# Monotonicity inequalities for ${L}_{p}$ Blaschke-Minkowski homomorphisms

## Abstract

Schuster introduced the notion of Blaschke-Minkowski homomorphism and considered its Shephard problems. Wang gave the definition of ${L}_{p}$ Blaschke-Minkowski homomorphisms and considered its Shephard problems for volume. In this paper, we obtain its Shephard type inequalities for the affine surface area and two monotonicity inequalities for ${L}_{p}$ Blaschke-Minkowski homomorphisms are established.

MSC:52A20, 52A40.

## 1 Introduction

Let ${\mathcal{K}}^{n}$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space ${\mathbb{R}}^{n}$. Let ${\mathcal{K}}_{o}^{n}$ denote the set of convex bodies and containing the origin in their interiors, and let ${\mathcal{K}}_{e}^{n}$ denote origin-symmetric convex bodies in ${\mathbb{R}}^{n}$. Let ${S}^{n-1}$ denote the unit sphere in ${\mathbb{R}}^{n}$, and let $V\left(K\right)$ denote the n-dimensional volume of body K.

If $K\in {\mathcal{K}}^{n}$, then its support function, ${h}_{K}=h\left(K,\cdot \right):{\mathbb{R}}^{n}\to \left(-\mathrm{\infty },+\mathrm{\infty }\right)$, is defined by (see [1, 2])

$h\left(K,x\right)=max\left\{x\cdot y:y\in K\right\},\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}^{n},$

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

A function Φ defined on ${\mathcal{K}}^{n}$ and taking values in an Ablelian semigroup is called a valuation if

$\mathrm{\Phi }\left(K\cup L\right)+\mathrm{\Phi }\left(K\cap L\right)=\mathrm{\Phi }K+\mathrm{\Phi }L,$

whenever K, L, $K\cup L$, $K\cap L\in {\mathcal{K}}^{n}$.

The theory of real valued valuations is at the center of convex geometry. A systematic study was initiated by Blaschke in the 1930s, and then Hadwiger  focused on classifying valuations on compact convex sets in ${\mathbb{R}}^{n}$ and obtained the famous Hadwiger’s characterization theorem. Schneider obtained first results on convex body valued valuations with Minkowski addition in 1970s. The survey [4, 5] and the book  are an excellent sources for the classical theory of valuations. Some more recent results can see [4, 5, 79]. Recently, Schuster in  gave the definition of Blaschke-Minkowski homomorphism as follows:

A map $\mathrm{\Phi }:{\mathcal{K}}^{n}\to {\mathcal{K}}^{n}$ is called Blaschke-Minkowski homomorphism if it satisfies the following conditions:

1. (a)

Φ is continuous.

2. (b)

Φ is a Blaschke-Minkowski addition, i.e., for all $K,L\in {\mathcal{K}}^{n}$

$\mathrm{\Phi }\left(K\mathrm{#}L\right)=\mathrm{\Phi }K+\mathrm{\Phi }L.$
3. (c)

Φ intertwines rotation, i.e., for all $K\in {\mathcal{K}}^{n}$ and $\vartheta \in SO\left(n\right)$

$\mathrm{\Phi }\left(\vartheta K\right)=\vartheta \mathrm{\Phi }K.$

Here $K\mathrm{#}L$ is the Blaschke sum of the convex bodies K and L, i.e., $S\left(K\mathrm{#}L,\cdot \right)=S\left(K,\cdot \right)+S\left(L,\cdot \right)$. $SO\left(n\right)$ is the group of rotation in n dimensions.

The ${L}_{p}$ Minkowski valuation was introduced by Ludwig (see ). A function $\mathrm{\Psi }:{\mathcal{K}}_{o}^{n}\to {\mathcal{K}}_{o}^{n}$ is called an ${L}_{p}$ Minkowski valuation if

$\mathrm{\Psi }\left(K\cup L\right)\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}\mathrm{\Psi }\left(K\cap L\right)=\mathrm{\Psi }K\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}\mathrm{\Psi }L,$

whenever K, L, $K\cup L\in {\mathcal{K}}_{o}^{n}$, and here ‘${+}_{p}$’ is ${L}_{p}$ Minkowski addition (see (2.2)).

Then, Wang in  introduced the ${L}_{p}$ Blaschke-Minkowski homomorphism and gave Theorem 1.A.

Definition 1.1 Let $p>1$, a map ${\mathrm{\Phi }}_{p}:{\mathcal{K}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$ satisfying the following properties (a), (b) and (c) is called an ${L}_{p}$ Blaschke-Minkowski homomorphism.

