p}$-centroid body, ${\mathrm{\Gamma}}_{-p}K$, of

*K*is defined by

*E*is an ellipsoid which is centered at the origin, then

Combined with (1.5) and (1.10), we have that ${\mathrm{\Gamma}}_{-p}E=E$. In particular, ${\mathrm{\Gamma}}_{-p}B=B$.

Si Lin gives the following dual inequality of inequality (1.8) (see [[13], p.9, Theorem 5.4]).

**Theorem C** (Dual ${L}_{p}$-Busemann-Petty centroid inequality)

*If*$K\in {\mathcal{K}}_{o}^{n}$

*and*$p\ge 1$,

*then*

*with equality if and only if* *K* *is an ellipsoid which is centered at the origin*.

*et al.*[14], Lu and Wang [15], Ma and Liu [16, 17] independently proposed the notion of ${L}_{p}$-mixed curvature function: Let $p\ge 1$, $i=0,1,\dots ,n-1$, a convex body $K\in {\mathcal{K}}_{o}^{n}$ is said to have an ${L}_{p}$-mixed curvature function ${f}_{p,i}(K,\cdot ):{S}^{n-1}\to \mathbb{R}$, if its ${L}_{p}$-mixed surface area measure ${S}_{p,i}(K,\cdot )$ is absolutely continuous with respect to spherical Lebesgue measure

*S*and has the Radon-Nikodym derivative

*S*, we have

*K*, by

*B*, we have ${\mathrm{\Lambda}}_{p,i}B=B$. For $K\in {\mathcal{F}}_{i,o}^{n}$, $n-i\ne p\ge 1$, $\lambda >0$,

Because the ${L}_{p}$-mixed curvature image belongs to star bodies, thus, ${\mathcal{C}}_{i,o}^{n}\subseteq {\mathcal{S}}_{o}^{n}$.

*K*is an origin-symmetric convex body whose support function is given by (see [18])

*B*, we have ${\mathrm{\Pi}}_{p,i}B=B$. For $K\in {\mathcal{K}}^{n}$, $\lambda >0$, $p\ge 1$ and $0\le i<n$, then

*i*be arbitrary real numbers, then the ${L}_{p}$-mixed centroid body, ${\mathrm{\Gamma}}_{p,i}K$, of

*K*is the origin-symmetric convex body whose support function is given by (see [19])

Obviously, ${\mathrm{\Gamma}}_{p,0}K={\mathrm{\Gamma}}_{p}K$, and for the standard unit ball *B*, we have ${\mathrm{\Gamma}}_{p,i}B=B$.

*K*are defined by:

Obviously, ${\mathrm{\Gamma}}_{-p,0}K={\mathrm{\Gamma}}_{-p}K$, and for the standard unit ball *B*, we have ${\mathrm{\Gamma}}_{-p,i}B=B$.

In this article, we will first introduce the concept of $(i,0)$-type ${L}_{p}$-mixed affine surface area of convex body as follows.

**Definition 1.1**For $K\in {\mathcal{F}}_{i,o}^{n}$, $i=0,1,\dots ,n-1$ and $p\ge 1$, the $(i,0)$-type ${L}_{p}$-mixed affine surface area, ${\mathrm{\Omega}}_{p}^{(i)}(K)$, of

*K*is defined by

Next, we have established an extension of ${L}_{p}$-Petty affine projection inequality (1.6) as follows.

**Theorem 1.1**

*Let*$K\in {\mathcal{F}}_{i,o}^{n}$, $i=0,1,\dots ,n-1$

*and*$p\ge 1$,

*then*

*with equality in inequality* (1.21) *for* $0<i<n-1$ *if and only if* *K* *is a ball which is centered at the origin*; *for* $i=0$ *if and only if* *K* *is an ellipsoid which is centered at the origin*.

Further, we obtain the following generalized ${L}_{p}$-Busemann-Petty centroid inequality.

**Theorem 1.2**

*Suppose that*$K\in {C}_{i,o}^{n}\subseteq {\mathcal{S}}_{o}^{n}$, $i=0,1,\dots ,n-1$

*and*$p\ge 1$,

*then*

*with equality in inequality* (1.22) *for* $0<i<n-1$ *if and only if* *K* *is a ball which is centered at the origin*; *for* $i=0$ *if and only if* *K* *is an ellipsoid which is centered at the origin*.

