Inequalities on General L_p-Mixed Chord Integral Difference

Abstract

In this article, we introduce the concept of general $L_p$-mixed chord integral difference of star bodies. Further, we establish the Brunn–Minkowski type, Aleksandrov–Fenchel type and cyclic inequalities for the $L_p$-mixed chord integral difference.

1. Introduction

The setting for this paper is n-dimensional Euclidean spaces ${\mathbb{R}}^n$$(n\ge 1)$. Let K and L be two convex bodies (compact, convex subsets with nonempty interiors) in ${\mathbb{R}}^n$. V denotes the volume. If K is a compact star-shaped (about the origin) set in ${\mathbb{R}}^n$, then its radial function, ${\rho }_K=\rho (K,·):{\mathbb{R}}^n\backslash \{0\}\to [0,\infty )$, is defined by (see [1]):

$$\rho (K,u)=max\{\lambda \ge 0,\lambda u\in K\},u\in S^{n-1}.$$

If ${\rho }_K$ is positive and continuous, K is called a star body (about the origin), and $S^n$ denotes the set of star bodies in ${\mathbb{R}}^n$. $S_0^n$ is the subset of $S^n$ containing the origin in their interiors. The unit sphere in ${\mathbb{R}}^n$ is denoted by $S^{n-1}$, and B denotes the standard unit ball in ${\mathbb{R}}^n$.

The classical Brunn–Minkowski inequality is (see [2])

$$V{(K+L)}^{\frac{1}{n}}\ge V{(K)}^{\frac{1}{n}}+V{(L)}^{\frac{1}{n}},$$

where + denotes vector or the Minkowski sum of two sets, i.e., $A+B=\{a+b:a\in A,b\in B\}.$

In 2004, Leng (see [3]) presented a new generalization of the Brunn–Minkowski inequality for the volume difference of convex bodies.

Theorem 1.

Suppose that $K,L$ and D are compact domains, and $D\subset K,D^{\prime }\subset L$, $D^{\prime }$ is a homothetic copy of D. Then

$${[V(K+L)-V(D+D^{\prime })]}^{\frac{1}{n}}\ge {[V(K)-V(D)]}^{\frac{1}{n}}+{[V(L)-V(D^{\prime })]}^{\frac{1}{n}}.$$

The equality holds if and only if K and L are homothetic and $(V(K),V(D))=\mu (V(L),V(D^{\prime })),$ where μ is a constant.

Leng’s result is a major extension of the classical Brunn–Minkowski inequality and attracts more and more attention (see [4,5,6]).

In 1977, Lutwak introduced the notion of a mixed width-integral of convex bodies (see [7]), and the dual notion, mixed chord-integrals of star bodies was defined by Lu (see [8]). Later, as a part of the asymmetric $L_p$ Brunn–Minkowski theory, which has its origins in the work of Ludwig, Haberl and Schuster (see [9,10,11,12,13]), Feng and Wang generalized the mixed chord-integrals to general mixed chord-integrals of star bodies (see [14]). For $K_1,\cdots ,K_n\in S_0^n$ and $\tau \in (-1,1)$, the general mixed chord-integral $C^{(\tau )}(K_1,\cdots ,K_n)$ is defined by

$$C^{(\tau )}(K_1,\cdots ,K_n)=\frac{1}{n}{\int }_{S^{n-1}}{c}^{(\tau )}(K_1,u)\cdots c^{(\tau )}(K_n,u)du,$$

here, $c^{(\tau )}(K,·)=f_1(\tau )\rho (K,·)+f_2(\tau )\rho (-K,·)$, and the functions $f_1(\tau )$ and $f_2(\tau )$ are defined as follows

$$f_1(\tau )=\frac{{(1+\tau )}^2}{2(1+{\tau }^2)},f_2(\tau )=\frac{{(1-\tau )}^2}{2(1+{\tau }^2)}.$$