1. (a)

${\mathrm{\Phi }}_{p}$ is continuous with respect to Hausdorff metric.

2. (b)

${\mathrm{\Phi }}_{p}\left(K{\mathrm{#}}_{p}L\right)={\mathrm{\Phi }}_{p}K\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}{\mathrm{\Phi }}_{p}L$ for all $K,L\in {\mathcal{K}}_{e}^{n}$.

3. (c)

${\mathrm{\Phi }}_{p}$ is $SO\left(n\right)$ equivariant, i.e., ${\mathrm{\Phi }}_{p}\left(\vartheta K\right)=\vartheta {\mathrm{\Phi }}_{p}K$ for all $\vartheta \in SO\left(n\right)$ and all $K\in {\mathcal{K}}_{e}^{n}$.

Here $K{\mathrm{#}}_{p}L$ denotes the ${L}_{p}$ Blaschke sum of $K,L\in {\mathcal{K}}_{e}^{n}$, i.e., ${S}_{p}\left(K{\mathrm{#}}_{p}L,\cdot \right)={S}_{p}\left(K,\cdot \right)\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}{S}_{p}\left(L,\cdot \right)$.

Theorem 1.A Let $p>1$ and $p\ne n$. If ${\mathrm{\Phi }}_{p}:{\mathcal{K}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$ is an ${L}_{p}$ Blaschke-Minkowski homomorphism, then there is a nonnegative function $g\in \mathcal{C}\left({S}^{n-1},\stackrel{ˆ}{e}\right)$, such that

${h}^{p}\left({\mathrm{\Phi }}_{p}K,\cdot \right)={S}_{p}\left(K,\cdot \right)\ast g.$

A map ${\mathrm{\Phi }}_{p}:{\mathcal{K}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$ is even because of ${\mathrm{\Phi }}_{p}\left(K\right)={\mathrm{\Phi }}_{p}\left(-K\right)$ for $K\in {\mathcal{K}}_{e}^{n}$.

A map ${\mathrm{\Phi }}_{p}:{\mathcal{K}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$ is an even ${L}_{p}$ Blaschke-Minkowski homomorphism, if and only if there is a convex body of revolution $F\in {\mathcal{K}}_{e}^{n}$, unique up to translation, such that

${h}^{p}\left({\mathrm{\Phi }}_{p}K,\cdot \right)={S}_{p}\left(K,\cdot \right)\ast h\left(F,\cdot \right).$
(1.1)

In , together with the ${L}_{p}$ Blaschke-Minkowski homomorphisms, Wang studied the Shephard problems of ${L}_{p}$ Blaschke-Minkowski homomorphisms.

Theorem 1.B Let ${\mathrm{\Phi }}_{p}:{\mathcal{K}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$ is an ${L}_{p}$ Blaschke-Minkowski homomorphism, $K\in {\mathcal{K}}_{e}^{n}$, $L\in {\mathrm{\Phi }}_{p}{\mathcal{K}}_{e}^{n}$ and p is not an even integer. If $1, then

${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}V\left(K\right)\le V\left(L\right).$

If $p>n$, then

${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}V\left(K\right)\ge V\left(L\right),$

and $V\left(K\right)=V\left(L\right)$, if and only if $K=L$.

In this article, we continuously study the ${L}_{p}$ Blaschke-Minkowski homomorphisms. Firstly, comparing with Theorem 1.B, we give the ${L}_{p}$-affine surface area of Shephard type inequalities for the ${L}_{p}$ Blaschke-Minkowski homomorphisms.

Theorem 1.1 Let $K\in {\mathcal{F}}_{e}^{n}$, $L\in {\omega }_{p}^{n}$ and $n\ne p>1$. If ${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L$, then

${\mathrm{\Omega }}_{p}\left(K\right)\le {\mathrm{\Omega }}_{p}\left(L\right),$

with equality if and only if K and L are dilates.

Here , where ${f}_{p}\left(N,\cdot \right)$ is the p-curvature function of N, ${\mathcal{F}}_{e}^{n}$ denotes the set of convex bodies in ${\mathcal{K}}_{e}^{n}$ with positive continuous curvature function and ${\mathcal{Z}}_{p}^{n}$ denotes the set of ${L}_{p}$ Blaschke-Minkowski homomorphisms. Besides, ${\mathrm{\Omega }}_{p}\left(K\right)$ denotes the ${L}_{p}$-affine surface area of $K\in {\mathcal{K}}_{o}^{n}$.

Actually, we will prove a more general result than Theorem 1.1 in Section 3.