Finally, we get the following dual inequality of the inequality (1.22).

**Theorem 1.3**

*Suppose that*$K\in {\mathcal{K}}_{o}^{n}$.

*If*${\mathrm{\Gamma}}_{-p,i}K\in {C}_{i,o}^{n}\subseteq {\mathcal{S}}_{o}^{n}$, $i=0,1,\dots ,n-1$

*and*$p\ge 1$,

*then*

*with equality in inequality* (1.23) *for* $0<i<n-1$ *if and only if* *K* *is a ball which is centered at the origin*; *for* $i=0$ *if and only if* *K* *is an ellipsoid which is centered at the origin*.

## 2 Preliminaries

### 2.1 Support function, radial function and polar of convex body

Obviously, if $K\in {\mathcal{K}}^{n}$, *λ* is a positive constant, $x\in {\mathbb{R}}^{n}$, then $h(\lambda K,x)=\lambda h(K,x)$.

*K*is a compact star-shaped (about the origin) in ${\mathbb{R}}^{n}$, its radial function, ${\rho}_{K}=\rho (K,\cdot ):{\mathbb{R}}^{n}\setminus \{0\}\to [0,\mathrm{\infty})$, is defined by (see [20, 21])

When ${\rho}_{K}$ is positive and continuous, *K* is called a star body (about the origin). Obviously, if $K\in {\mathcal{S}}_{o}^{n}$, $\alpha >0$, $x\in {\mathbb{R}}^{n}$, then $\rho (K,\alpha x)={\alpha}^{-1}\rho (K,x)$ and $\rho (\alpha K,x)=\alpha \rho (K,x)$. Two star bodies *K* and *L* are said to be dilates (of one another) if ${\rho}_{K}(u)/{\rho}_{L}(u)$ is independent on $u\in {S}^{n-1}$.

*K*is defined by (see [20, 21])

*K*are defined respectively by (see [20, 21])

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

### 2.2 The quermassintegrals, ${L}_{p}$-mixed quermassintegrals and ${L}_{p}$-mixed volume

*K*are defined by (see [20, 21])

where ‘⋅’ in $\epsilon \cdot L$ denotes the Firey scalar multiplication.

*K*and

*L*($i=0,1,\dots ,n-1$) are defined by (see [1])

with equality for $p=1$ and $0\le i<n-1$ if and only if *K* and *L* are homothetic; for $p>1$ and $0\le i\le n-1$ if and only if *K* and *L* are dilates. For $p=1$ and $i=n-1$, inequality (2.11) is identical.

### 2.3 Dual quermassintegrals and ${L}_{p}$-dual mixed quermassintegrals

*i*, the dual quermassintegrals, ${\tilde{W}}_{i}(K)$, of

*K*are defined by (see [20, 21])

Note that here ‘$\epsilon \cdot L$’ is different from ‘$\epsilon \cdot L$’ in the Firey ${L}_{p}$-combination.

*K*and

*L*are defined by [25]

For $n<i<n+p$, inequality (2.18) is reversed. With equality in every inequality if and only if *K* and *L* are dilates.

## 3 The $(i,j)$-type ${L}_{p}$-mixed affine surface area

In this section, we further propose the concept of $(i,j)$-type ${L}_{p}$-mixed affine surface area as follows.

**Definition 3.1**For $K\in {\mathcal{F}}_{i,o}^{n}$, $i=0,1,\dots ,n-1$, $j\in \mathbb{R}$ and $p\ge 1$, the $(i,j)$-type ${L}_{p}$-mixed affine surface area, ${\mathrm{\Omega}}_{p,j}^{(i)}(K)$, of

*K*is defined by

Obviously, ${\mathrm{\Omega}}_{p}^{(0)}(K)={\mathrm{\Omega}}_{p}(K)$ and ${\mathrm{\Omega}}_{p,0}^{(i)}(K)={\mathrm{\Omega}}_{p}^{(i)}(K)$.