In 2016, Li and Wang extended the general mixed chord-integral to the general $L_p$-mixed chord integral of star bodies (see [15]): For $K_1,\cdots ,K_n\in S_0^n$, $p>0$ and $\tau \in (-1,1)$, the general $L_p$-mixed chord integral $C_p^{(\tau )}(K_1,\cdots ,K_n)$ of $K_1,\cdots ,K_n$ is defined by

$$C_p^{(\tau )}(K_1,\cdots ,K_n)=\frac{1}{n}{\int }_{S^{n-1}}{c}_p^{(\tau )}(K_1,u)\cdots c_p^{(\tau )}(K_n,u)du.$$

(1a)

Here, $c_p^{(\tau )}(K,·)$ is defined by

$$c_p^{(\tau )}(K,u)={\left(f_1(\tau ){\rho }^p(K,u)+f_2(\tau ){\rho }^p(-K,u)\right)}^{\frac{1}{p}},$$

for any $u\in S^{n-1}$, and $f_1(\tau )$ and $f_2(\tau )$ are chosen as (see [16])

$$f_1(\tau )=\frac{{(1+\tau )}^p}{{(1+\tau )}^p+{(1-\tau )}^p},f_2(\tau )=\frac{{(1-\tau )}^p}{{(1+\tau )}^p+{(1-\tau )}^p}.$$

Obviously, $f_1(\tau )$ and $f_2(\tau )$ satisfy

$$f_1(\tau )+f_2(\tau )=1,$$

$$f_1(-\tau )=f_2(\tau ),f_2(-\tau )=f_1(\tau ).$$

$C_{p,i}^{(\tau )}(K,L)$ denotes that K appears $n-i$ times, and L appears i times, which is

$$C_{p,i}^{(\tau )}(K,L)=\frac{1}{n}{\int }_{S^{n-1}}{c}_p^{(\tau )}{(K,u)}^{n-i}{c}_p^{(\tau )}{(L,u)}^{i}du.$$

If constants ${\lambda }_1,\cdots ,{\lambda }_n>0$ exist such that ${\lambda }_1{c}_p^{(\tau )}(K_1,u)=\cdots ={\lambda }_n{c}_p^{(\tau )}(K_n,u)$ for all $u\in S^{n-1}$, star bodies $K_1,\cdots ,K_n$ are said to have a similar general $L_p$-chord. For this general $L_p$-chord integral, Li and Wang gave the following inequalities (see [15]).

Theorem 2.

If $K,L\in S_o^n$ and $\tau \in (-1,1),p>0$, then for $i\le n-p,$

$$C_{p,i}^{(\tau )}{(K{\widetilde{+}}_{p}L)}^{\frac{p}{n-i}}\le C_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+C_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}},$$

(1b)

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

$$C_{p,i}^{(\tau )}{(K{\widetilde{+}}_{p}L)}^{\frac{p}{n-i}}\ge C_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+C_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}},$$

(1c)

with equality in each inequality if and only if K and L have a similar general $L_p$-chord. Here and in the following Theorems, $K{\widetilde{+}}_{p}L$ denotes the $L_p$-radial Minkowski combination of K and L.

Theorem 3.

If $K_1,\cdots ,K_n\in S_o^n$ and $\tau \in (-1,1),p>0$, then for $1<m\le n$,

$$C_p^{(\tau )}{(K_1,\cdots ,K_n)}^m\le \prod _{i=1}^m{C}_p^{(\tau )}(K_1,\cdots ,K_{n-m},K_{n-i+1},K_{n-i+1},\cdots ,K_{n-i+1}),$$

(1d)

with equality if and only if $K_{n-m+1},\cdots ,K_n$ all have a similar general $L_p$-chord.

Theorem 4.

If $K,L\in S_o^n$ and $\tau \in (-1,1),p>0$, then for $i<j<k$,

$$C_{p,j}^{(\tau )}{(K,L)}^{k-i}\le C_{p,i}^{(\tau )}{(K,L)}^{k-j}{C}_{p,k}^{(\tau )}{(K,L)}^{j-i},$$

(1e)

with equality if and only if K and L have a similar general $L_p$-chord.

2. Main Results

Inspired by Leng’s idea, this article deals with the general $L_p$-chord integral of star bodies and gives some inequalities for the general $L_p$-chord integral difference.

Theorem 5.