Further, associated with the ${L}_{p}$ Blaschke-Minkowski homomorphisms, we establish the following monotonicity inequalities.

Theorem 1.2 Let $K,L\in {K}_{e}^{n}$, $n\ne p>1$. If for every $Q\in {K}_{e}^{n}$, ${V}_{p}\left(K,Q\right)\le {V}_{p}\left(L,Q\right)$, then

$V\left({\mathrm{\Phi }}_{p}K\right)\le V\left({\mathrm{\Phi }}_{p}L\right),$

with equality if and only if K and L are dilates.

Theorem 1.3 Let $K,L\in {\mathcal{K}}_{e}^{n}$, $n\ne p>1$. If for every $Q\in {K}_{e}^{n}$, ${V}_{p}\left(K,Q\right)\le {V}_{p}\left(L,Q\right)$, then

$V\left({\mathrm{\Phi }}_{p}^{\ast }L\right)\le V\left({\mathrm{\Phi }}_{p}^{\ast }K\right),$

with equality if and only if K and L are dilates.

Here and the following we write ${\mathrm{\Phi }}_{p}^{\ast }K$ for the polar of ${\mathrm{\Phi }}_{p}K$.

## 2 Notations and background materials

If K is a compact star-shaped (about the origin) in ${\mathbb{R}}^{n}$, its radial function, ${\rho }_{K}=\rho \left(K,\cdot \right):{\mathbb{R}}^{n}\mathrm{\setminus }\left\{0\right\}⟶\left[0,+\mathrm{\infty }\right)$, is defined by (see )

$\rho \left(K,x\right)=max\left\{\lambda \ge 0:\lambda x\in K\right\},\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}^{n}\mathrm{\setminus }\left\{0\right\}.$

If ${\rho }_{K}$ is positive and continuous, K will be called a star body (about the origin). Two star bodies K and L are said to be dilates (of one another) if ${\rho }_{K}\left(u\right)/{\rho }_{L}\left(u\right)$ is independent of $u\in {S}^{n-1}$. Let ${\mathcal{S}}_{o}^{n}$ denote the set of star bodies (about the origin), and let ${\mathcal{S}}_{e}^{n}$ denote the set of origin-symmetric star bodies.

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

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

If $K\in {\mathcal{K}}_{o}^{n}$, then the support function and radial function of ${K}^{\ast }$, the polar body of K, are given (see ), respectively, by

${h}_{{K}^{\ast }}=\frac{1}{{\rho }_{K}},\phantom{\rule{2em}{0ex}}{\rho }_{{K}^{\ast }}=\frac{1}{{h}_{K}}.$
(2.1)

### 2.1 ${L}_{p}$-mixed volume

For $K,L\in {K}_{o}^{n}$, $p\ge 1$ and $\lambda ,\mu \ge 0$ (not both zero), the Firey ${L}_{p}$-combination, $\lambda \cdot K\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}\mu \cdot L\in {\mathcal{K}}_{o}^{n}$, of K and L is defined by (see )

$h{\left(\lambda \cdot K\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}\mu \cdot L,\cdot \right)}^{p}=\lambda h{\left(K,\cdot \right)}^{p}+\mu h{\left(L,\cdot \right)}^{p},$
(2.2)

where ‘’ in $\lambda \cdot K$ denotes the Firey scalar multiplication.

Associated with Firey ${L}_{p}$-combination (2.2) of convex bodies, Lutwak (see ) introduced the following. For $K,L\in {\mathcal{K}}_{o}^{n}$, $\epsilon >0$ and $p\ge 1$, the ${L}_{p}$-mixed volume, ${V}_{p}\left(K,L\right)$, of K and L is defined by

$\frac{n}{p}{V}_{p}\left(K,L\right)=\underset{\epsilon ⟶{0}^{+}}{lim}\frac{V\left(K\phantom{\rule{0.2em}{0ex}}{+}_{p}\phantom{\rule{0.2em}{0ex}}\epsilon \cdot L\right)-V\left(K\right)}{\epsilon }.$

It was shown in  that corresponding to each $K\in {\mathcal{K}}_{o}^{n}$, there exists a positive Borel measure on ${S}^{n-1}$, ${S}_{p}\left(K,\cdot \right)$ of K, such that for each $L\in {\mathcal{K}}_{o}^{n}$,

${V}_{p}\left(K,L\right)=\frac{1}{n}{\int }_{{S}^{n-1}}{h}_{L}^{p}\left(v\right)\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(K,v\right).$
(2.3)