Next, we introduce the concept of $(i,0)$-type ${L}_{p}$-mixed affine surface area of the convex bodies ${K}_{1},{K}_{2},\dots ,{K}_{n-i}$ as follows.

**Definition 3.2**For $p\ge 1$, $i=0,1,\dots ,n-1$, the $(i,0)$-type ${L}_{p}$-mixed affine surface area, ${\mathrm{\Omega}}_{p}^{(i)}({K}_{1},\dots ,{K}_{n-i})$, of ${K}_{1},\dots ,{K}_{n-i}\in {\mathcal{F}}_{i,o}^{n}$ is defined by

Let ${K}_{1}=\cdots ={K}_{n-i-j}=K$ and ${K}_{n-i-j+1}=\cdots ={K}_{n-i}=L$ ($j=0,\dots ,n-i$), we define ${\mathrm{\Omega}}_{p,j}^{(i)}(K,L)={\mathrm{\Omega}}_{p}^{(i)}(K,\dots ,K,L,\dots ,L)$ with $n-i-j$ copies of *K* and *j* copies of *L*. From this, if *j* is any real number, we can define the following.

**Definition 3.3**For $K,L\in {\mathcal{F}}_{i,o}^{n}$, $i=0,\dots ,n-1$, $p\ge 1$, $j\in \mathbb{R}$, the $(i,j)$-type ${L}_{p}$-mixed affine surface area, ${\mathrm{\Omega}}_{p,j}^{(i)}(K,L)$, of

*K*,

*L*is defined by

and ${\mathrm{\Omega}}_{p,j}^{(i)}(K)$ is called the $(i,j)$-type ${L}_{p}$-mixed affine surface area of $K\in {\mathcal{F}}_{i,o}^{n}$. In particular, ${\mathrm{\Omega}}_{p,j}^{(o)}(K)={\mathrm{\Omega}}_{p,j}(K)$ is called the *j* th ${L}_{p}$-mixed affine surface area of $K\in {\mathcal{F}}_{o}^{n}$ (see [26]).

Next, we give some propositions of ${L}_{p}$-mixed curvature image and $(i,j)$-type ${L}_{p}$-mixed affine surface area.

**Proposition 3.1**

*Let*$K\in {\mathcal{F}}_{i,o}^{n}$, $i=0,1,\dots ,n-1$, $j\in \mathbb{R}$.

*Then*

*In particular*,

*take*$j=0$

*in*(3.7),

*then*

*Proof*From (3.6), (1.14) and (2.12), we have

□

**Proposition 3.2**

*Let*$p\ge 1$, $K\in {\mathcal{F}}_{i,o}^{n}$

*and*$i=0,1,\dots ,n-1$.

*Then*

*for each* $Q\in {\mathcal{K}}_{o}^{n}$.

*Proof*For each $Q\in {\mathcal{K}}_{o}^{n}$, from (2.10), (1.14), (2.2) and (2.16), we have

□

**Proposition 3.3**

*If*$p\ge 1$, $L\in {\mathcal{F}}_{i,o}^{n}$,

*then*

*for all* $K\in {\mathcal{S}}_{o}^{n}$ *with equality if and only if* *K* *and* ${\mathrm{\Lambda}}_{p,i}L$ *are dilates*.

*Proof*Let $L\in {\mathcal{F}}_{i,o}^{n}$ and each $K\in {\mathcal{S}}_{o}^{n}$, then from (1.20), (2.2), (2.7), (2.12) and Hölder’s inequality, we have

From this, we immediately get (3.10).

*c*is a constant. Combined with the definition of ${L}_{p}$-mixed curvature image, for any $u\in {S}^{n-1}$, we have

this shows that *K* and ${\mathrm{\Lambda}}_{p,i}L$ are dilates. Therefore, the equality holds in inequality (3.10) if and only if *K* and ${\mathrm{\Lambda}}_{p,i}L$ are dilates. The proof is complete. □

Now, according to Proposition 3.3, we can give an expansion of the definition of the $(i,0)$-type ${L}_{p}$-mixed affine surface area of $K\in {\mathcal{K}}_{o}^{n}$ as follows.

