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 $V(K)$ denote the n-dimensional volume of body K.

If $K\in {\mathcal{K}}^n$, then its support function, $h_K=h(K,\cdot ):{\mathbb{R}}^n\to (-\mathrm{\infty },+\mathrm{\infty })$, 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.

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

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

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 [3] 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 [6] are an excellent sources for the classical theory of valuations. Some more recent results can see [4, 5, 7–9]. Recently, Schuster in [10] 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:

• (a) Φ is continuous.

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

  $$\mathrm{\Phi }(K\mathrm{\#}L)=\mathrm{\Phi }K+\mathrm{\Phi }L.$$

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

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

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

The $L_p$ Minkowski valuation was introduced by Ludwig (see [11]). A function $\mathrm{\Psi }:{\mathcal{K}}_o^n\to {\mathcal{K}}_o^n$ is called an $L_p$ Minkowski valuation if

$$\mathrm{\Psi }(K\cup L){+}_p\mathrm{\Psi }(K\cap L)=\mathrm{\Psi }K{+}_p\mathrm{\Psi }L,$$

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 [12] 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.

• (a) ${\mathrm{\Phi }}_p$ is continuous with respect to Hausdorff metric.

• (b) ${\mathrm{\Phi }}_p(K{\mathrm{\#}}_{p}L)={\mathrm{\Phi }}_{p}K{+}_p{\mathrm{\Phi }}_{p}L$ for all $K,L\in {\mathcal{K}}_e^n$.

• (c) ${\mathrm{\Phi }}_p$ is $SO(n)$ equivariant, i.e., ${\mathrm{\Phi }}_p(\vartheta K)=\vartheta {\mathrm{\Phi }}_{p}K$ for all $\vartheta \in SO(n)$ 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., $S_p(K{\mathrm{\#}}_{p}L,\cdot )=S_p(K,\cdot ){+}_p{S}_p(L,\cdot )$.

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 $g\in \mathcal{C}(S^{n-1},\hat{e})$, such that

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

A map ${\mathrm{\Phi }}_p:{\mathcal{K}}_e^n\to {\mathcal{K}}_e^n$ is even because of ${\mathrm{\Phi }}_p(K)={\mathrm{\Phi }}_p(-K)$ 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

$$h^p({\mathrm{\Phi }}_{p}K,\cdot )=S_p(K,\cdot )\ast h(F,\cdot ).$$

(1.1)

In [12], 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 $1<p<n$, then

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\Rightarrow V(K)\le V(L).$$

If $p>n$, then

$${\mathrm{\Phi }}_{p}K\subseteq {\mathrm{\Phi }}_{p}L\Rightarrow V(K)\ge V(L),$$

and $V(K)=V(L)$, 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

$${\mathrm{\Omega }}_p(K)\le {\mathrm{\Omega }}_p(L),$$

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

Here ${\omega }_p^n=\{N\in {\mathcal{F}}_e^n:\text{ there exists }Z\in {\mathcal{Z}}_p^n\text{ with }{f}_p(N,\cdot )=h{(Z,\cdot )}^{-(n+p)}\}$, where $f_p(N,\cdot )$ 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, ${\mathrm{\Omega }}_p(K)$ 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 $Q\in K_e^n$, $V_p(K,Q)\le V_p(L,Q)$, then

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

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 $Q\in K_e^n$, $V_p(K,Q)\le V_p(L,Q)$, 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, ${\rho }_K=\rho (K,\cdot ):{\mathbb{R}}^n\mathrm{\setminus }\{0\}⟶[0,+\mathrm{\infty })$, is defined by (see [1])

$$\rho (K,x)=max\{\lambda \ge 0:\lambda x\in K\},x\in {\mathbb{R}}^n\mathrm{\setminus }\{0\}.$$

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 ${\rho }_K(u)/{\rho }_L(u)$ 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 [1])

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

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 [1]), respectively, by

$$h_{K^{\ast }}=\frac{1}{{\rho }_K},{\rho }_{K^{\ast }}=\frac{1}{h_K}.$$

(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, $\lambda \cdot K{+}_p\mu \cdot L\in {\mathcal{K}}_o^n$, of K and L is defined by (see [13])

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

(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 [14]) introduced the following. For $K,L\in {\mathcal{K}}_o^n$, $\epsilon >0$ and $p\ge 1$, the $L_p$-mixed volume, $V_p(K,L)$, of K and L is defined by

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

It was shown in [14] that corresponding to each $K\in {\mathcal{K}}_o^n$, there exists a positive Borel measure on $S^{n-1}$, $S_p(K,\cdot )$ of K, such that for each $L\in {\mathcal{K}}_o^n$,

$$V_p(K,L)=\frac{1}{n}{\int }_{S^{n-1}}{h}_L^p(v)d{S}_p(K,v).$$

(2.3)