Let $K,L,M,M^{\prime }\in S_o^n$ and $\tau \in (-1,1),p>0$. If K and L have similar general $L_p$-chord and $M\subseteq K$, $M^{\prime }\subseteq L$, then for $i\le n-p,$

$${[C_{p,i}^{(\tau )}(K{\widetilde{+}}_{p}L)-C_{p,i}^{(\tau )}(M{\widetilde{+}}_p{M}^{\prime })]}^{\frac{p}{n-i}}\ge {[C_{p,i}^{(\tau )}(K)-C_{p,i}^{(\tau )}(M)]}^{\frac{p}{n-i}}+{[C_{p,i}^{(\tau )}(L)-C_{p,i}^{(\tau )}(M^{\prime })]}^{\frac{p}{n-i}},$$

(1f)

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

$${[C_{p,i}^{(\tau )}(K{\widetilde{+}}_{p}L)-C_{p,i}^{(\tau )}(M{\widetilde{+}}_p{M}^{\prime })]}^{\frac{p}{n-i}}\le {[C_{p,i}^{(\tau )}(K)-C_{p,i}^{(\tau )}(M)]}^{\frac{p}{n-i}}+{[C_{p,i}^{(\tau )}(L)-C_{p,i}^{(\tau )}(M^{\prime })]}^{\frac{p}{n-i}},$$

(1g)

with equality in each inequality if and only if M and $M^{\prime }$ have a similar general $L_p$-chord.

Theorem 6.

Let $K_1,\cdots ,K_n$ and $M_1,\cdots ,M_n\in S_o^n$, and $\tau \in (-1,1),p>0$. If $M_i\subseteq K_i,i=1,2,\cdots ,n$, $K_1,\cdots K_n$ have similar general $L_p$-chord, then for $1<m\le n$,

$${[C_p^{(\tau )}(K_1,\cdots ,K_n)-C_p^{(\tau )}(M_1,\cdots ,M_n)]}^m\ge$$

$$\prod _{i=1}^m[C_p^{(\tau )}(K_1,\cdots ,K_{n-m},K_{n-i+1},K_{n-i+1},\cdots ,K_{n-i+1})-C_p^{(\tau )}(M_1,\cdots ,M_{n-m},K_{n-i+1},M_{n-i+1},\cdots ,M_{n-i+1})],$$

(1h)

with equality if and only if $M_1,\cdots ,M_n$ all have a similar general $L_p$-chord.

Theorem 7.

Let $K,L,M,M^{\prime }\in S_o^n$ and $\tau \in (-1,1),p>0$. If K and L have similar general $L_p$-chord, then for $i<j<k$,

$${[C_{p,j}^{(\tau )}(K,L)-C_{p,j}^{(\tau )}(M,M^{\prime })]}^{k-i}\ge {[C_{p,i}^{(\tau )}(K,L)-C_{p,i}^{(\tau )}(M,M^{\prime })]}^{k-j}{[C_{p,k}^{(\tau )}(K,L)-C_{p,k}^{(\tau )}(M,M^{\prime })]}^{j-i},$$

(1i)

with equality if and only if K and L have a similar general $L_p$-chord.

3. Preliminaries

For $K,L\in S^n$, the radial Blaschke linear combination $K\check{+}L$ and the radial Minkowski linear combination are defined by Lutwak (see [17]), respectively:

$$\rho {(K\check{+}L,u)}^{n-1}=\rho {(K,u)}^{n-1}+\rho {(L,u)}^{n-1},$$

(2a)

and

$$\rho (K\widetilde{+}L,u)=\rho (K,u)+\rho (L,u).$$

(2b)

In 2007, Schuster introduced the notion of radial Blaschke–Minkowski homomorphism (see [18,19,20,21,22]) as follows.

Definition 1.

A map $\mathsf{\Psi }:S^n\to S^n$ is called a radial Blaschke–Minkowski homomorphism if it satisfies the following conditions:

(1)

$\mathsf{\Psi }$ is coninuous;

(2)

$\mathsf{\Psi }$ is radial Blaschke Minkowski additive, i.e., $\mathsf{\Psi }(K\check{+}L)=\mathsf{\Psi }K\widetilde{+}\mathsf{\Psi }L$ for all $K,L\in S^n;$

(3)

$\mathsf{\Psi }$ intertwines rotations, i.e., $\mathsf{\Psi }(\varphi K)=\varphi \mathsf{\Psi }K$, for all $\varphi \in SO(n)$ and $K\in S^n$.