The measure ${S}_{p}\left(K,\cdot \right)$ is just the ${L}_{p}$ surface area measure of K, which is absolutely continuous with respect to classical surface area measure $S\left(K,\cdot \right)$ and has a Radon-Nikodym derivative

$\frac{d{S}_{p}\left(K,\cdot \right)}{dS\left(K,\cdot \right)}=h{\left(K,\cdot \right)}^{1-p}.$

Obviously, from (2.3), it follows immediately that, for each $K\in {\mathcal{K}}_{o}^{n}$,

${V}_{p}\left(K,K\right)=V\left(K\right).$
(2.4)

The Minkowski inequality for the ${L}_{p}$-mixed volume is called ${L}_{p}$-Minkowski inequality. The ${L}_{p}$-Minkowski inequality can be stated that (see ): If $K,L\in {\mathcal{K}}_{o}^{n}$ and $p\ge 1$, then

${V}_{p}\left(K,L\right)\ge V{\left(K\right)}^{\frac{n-p}{n}}V{\left(L\right)}^{\frac{p}{n}},$
(2.5)

with equality for $p=1$ if and only if K and L are homothetic, for $p>1$ if and only if K and L are dilates.

A convex body $K\in {\mathcal{K}}_{o}^{n}$ is said to have a ${L}_{p}$-curvature function (see ) ${f}_{p}\left(K,\cdot \right):{S}^{n-1}\to \mathbb{R}$, if its ${L}_{p}$ surface area measure ${S}_{p}\left(K,\cdot \right)$ is absolutely continuous with respect to spherical Lebesgue measure S and

$\frac{d{S}_{p}\left(K,\cdot \right)}{dS}={f}_{p}\left(K,\cdot \right).$
(2.6)

### 2.2 ${L}_{p}$-dual mixed volume

For $K,L\in {S}_{o}^{n}$, $p\ge 1$ and $\lambda ,\mu \ge 0$ (not both zero), the ${L}_{p}$-harmonic radial combination, $\lambda \star K\phantom{\rule{0.2em}{0ex}}{+}_{-p}\phantom{\rule{0.2em}{0ex}}\mu \star L\in {\mathcal{S}}_{o}^{n}$, of K and L is defined by (see )

$\rho {\left(\lambda \star K\phantom{\rule{0.2em}{0ex}}{+}_{-p}\phantom{\rule{0.2em}{0ex}}\mu \star L,\cdot \right)}^{-p}=\lambda \rho {\left(K,\cdot \right)}^{-p}+\mu \rho {\left(L,\cdot \right)}^{-p}.$
(2.7)

Using the ${L}_{p}$-harmonic radial combination (2.7), Lutwak (see ) introduced the notion of ${L}_{p}$-dual mixed volume. For $K,L\in {S}_{o}^{n}$ and $p\ge 1$, the ${L}_{p}$-dual mixed volume, ${\stackrel{˜}{V}}_{-p}\left(K,L\right)$, of K and L is defined by

$\frac{n}{-p}{\stackrel{˜}{V}}_{-p}\left(K,L\right)=\underset{\epsilon \to {0}^{+}}{lim}\frac{V\left(K{+}_{-p}\epsilon \star L\right)-V\left(K\right)}{\epsilon }.$

The definition above and the polar coordinate formula for volume give the following integral representation of the ${L}_{p}$-dual mixed volume:

${\stackrel{˜}{V}}_{-p}\left(K,L\right)=\frac{1}{n}{\int }_{{S}^{n-1}}{\rho }_{K}^{n+p}\left(\upsilon \right){\rho }_{L}^{-p}\left(\upsilon \right)\phantom{\rule{0.2em}{0ex}}dS\left(\upsilon \right),$
(2.8)

where the integration is with respect to spherical Lebesgue measure S on ${S}^{n-1}$.

From (2.8), it follows that for each $K\in {S}_{o}^{n}$ and $p\ge 1$,

${\stackrel{˜}{V}}_{-p}\left(K,K\right)=V\left(K\right)=\frac{1}{n}{\int }_{{S}^{n-1}}{\rho }_{K}^{n}\left(\upsilon \right)\phantom{\rule{0.2em}{0ex}}dS\left(\upsilon \right).$
(2.9)

Lutwak in  established the ${L}_{p}$-dual Minkowski inequality: If $K,L\in {S}_{o}^{n}$, and $p\ge 1$, then

${\stackrel{˜}{V}}_{-p}\left(K,L\right)\ge V{\left(K\right)}^{\frac{n+p}{n}}V{\left(L\right)}^{\frac{-p}{n}},$
(2.10)

with equality if and only if K and L are dilates.