**Definition 3.4**If $K\in {\mathcal{K}}_{o}^{n}$, $p\ge 1$, then the $(i,0)$-type ${L}_{p}$-mixed affine surface area, ${\mathrm{\Omega}}_{p}^{(i)}(K)$, of

*K*is defined by

For $i=0$, the definition is just the definition of ${L}_{p}$-affine surface area by Lutwak proposed in [2].

## 4 Generalized ${L}_{p}$-Petty affine projection inequality

In this section, we complete the proof of Theorem 1.1 in the introduction. In fact, we prove the following more general conclusion.

**Theorem 4.1**

*Let*$K\in {\mathcal{K}}_{o}^{n}$, $L\in {\mathcal{F}}_{i,o}^{n}$, $p\ge 1$, $0\le i<n-1$,

*then*

*with equality in inequality* (4.1) *for* $0<i<n-1$ *if and only if* *K* *and* *L* *are balls of dilates which are centered at the origin*; *for* $i=0$ *if and only if* *K* *and* *L* *are ellipsoids of dilates which are centered at the origin*.

In order to prove the theorems above, we first give the following three lemmas.

**Lemma 4.1** (See [27])

*Suppose that*$K\in {\mathcal{K}}_{o}^{n}$, $i\in \mathbb{R}$

*and*$0\le i<n$,

*then*

*with the equality for* $0<i<n$ *if and only if* *K* *is a ball which is centered at the origin*. *If* $i=0$, *then* (4.2) *is identical*.

**Lemma 4.2** (See [18])

*Suppose that*$K\in {\mathcal{K}}_{o}^{n}$, $p>1$

*and*$0<i<n-1$,

*i*

*is a positive integer*,

*then*

*with equality if and only if* *K* *is a ball which is centered at the origin*.

**Remark 4.1** The conditions of inequality (4.3) can be relaxed to $p\ge 1$ and $0\le i<n-1$, while the conditions of the equality that holds can be given separately. For $0<i<n-1$ and $p=1$, the inequality (4.3) is proved by Lutwak with the equality holding if and only if *K* is a ball (see [7]). For $i=0$ and $p>1$, inequality (4.3) is proved by Lutwak *et al.* with the equality that holds if and only if *K* is an ellipsoid which is centered at the origin (see [6]). For $i=0$ and $p=1$, then (4.3) is the famous Petty projection inequality (see [28]), with the equality that holds if and only if *K* is an ellipsoid.

**Lemma 4.3**

*If*$K,L\in {\mathcal{K}}_{o}^{n}$, $p\ge 1$, $i=0,1,\dots ,n-1$,

*then*

*Proof* From (1.16), (2.10) and the Fubini theorem, it is easy to prove Lemma 4.3. □

*Proof of Theorem 4.1*For $L\in {\mathcal{F}}_{i,o}^{n}$ and any $Q\in {\mathcal{K}}_{o}^{n}$, by inequality (3.10) and Lemma 4.1, we have

with equality for $0<i\le n-1$ if and only if ${\mathrm{\Lambda}}_{p,i}L$ and ${Q}^{\ast}$ are centered balls of dilates; for $i=0$ if and only if ${\mathrm{\Lambda}}_{p,i}L$ and ${Q}^{\ast}$ are dilates.

with equality for $0<i\le n-1$ if and only if ${\mathrm{\Lambda}}_{p,i}L$ and ${\mathrm{\Pi}}_{p,i}^{\ast}K$ are centered balls of dilates; for $i=0$ if and only if ${\mathrm{\Lambda}}_{p}L$ and ${\mathrm{\Pi}}_{p}^{\ast}K$ are dilates.

which implies that inequality (4.1) holds.

Next, we discuss the conditions of equality that holds in inequality (4.1).