The measure $S_p(K,\cdot )$ is just the $L_p$ surface area measure of K, which is absolutely continuous with respect to classical surface area measure $S(K,\cdot )$ and has a Radon-Nikodym derivative

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

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

$$V_p(K,K)=V(K).$$

(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 [14]): If $K,L\in {\mathcal{K}}_o^n$ and $p\ge 1$, then

$$V_p(K,L)\ge V{(K)}^{\frac{n-p}{n}}V{(L)}^{\frac{p}{n}},$$

(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 [14]) $f_p(K,\cdot ):S^{n-1}\to \mathbb{R}$, if its $L_p$ surface area measure $S_p(K,\cdot )$ is absolutely continuous with respect to spherical Lebesgue measure S and

$$\frac{d{S}_p(K,\cdot )}{dS}=f_p(K,\cdot ).$$

(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, $\lambda \star K{+}_{-p}\mu \star L\in {\mathcal{S}}_o^n$, of K and L is defined by (see [14])

$$\rho {(\lambda \star K{+}_{-p}\mu \star L,\cdot )}^{-p}=\lambda \rho {(K,\cdot )}^{-p}+\mu \rho {(L,\cdot )}^{-p}.$$

(2.7)

Using the $L_p$-harmonic radial combination (2.7), Lutwak (see [14]) 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, ${\tilde{V}}_{-p}(K,L)$, of K and L is defined by

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

The definition above and the polar coordinate formula for volume give the following integral representation of the $L_p$-dual mixed volume:

$${\tilde{V}}_{-p}(K,L)=\frac{1}{n}{\int }_{S^{n-1}}{\rho }_K^{n+p}(\upsilon ){\rho }_L^{-p}(\upsilon )dS(\upsilon ),$$

(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 $p\ge 1$,

$${\tilde{V}}_{-p}(K,K)=V(K)=\frac{1}{n}{\int }_{S^{n-1}}{\rho }_K^n(\upsilon )dS(\upsilon ).$$

(2.9)

Lutwak in [14] established the $L_p$-dual Minkowski inequality: If $K,L\in S_o^n$, and $p\ge 1$, then

$${\tilde{V}}_{-p}(K,L)\ge V{(K)}^{\frac{n+p}{n}}V{(L)}^{\frac{-p}{n}},$$

(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 [15]) 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, ${\mathrm{\Omega }}_i(K,L)$, of K and L is defined by

$${\mathrm{\Omega }}_i(K,L)={\int }_{S^{n-1}}f{(K,u)}^{\frac{n-i}{n+1}}f{(L,u)}^{\frac{i}{n+1}}dS(u).$$

For $K,L\in {\mathcal{F}}_o^n$, $p\ge 1$ and $i\in \mathbb{R}$, the $L_p$-mixed affine surface area, ${\mathrm{\Omega }}_{p,i}(K,L)$, of K and L is defined by Wang and Leng (see [16])

$${\mathrm{\Omega }}_{p,i}(K,L)={\int }_{S^{n-1}}{f}_p{(K,u)}^{\frac{n-i}{n+p}}{f}_p{(L,u)}^{\frac{i}{n+p}}dS(u).$$

(2.11)

Obviously, from (2.11), we have

$${\mathrm{\Omega }}_{p,i}(K,K)={\mathrm{\Omega }}_p(K).$$

(2.12)

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

$$\begin{array}{rl}{\mathrm{\Omega }}_{-p}(K,L)& ={\int }_{S^{n-1}}{f}_p(K,u)f_p{(L,u)}^{\frac{-p}{n+p}}dS(u)\\ ={\int }_{S^{n-1}}{f}_p{(L,u)}^{\frac{-p}{n+p}}d{S}_p(K,u).\end{array}$$

(2.13)

The Minkowski inequality for the $L_p$-mixed affine surface area was given by Wang and Leng (see [16]): If $K,L\in {\mathcal{F}}_o^n$, $p\ge 1$ and $i\in \mathbb{R}$, then for $i<0$ or $i>n$,

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

(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 $0<i<n$, (2.14) is reverse; for $i=0$ or $i=n$, (2.14) is identical.