### 2.3 ${L}_{p}$-mixed affine surface area

Let ${\mathcal{F}}^{n}$, ${\mathcal{F}}_{o}^{n}$ denote the set of convex bodies in ${\mathcal{K}}^{n}$, ${\mathcal{K}}_{o}^{n}$ with positive continuous curvature function.

Lutwak (see ) defined the i th mixed affine surface area as follows: For $K,L\in {\mathcal{F}}^{n}$ and $i\in \mathbb{R}$, the i th mixed affine surface area, ${\mathrm{\Omega }}_{i}\left(K,L\right)$, of K and L is defined by

${\mathrm{\Omega }}_{i}\left(K,L\right)={\int }_{{S}^{n-1}}f{\left(K,u\right)}^{\frac{n-i}{n+1}}f{\left(L,u\right)}^{\frac{i}{n+1}}\phantom{\rule{0.2em}{0ex}}dS\left(u\right).$

For $K,L\in {\mathcal{F}}_{o}^{n}$, $p\ge 1$ and $i\in \mathbb{R}$, the ${L}_{p}$-mixed affine surface area, ${\mathrm{\Omega }}_{p,i}\left(K,L\right)$, of K and L is defined by Wang and Leng (see )

${\mathrm{\Omega }}_{p,i}\left(K,L\right)={\int }_{{S}^{n-1}}{f}_{p}{\left(K,u\right)}^{\frac{n-i}{n+p}}{f}_{p}{\left(L,u\right)}^{\frac{i}{n+p}}\phantom{\rule{0.2em}{0ex}}dS\left(u\right).$
(2.11)

Obviously, from (2.11), we have

${\mathrm{\Omega }}_{p,i}\left(K,K\right)={\mathrm{\Omega }}_{p}\left(K\right).$
(2.12)

Specially, for the case $i=-p$, we write ${\mathrm{\Omega }}_{p,-p}\left(K,L\right)={\mathrm{\Omega }}_{-p}\left(K,L\right)$. Associated with (2.6), then

$\begin{array}{rl}{\mathrm{\Omega }}_{-p}\left(K,L\right)& ={\int }_{{S}^{n-1}}{f}_{p}\left(K,u\right){f}_{p}{\left(L,u\right)}^{\frac{-p}{n+p}}\phantom{\rule{0.2em}{0ex}}dS\left(u\right)\\ ={\int }_{{S}^{n-1}}{f}_{p}{\left(L,u\right)}^{\frac{-p}{n+p}}\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(K,u\right).\end{array}$
(2.13)

The Minkowski inequality for the ${L}_{p}$-mixed affine surface area was given by Wang and Leng (see ): If $K,L\in {\mathcal{F}}_{o}^{n}$, $p\ge 1$ and $i\in \mathbb{R}$, then for $i<0$ or $i>n$,

${\mathrm{\Omega }}_{p,i}{\left(K,L\right)}^{n}\ge {\mathrm{\Omega }}_{p}{\left(K\right)}^{n-i}{\mathrm{\Omega }}_{p}{\left(L\right)}^{i},$
(2.14)

with equality for $p=1$ if and only if K and L are homothetic, for $n\ne p>1$ if and only if K and L are dilates; for $0, (2.14) is reverse; for $i=0$ or $i=n$, (2.14) is identical.

Combining with (2.14), they in  obtain the following result. If $K,L\in {\mathcal{F}}_{o}^{n}$ and $p\ge 1$,

${\mathrm{\Omega }}_{-p}\left(K,L\right)={\mathrm{\Omega }}_{p,-p}\left(K,L\right)\ge {\mathrm{\Omega }}_{p}{\left(K\right)}^{\frac{n+p}{n}}{\mathrm{\Omega }}_{p}{\left(L\right)}^{\frac{-p}{n}},$
(2.15)

with equality for $n\ne p>1$ if and only if K and L are dilates, for $p=1$ if and only if K and L are homothetic.

### 2.4 Spherical convolution and spherical harmonics

In the following we state some material on convolution and spherical harmonics, and they can be found in the references (see [17, 18]).