- (1)
For the case $p>1$ and $0<i<n-1$, the equality holds in (4.1) if and only if ${\mathrm{\Pi}}_{p,i}^{\ast}K$ and ${\mathrm{\Lambda}}_{p,i}L$ are balls of dilates which are centered at the origin, and

*K*is a ball which is centered at the origin. Together with ${\mathrm{\Lambda}}_{p,i}B=B$ and ${\mathrm{\Pi}}_{p,i}B=B$, we know that*K*and*L*are balls of dilates which are centered at the origin. - (2)
For the case $p=1$ and $0<i<n-1$, the equality holds in (4.1) if and only if ${\mathrm{\Pi}}_{1,i}^{\ast}K$ and ${\mathrm{\Lambda}}_{1,i}L$ are balls of dilates which are centered at the origin, and

*K*is a ball. By using ${\mathrm{\Pi}}_{p,i}B=B$ and (1.17), it is obtained that ${\mathrm{\Pi}}_{1,i}^{\ast}(\lambda B)={\lambda}^{1+i-n}{\mathrm{\Pi}}_{1,p}^{\ast}B={\lambda}^{1+i-n}B$ ($\lambda >0$) is a ball which is centered at the origin. Because ${\mathrm{\Pi}}_{1,i}^{\ast}K$ and ${\mathrm{\Lambda}}_{1,i}L$ are balls of dilates which are centered at the origin, then ${\mathrm{\Lambda}}_{1,i}L$ is a ball of dilates which are centered at the origin, and together with ${\mathrm{\Lambda}}_{p,i}B=B$ and (1.15),*L*is a ball which is centered at the origin. However,*K*is a ball, so the equality holds in (4.1) if and only if*K*and*L*are balls of dilates which are centered at the origin. - (3)
For the case $p>1$ and $i=0$, the equality holds in (4.1) if and only if ${\mathrm{\Pi}}_{p}^{\ast}K$ and ${\mathrm{\Lambda}}_{p}L$ are dilates and

*K*is an ellipsoid which is centered at the origin. Let $K=E$ be an ellipsoid which is centered at the origin, from (1.5), we know that ${\mathrm{\Pi}}_{p}^{\ast}E={({\omega}_{n}/V(E))}^{\frac{1}{p}}E$ is an ellipsoid which is centered at the origin. Other, from the literature [2], we know that*L*is an ellipsoid*E*which is centered at the origin if and only if ${\mathrm{\Lambda}}_{p}L$ are dilates of polar body ${E}^{\ast}$ of this*E*. So we know that the equality holds in (4.1) if and only if*L*and*K*are ellipsoids which are centered at the origin and both are dilates. - (4)
For the case $p=1$ and $i=0$, the equality holds in (4.1) if and only if ${\mathrm{\Lambda}}_{1}L$ and ${\mathrm{\Pi}}_{1}^{\ast}K$ are dilates, and

*K*is an ellipsoid. Suppose that $K=\lambda E+{x}_{0}$ with $\lambda >0$, ${x}_{0}\in {\mathbb{R}}^{n}$, and*E*is an ellipsoid which is centered at the origin, noting that $S(\lambda E+{x}_{0},\cdot )=S(\lambda E,\cdot )={\lambda}^{n-1}S(E,\cdot )$ (see [29]), this together with (1.5) ${\mathrm{\Pi}}_{1}^{\ast}K={\mathrm{\Pi}}_{1}^{\ast}(\lambda E+{x}_{0})={\mathrm{\Pi}}_{1}^{\ast}\lambda E={\lambda}^{1-n}{\mathrm{\Pi}}_{1}^{\ast}E={\lambda}^{1-n}({\omega}_{n}/V(E))E$ is an ellipsoid which is centered at the origin. Because ${\mathrm{\Lambda}}_{1}L$ and ${\mathrm{\Pi}}_{1}^{\ast}K$ are dilates, then ${\mathrm{\Lambda}}_{1}L$ is an ellipsoid which is centered at the origin. However, from [2], we know that*L*is an ellipsoid*E*which is centered at the origin if and only if ${\mathrm{\Lambda}}_{1}L$ are dilates of polar body ${E}^{\ast}$ of this*E*. Therefore, the equality holds in (4.1) if and only if*K*and*L*are ellipsoids of the dilates which are centered at the origin.