Combining with (2.14), they in [16] obtain the following result. If $K,L\in {\mathcal{F}}_o^n$ and $p\ge 1$,

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

(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 $SO(n)$ and $S^{n-1}$. As usual, $SO(n)$ and $S^{n-1}$ will be equipped with invariant probability measures. Let $\mathcal{C}(SO(n))$, $\mathcal{C}(S^{n-1})$ be the spaces of continuous function on $SO(n)$ and $S^{n-1}$ with uniform topology and $\mathcal{M}(SO(n))$, $\mathcal{M}(S^{n-1})$ their dual spaces of signed finite Borel measures with weak topology. If $\mu ,\sigma \in \mathcal{M}(SO(n))$, the convolution $\mu \ast \sigma$ is defined by

$${\int }_{SO(n)}f(\vartheta )d(\mu \ast \sigma )(\vartheta )={\int }_{SO(n)}{\int }_{SO(n)}f(\eta \tau )d\mu (\eta )d\sigma (\tau ),$$

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

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

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

(2.16)

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

$$〈\mu ,f〉=〈f,\mu 〉={\int }_{S^{n-1}}f(u)d\mu (u).$$

(2.17)

From (2.16) and (2.17), it follows that (see [18]) if $\mu ,\nu \in \mathcal{M}(S^{n-1})$ and $f\in \mathcal{C}(S^{n-1})$, 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 $V_p(Q,{\mathrm{\Phi }}_{p}K)\le V_p(Q,{\mathrm{\Phi }}_{p}L)$, then

$${\mathrm{\Omega }}_p(K)\le {\mathrm{\Omega }}_p(L),$$

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

Wang in [12] 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 $K,L\in {\mathcal{K}}_e^n$,

$$V_p(K,{\mathrm{\Phi }}_{p}L)=V_p(L,{\mathrm{\Phi }}_{p}K).$$

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

$$h(Z,\cdot )=f_p{(N,\cdot )}^{\frac{-1}{n+p}}.$$

(3.1)

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

$$\begin{array}{rl}\frac{{\mathrm{\Omega }}_{-p}(L,N)}{{\mathrm{\Omega }}_{-p}(K,N)}& =\frac{{\int }_{S^{n-1}}{f}_p{(N,u)}^{\frac{-p}{n+p}}d{S}_p(L,u)}{{\int }_{S^{n-1}}{f}_p{(N,u)}^{\frac{-p}{n+p}}d{S}_p(K,u)}\\ =\frac{{\int }_{S^{n-1}}h{(Z,\cdot )}^{p}d{S}_p(L,u)}{{\int }_{S^{n-1}}h{(Z,\cdot )}^{p}d{S}_p(K,u)}\\ =\frac{V_p(L,Z)}{V_p(K,Z)}.\end{array}$$

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

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

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

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

(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

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

i.e.,

$${\mathrm{\Omega }}_p(K)\le {\mathrm{\Omega }}_p(L).$$

(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

$$V_p(K,Q)\le V_p(L,Q)$$

can be written as

$$V_p(K,{\mathrm{\Phi }}_{p}M)\le V_p(L,{\mathrm{\Phi }}_{p}M),$$

then from Lemma 3.1, it follows that

$$V_p(M,{\mathrm{\Phi }}_{p}K)\le V_p(M,{\mathrm{\Phi }}_{p}L).$$

(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

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

(3.5)

such that

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

(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

$$h^p(M_{{\mathrm{\Phi }}_p}L,\cdot )={\rho }^{n+p}(L,\cdot )\ast h(F,\cdot ).$$

(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

$${\tilde{V}}_{-p}(L,{\mathrm{\Phi }}_p^{\ast }K)=V_p(K,M_{{\mathrm{\Phi }}_{p}L}).$$

(3.8)

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

$$\begin{array}{rl}{\tilde{V}}_{-p}(L,{\mathrm{\Phi }}_p^{\ast }K)& =\frac{1}{n}〈{\rho }_L^{n+p}(u),{\rho }_{{\mathrm{\Phi }}_p^{\ast }K}^{-p}(u)〉\\ =\frac{1}{n}〈{\rho }_L^{n+p}(u),h_{{\mathrm{\Phi }}_{p}K}^p(u)〉\\ =\frac{1}{n}〈{\rho }_L^{n+p}(u),(S_p(K,u)\ast h(F,u))〉\\ =\frac{1}{n}〈{\rho }_L^{n+p}(u)\ast h(F,u),S_p(K,u)〉\\ =\frac{1}{n}〈h^p(M_{{\mathrm{\Phi }}_p}L,u),S_p(K,u)〉\\ =\frac{1}{n}{\int }_{S^{n-1}}{h}^p(M_{{\mathrm{\Phi }}_p}L,u)d{S}_p(K,u)\\ =V_p(K,M_{{\mathrm{\Phi }}_{p}L}).\end{array}$$

 □

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

$$V_p(K,Q)\le V_p(L,Q)$$

can be written as

$$V_p(K,M_{{\mathrm{\Phi }}_p}N)\le V_p(L,M_{{\mathrm{\Phi }}_p}N).$$

(3.9)

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

$${\tilde{V}}_{-p}(N,{\mathrm{\Phi }}_p^{\ast }K)\le {\tilde{V}}_{-p}(N,{\mathrm{\Phi }}_p^{\ast }L).$$

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

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

(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. □