In order to state the material on spherical harmonics, we first introduce further basic notions connected to $SO\left(n\right)$ and ${S}^{n-1}$. As usual, $SO\left(n\right)$ and ${S}^{n-1}$ will be equipped with invariant probability measures. Let $\mathcal{C}\left(SO\left(n\right)\right)$, $\mathcal{C}\left({S}^{n-1}\right)$ be the spaces of continuous function on $SO\left(n\right)$ and ${S}^{n-1}$ with uniform topology and $\mathcal{M}\left(SO\left(n\right)\right)$, $\mathcal{M}\left({S}^{n-1}\right)$ their dual spaces of signed finite Borel measures with weak topology. If $\mu ,\sigma \in \mathcal{M}\left(SO\left(n\right)\right)$, the convolution $\mu \ast \sigma$ is defined by

${\int }_{SO\left(n\right)}f\left(\vartheta \right)\phantom{\rule{0.2em}{0ex}}d\left(\mu \ast \sigma \right)\left(\vartheta \right)={\int }_{SO\left(n\right)}{\int }_{SO\left(n\right)}f\left(\eta \tau \right)\phantom{\rule{0.2em}{0ex}}d\mu \left(\eta \right)\phantom{\rule{0.2em}{0ex}}d\sigma \left(\tau \right),$

for every $f\in \mathcal{C}\left(SO\left(n\right)\right)$ and $\vartheta \in SO\left(n\right)$. The sphere ${S}^{n-1}$ is identical with the honogeneous space $SO\left(n\right)/SO\left(n-1\right)$, where $SO\left(n-1\right)$ denotes the subgroup of rotations leaving the pole $\stackrel{ˆ}{e}$ of ${S}^{n-1}$ fixed.

For $\mu \in \mathcal{M}\left(SO\left(n\right)\right)$, the convolutions $\mu \ast f\in \mathcal{C}\left(SO\left(n\right)\right)$ and $f\ast \mu \in \mathcal{C}\left(SO\left(n\right)\right)$ with a function $f\in \mathcal{C}\left(SO\left(n\right)\right)$ are defined by

$\begin{array}{r}\left(f\ast \mu \right)\left(\eta \right)={\int }_{SO\left(n\right)}f\left(\eta {\vartheta }^{-1}\right)\phantom{\rule{0.2em}{0ex}}d\mu \left(\vartheta \right),\\ \left(\mu \ast f\right)\left(\eta \right)={\int }_{SO\left(n\right)}\vartheta f\left(\eta \right)\phantom{\rule{0.2em}{0ex}}d\mu \left(\vartheta \right).\end{array}$
(2.16)

The canonical pairing of $f\in \mathcal{C}\left({S}^{n-1}\right)$ and $\mu \in \mathcal{M}\left({S}^{n-1}\right)$ is defined by

$〈\mu ,f〉=〈f,\mu 〉={\int }_{{S}^{n-1}}f\left(u\right)\phantom{\rule{0.2em}{0ex}}d\mu \left(u\right).$
(2.17)

From (2.16) and (2.17), it follows that (see ) if $\mu ,\nu \in \mathcal{M}\left({S}^{n-1}\right)$ and $f\in \mathcal{C}\left({S}^{n-1}\right)$, then

$〈\mu \ast \nu ,f〉=〈\mu ,f\ast \nu 〉.$
(2.18)

## 3 Proofs of theorems

In this section, firstly, we will prove the general form of Theorem 1.1.

Theorem 3.1 Let $K\in {\mathcal{F}}_{e}^{n}$, $L\in {\omega }_{p}^{n}$ and $n\ne p>1$. For every $Q\in {\mathcal{K}}_{e}^{n}$, if ${V}_{p}\left(Q,{\mathrm{\Phi }}_{p}K\right)\le {V}_{p}\left(Q,{\mathrm{\Phi }}_{p}L\right)$, then

${\mathrm{\Omega }}_{p}\left(K\right)\le {\mathrm{\Omega }}_{p}\left(L\right),$

with equality if and only if K and L are dilates.

Wang in  gave the following conclusion; this result is a very useful tool for the following proofs.

Lemma 3.1 If ${\mathrm{\Phi }}_{p}:{\mathcal{K}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$, is an ${L}_{p}$ Blaschke-Minkowski homomorphism, then for $K,L\in {\mathcal{K}}_{e}^{n}$,

${V}_{p}\left(K,{\mathrm{\Phi }}_{p}L\right)={V}_{p}\left(L,{\mathrm{\Phi }}_{p}K\right).$

Proof of Theorem 3.1 Since $N\in {\omega }_{p}^{n}$, then there exists $Z\in {\mathcal{Z}}_{p}^{n}$ such that

$h\left(Z,\cdot \right)={f}_{p}{\left(N,\cdot \right)}^{\frac{-1}{n+p}}.$
(3.1)