To sum up, the equality holds in (4.1) for $p\ge 1$ and $0<i<n-1$ if and only if *K* and *L* are balls of the dilates which are centered at the origin; for $p\ge 1$ and $i=0$ if and only if *K* and *L* are ellipsoids of the dilates which are centered at the origin. The proof is complete. □

*Proof of Theorem 1.1*Exchange

*K*and

*L*in inequality (4.1), we have for $L\in {\mathcal{K}}_{o}^{n}$, $K\in {\mathcal{F}}_{i,o}^{n}$, $p\ge 1$, $0\le i<n-1$

Taking $L={\mathrm{\Pi}}_{p,i}K$ in the inequality above, we immediately obtain inequality (1.21). The proof is complete. □

Combining with Theorem 1.1 and (3.8), we immediately obtain the following.

**Corollary 4.1**

*If*$K\in {\mathcal{F}}_{i,o}^{n}$, $i=0,1,\dots ,n-1$

*and*$p\ge 1$,

*then*

*with the equality in inequality* (4.8) *for* $0<i<n-1$ *if and only if* *K* *is a ball which is centered at the origin*; *for* $i=0$ *if and only if* *K* *is an ellipsoid which is centered at the origin*.

Further, we have established the following results.

**Theorem 4.2**

*Let*$K,L\in {\mathcal{F}}_{i,o}^{n}$, $0\le i<n-1$, $p\ge 1$,

*then*

*with the equality in inequality* (4.9) *for* $0<i<n-1$ *if and only if* *K* *and* *L* *are balls of dilates which are centered at the origin*; *for* $i=0$ *if and only if* *K* *and* *L* *are ellipsoids of dilates which are centered at the origin*.

*Proof*From inequality (4.1), we know that for $Q\in {\mathcal{K}}_{o}^{n}$, $L\in {\mathcal{F}}_{i,o}^{n}$, $p\ge 1$, $0\le i\le n-1$,

Combining inequality (4.11) with (3.8), we immediately obtain inequality (4.8). According to the condition of the equality holding in inequalities (4.1) and (4.9), the condition of the equality that holds in inequality (4.8) is easily obtained. The proof is complete. □

## 5 Generalized ${L}_{p}$-Busemann-Petty centroid inequality and dual inequality

In this section, we give the extension of the well-known ${L}_{p}$-Busemann-Petty centroid inequality (1.8). Namely, we complete the proof of Theorem 1.2 and Theorem 1.3 (*i.e.*, dual inequality of Theorem 1.2) in the introduction.

**Lemma 5.1**

*If*$K\in {\mathcal{F}}_{i,o}^{n}$, $0=0,1,\dots ,n-1$, $p\ge 1$,

*then*

*Proof* Using definition (1.16) of ${L}_{p}$-mixed projection body and definition (1.14) of ${L}_{p}$-mixed curvature image, it is easy to prove (5.1). □

**Lemma 5.2**

*If*$K\in {\mathcal{K}}_{o}^{n}$, $L\in {\mathcal{S}}_{o}^{n}$, $p\ge 1$, $i=0,1,\dots ,n-1$,

*then*

*Proof* By (1.18), (1.19), (2.3), (2.7), (2.12) and (2.16), it is easy to prove (5.2). □

*Proof of Theorem 1.2*For $L\in {\mathcal{F}}_{i,o}^{n}$, using (5.1) and Corollary 4.1, we have

taking $K={\mathrm{\Lambda}}_{p,i}L$ in the inequality above, we immediately get inequality (1.22).

According to the condition of the equality that holds in inequalities (4.8) and (1.8), and noting that ${\mathrm{\Lambda}}_{p,i}B=B$ and (1.15), we know with the equality in inequality (1.22) for $p\ge 1$ and $0<i<n-1$ if and only if *K* is a ball which is centered at the origin; for $p\ge 1$ and $i=0$ if and only if *K* is an ellipsoid which is centered at the origin. □

*Proof of Theorem 1.3*Take $L={\mathrm{\Gamma}}_{-p,i}K$ in (5.2), we have

from this, we can get inequality (1.23).