By (2.3), (2.13), and (3.1), we consider

$\begin{array}{rl}\frac{{\mathrm{\Omega }}_{-p}\left(L,N\right)}{{\mathrm{\Omega }}_{-p}\left(K,N\right)}& =\frac{{\int }_{{S}^{n-1}}{f}_{p}{\left(N,u\right)}^{\frac{-p}{n+p}}\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(L,u\right)}{{\int }_{{S}^{n-1}}{f}_{p}{\left(N,u\right)}^{\frac{-p}{n+p}}\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(K,u\right)}\\ =\frac{{\int }_{{S}^{n-1}}h{\left(Z,\cdot \right)}^{p}\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(L,u\right)}{{\int }_{{S}^{n-1}}h{\left(Z,\cdot \right)}^{p}\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(K,u\right)}\\ =\frac{{V}_{p}\left(L,Z\right)}{{V}_{p}\left(K,Z\right)}.\end{array}$

Since $Z\in {\mathcal{Z}}_{p}^{n}$, letting $Z={\mathrm{\Phi }}_{p}Q$ for $Q\in {\mathcal{K}}_{e}^{n}$, combining with Lemma 3.1, we obtain

$\frac{{V}_{p}\left(L,Z\right)}{{V}_{p}\left(K,Z\right)}=\frac{{V}_{p}\left(L,{\mathrm{\Phi }}_{p}Q\right)}{{V}_{p}\left(K,{\mathrm{\Phi }}_{p}Q\right)}=\frac{{V}_{p}\left(Q,{\mathrm{\Phi }}_{p}L\right)}{{V}_{p}\left(Q,{\mathrm{\Phi }}_{p}K\right)}.$

Therefore, if ${V}_{p}\left(Q,{\mathrm{\Phi }}_{p}K\right)\le {V}_{p}\left(Q,{\mathrm{\Phi }}_{p}L\right)$, then we have

${\mathrm{\Omega }}_{-p}\left(L,N\right)\ge {\mathrm{\Omega }}_{-p}\left(K,N\right).$
(3.2)

Due to $L\in {\omega }_{p}^{n}$, taking $N=L$ in (3.2), and together with (2.12) and inequality (2.15), we get

${\mathrm{\Omega }}_{p}\left(L\right)\ge {\mathrm{\Omega }}_{-p}\left(K,L\right)\ge {\mathrm{\Omega }}_{p}{\left(K\right)}^{\frac{n+p}{n}}{\mathrm{\Omega }}_{p}{\left(L\right)}^{\frac{-p}{n}},$

i.e.,

${\mathrm{\Omega }}_{p}\left(K\right)\le {\mathrm{\Omega }}_{p}\left(L\right).$
(3.3)

According to the equality conditions of (2.15) and (3.2), we see that equality holds in (3.3) for $n\ne p>1$ if and only if K and L are dilates. □

Proof of Theorem 1.2 Since $Q\in {K}_{e}^{n}$, taking $Q={\mathrm{\Phi }}_{p}M$ for $M\in {K}_{e}^{n}$, then

${V}_{p}\left(K,Q\right)\le {V}_{p}\left(L,Q\right)$

can be written as

${V}_{p}\left(K,{\mathrm{\Phi }}_{p}M\right)\le {V}_{p}\left(L,{\mathrm{\Phi }}_{p}M\right),$

then from Lemma 3.1, it follows that

${V}_{p}\left(M,{\mathrm{\Phi }}_{p}K\right)\le {V}_{p}\left(M,{\mathrm{\Phi }}_{p}L\right).$
(3.4)

Since ${\mathrm{\Phi }}_{p}L\in {K}_{e}^{n}$, let $M={\mathrm{\Phi }}_{p}L$ in (3.4), together with (2.4) and (2.5), we can get

$V\left({\mathrm{\Phi }}_{p}L\right)\ge {V}_{p}\left({\mathrm{\Phi }}_{p}L,{\mathrm{\Phi }}_{p}K\right)\ge V{\left({\mathrm{\Phi }}_{p}L\right)}^{\frac{n-p}{n}}V{\left({\mathrm{\Phi }}_{p}K\right)}^{\frac{p}{n}},$
(3.5)

such that

$V\left({\mathrm{\Phi }}_{p}K\right)\le V\left({\mathrm{\Phi }}_{p}L\right).$
(3.6)

According to the equality conditions of (2.5) and (3.5), we see that equality holds in (3.6) for $n\ne p>1$ if and only if K and L are dilates. □

We turn now to proof of Theorem 1.3. To this end, associate with the ${L}_{p}$ Blaschke-Minkowski homomorphism ${\mathrm{\Phi }}_{p}$, we define a new operator ${M}_{{\mathrm{\Phi }}_{p}}:{\mathcal{S}}_{e}^{n}\to {\mathcal{K}}_{e}^{n}$ by

${h}^{p}\left({M}_{{\mathrm{\Phi }}_{p}}L,\cdot \right)={\rho }^{n+p}\left(L,\cdot \right)\ast h\left(F,\cdot \right).$
(3.7)

By (2.16), the operator ${M}_{{\mathrm{\Phi }}_{p}}$ is well defined.

Lemma 3.2 If $K\in {K}_{e}^{n}$, $L\in {S}_{e}^{n}$, $n\ne p>1$, then

${\stackrel{˜}{V}}_{-p}\left(L,{\mathrm{\Phi }}_{p}^{\ast }K\right)={V}_{p}\left(K,{M}_{{\mathrm{\Phi }}_{p}L}\right).$
(3.8)

Proof By (1.1), (2.1), (2.3), (2.8), (2.18), and (3.7), we have

$\begin{array}{rl}{\stackrel{˜}{V}}_{-p}\left(L,{\mathrm{\Phi }}_{p}^{\ast }K\right)& =\frac{1}{n}〈{\rho }_{L}^{n+p}\left(u\right),{\rho }_{{\mathrm{\Phi }}_{p}^{\ast }K}^{-p}\left(u\right)〉\\ =\frac{1}{n}〈{\rho }_{L}^{n+p}\left(u\right),{h}_{{\mathrm{\Phi }}_{p}K}^{p}\left(u\right)〉\\ =\frac{1}{n}〈{\rho }_{L}^{n+p}\left(u\right),\left({S}_{p}\left(K,u\right)\ast h\left(F,u\right)\right)〉\\ =\frac{1}{n}〈{\rho }_{L}^{n+p}\left(u\right)\ast h\left(F,u\right),{S}_{p}\left(K,u\right)〉\\ =\frac{1}{n}〈{h}^{p}\left({M}_{{\mathrm{\Phi }}_{p}}L,u\right),{S}_{p}\left(K,u\right)〉\\ =\frac{1}{n}{\int }_{{S}^{n-1}}{h}^{p}\left({M}_{{\mathrm{\Phi }}_{p}}L,u\right)\phantom{\rule{0.2em}{0ex}}d{S}_{p}\left(K,u\right)\\ ={V}_{p}\left(K,{M}_{{\mathrm{\Phi }}_{p}L}\right).\end{array}$

□

Proof of Theorem 1.3 Since $Q\in {\mathcal{K}}_{e}^{n}$, taking $Q={M}_{{\mathrm{\Phi }}_{p}}N$ for any $N\in {S}_{e}^{n}$, then

${V}_{p}\left(K,Q\right)\le {V}_{p}\left(L,Q\right)$

can be written as

${V}_{p}\left(K,{M}_{{\mathrm{\Phi }}_{p}}N\right)\le {V}_{p}\left(L,{M}_{{\mathrm{\Phi }}_{p}}N\right).$
(3.9)

Combining with (3.8), (3.9) can be written as

${\stackrel{˜}{V}}_{-p}\left(N,{\mathrm{\Phi }}_{p}^{\ast }K\right)\le {\stackrel{˜}{V}}_{-p}\left(N,{\mathrm{\Phi }}_{p}^{\ast }L\right).$

But $N\in {S}_{e}^{n}$, taking $N={\mathrm{\Phi }}_{p}^{\ast }L$, together with (2.9) and inequality (2.10), we get

$\begin{array}{rl}V\left({\mathrm{\Phi }}_{p}^{\ast }L\right)& \ge {\stackrel{˜}{V}}_{-p}\left({\mathrm{\Phi }}_{p}^{\ast }L,{\mathrm{\Phi }}_{p}^{\ast }K\right)\\ \ge V{\left({\mathrm{\Phi }}_{p}^{\ast }L\right)}^{\frac{n+p}{n}}V{\left({\mathrm{\Phi }}_{p}^{\ast }K\right)}^{\frac{-p}{n}},\end{array}$
(3.10)

such that

$V\left({\mathrm{\Phi }}_{p}^{\ast }L\right)\le V\left({\mathrm{\Phi }}_{p}^{\ast }K\right).$
(3.11)

According to the equality conditions of (2.9) and (3.10), we see that equality holds in (3.11) for $n\ne p>1$ if and only if K and L are dilates. □

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## Acknowledgements

The authors would like to deeply thank the referees for very valuable and helpful comments and suggestions, which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224).

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Li, Y., Wang, W. Monotonicity inequalities for ${L}_{p}$ Blaschke-Minkowski homomorphisms. J Inequal Appl 2014, 131 (2014). https://doi.org/10.1186/1029-242X-2014-131 