- (1)
For the case $p>1$ and $0<i<n-1$, the equality holds in (1.23) if and only if

*K*and ${\mathrm{\Gamma}}_{p,i}{\mathrm{\Gamma}}_{-p,i}K$ are dilates, and ${\mathrm{\Gamma}}_{-p,i}K$ is a ball which is centered at the origin. Because ${\mathrm{\Gamma}}_{-p,i}B=B$, then ${\mathrm{\Gamma}}_{-p,i}K$ is a ball which is centered at the origin if and only if*K*is a ball which is centered at the origin. Therefore,*K*is a ball which is centered at the origin. While the equality ${\mathrm{\Gamma}}_{p,i}B=B$ shows that ${\mathrm{\Gamma}}_{-p,i}K$ is a ball which is centered at the origin if and only if ${\mathrm{\Gamma}}_{p,i}{\mathrm{\Gamma}}_{-p,i}K$ is a ball which is centered at the origin. From this, for $p>1$ and $0<i<n-1$ the equality holds in inequality (1.23) if and only if*K*is a ball which is centered at the origin. - (2)
For the case $p>1$ and $i=0$, the equality holds in (1.23) if and only if

*K*and ${\mathrm{\Gamma}}_{p}{\mathrm{\Gamma}}_{-p}K$ are dilates and ${\mathrm{\Gamma}}_{-p}K$ is an ellipsoid which is centered at the origin. Because ${\mathrm{\Gamma}}_{-p}K$ is an ellipsoid which is centered at the origin, and together with (1.10),*K*is an origin-symmetric ellipsoid*E*if and only if ${\mathrm{\Gamma}}_{-p}K$ is an origin-symmetric ellipsoid. On the other hand, the literature [11] tells us that if*E*is an ellipsoid which is centered at the origin, then ${\mathrm{\Gamma}}_{p}E=E$. From this, ${\mathrm{\Gamma}}_{p}{\mathrm{\Gamma}}_{-p}K$ is an ellipsoid which is centered at the origin. Therefore, for $p>1$ and $i=0$, the equality holds in inequality (1.23) if and only if*K*is an ellipsoid which is centered at the origin. - (3)
For the case $p=1$ and $0<i<n-1$, the equality holds in (1.23) if and only if

*K*and ${\mathrm{\Gamma}}_{1,i}{\mathrm{\Gamma}}_{-1,i}K$ are homothetic, and ${\mathrm{\Gamma}}_{-1,i}K$ is a ball which is centered at the origin. From ${\mathrm{\Gamma}}_{-p,i}B=B$, we know that ${\mathrm{\Gamma}}_{-1,i}B=B$, then*K*is a ball which is centered at the origin. This ${\mathrm{\Gamma}}_{-1,i}B=B$ together with ${\mathrm{\Gamma}}_{1,i}B=B$, then ${\mathrm{\Gamma}}_{1,i}{\mathrm{\Gamma}}_{-1,i}K$ is a ball which is centered at the origin. Therefore, for $p=1$ and $0<i<n-1$, the equality holds in inequality (1.23) if and only if*K*is a ball which is centered at the origin. - (4)
For the case $p=1$ and $i=0$, the equality holds in (1.23) if

*K*and $\mathrm{\Gamma}{\mathrm{\Gamma}}_{-1}K$ are homothetic, and ${\mathrm{\Gamma}}_{-1}K$ is an ellipsoid which is centered at the origin. Because ${\mathrm{\Gamma}}_{-1}K$ is an origin-symmetric ellipsoid*E*if and only if*K*is an origin-symmetric ellipsoid. On the other hand, from ${\mathrm{\Gamma}}_{-p}E=E$, we know that ${\mathrm{\Gamma}}_{-1}K$ is an ellipsoid which is centered at the origin if and only if ${\mathrm{\Gamma}}_{1}{\mathrm{\Gamma}}_{-1}K$ is an ellipsoid which is centered at the origin. Therefore, for $p=1$ and $i=0$, the equality holds in inequality (1.23) if and only if*K*is an ellipsoid which is centered at the origin.

To sum up, the equality holds in (1.23) for $p\ge 1$ and $0<i<n-1$ if and only if *K* is a ball which is centered at the origin; for $p\ge 1$ and $i=0$ if and only if *K* is an ellipsoid which is centered at the origin. The proof is complete. □

## Declarations

### Acknowledgements

The author is supported by the NNSF of China (11161019, 11371224).

## Authors’ Affiliations

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