Here, $\mathsf{\Psi }K\widetilde{+}\mathsf{\Psi }L$ denotes the radial sum of $\mathsf{\Psi }K$ and $\mathsf{\Psi }L$, and $K\check{+}L$ is the radial Blaschke sum of the star bodies K and L.

In 2011, Wang et al. (see [23]) extended the notion of radial Blaschke–Minkowski homomorphism to $L_p$-radial Minkowski homomorphism as follows.

Definition 2.

A map ${\mathsf{\Psi }}_p:S^n\to S^n$ is called an $L_p$-radial Minkowski homomorphism if it satisfies the following conditions:

(1)

${\mathsf{\Psi }}_p$ is coninuous;

(2)

${\mathsf{\Psi }}_p$ is radial Minkowski additive, i.e., ${\mathsf{\Psi }}_p(K{\widetilde{+}}_{n-p}L)={\mathsf{\Psi }}_{p}K{\widetilde{+}}_p{\mathsf{\Psi }}_{p}L$ for all $K,L\in S^n;$

(3)

${\mathsf{\Psi }}_p$ intertwines rotations, i.e., ${\mathsf{\Psi }}_p(\varphi K)=\varphi {\mathsf{\Psi }}_{p}K$, for all ${\varphi }_p\in SO(n)$ and $K\in S^n$.

Here, ${\mathsf{\Psi }}_{p}K{\widetilde{+}}_{n-p}{\mathsf{\Psi }}_{p}L$ denotes the $L_{n-p}$ radial sum of ${\mathsf{\Psi }}_{p}K$ and ${\mathsf{\Psi }}_{p}L$, i.e., (see [9,24])

$$\rho {({\mathsf{\Psi }}_{p}K{\widetilde{+}}_{n-p}{\mathsf{\Psi }}_{p}L,u)}^{n-p}=\rho {({\mathsf{\Psi }}_{p}K,u)}^{n-p}+\rho {({\mathsf{\Psi }}_{p}L,u)}^{n-p}.$$

(2c)

For $0<p<n$, the $L_p$-radial Blaschke linear combination $K{\check{+}}_{p}L$ was defined by Wang (see [25]):

$$\rho {(K{\check{+}}_{p}L,u)}^{n-p}=\rho {(K,u)}^{n-p}+\rho {(L,u)}^{n-p}.$$

(2d)

From Equations (2c) and (2d), we easily obtain

$$K{\widetilde{+}}_{n-p}L=K{\check{+}}_{p}L.$$

(2e)

Here, we recall a special $L_p$-radial Minkowski homomorphism. In 2007, Yu, Wu and Leng (see [26]) introduced the quasi-$L_p$ intersection body $I_{p}K$ of a star body. Let K be a star body in ${\mathbb{R}}^n$, then the quasi-$L_p$ intersection body $I_{p}K$ of K is defined by:

$$\rho {(I_{p}K,u)}^p={\int }_{S^{n-1}\bigcap u^{\perp }}\rho {(K,u)}^{n-p}du.$$

Further, Wang (see [23]) proved that the operator $I_p:S^n\to S^n$ has the following properties: $(1)$$I_p$ is continuous with respect to radial metric; $(2)$$I_p(K{\widetilde{+}}_{n-p}L)=I_{p}K{\widetilde{+}}_p{I}_{p}L$ for all $K,L\in S^n;$ $(3)$$I_p$ intertwines rotations, i.e., ${\mathsf{\Psi }}_p(\varphi K)=\varphi {\mathsf{\Psi }}_{p}K$, for all ${\varphi }_p\in SO(n)$ and $K\in S^n$, which means that the operator $I_p$ is a special $L_p$-radial Minkowski homomorphism.

Now, we list three Lemmas useful in the proof of Theorems 5–7.

In 1997, Losonczi and Páles (see [27]) extended Bellman’s inequality as follows:

Lemma 1.

Let $a=\{a_1,a_2,\cdots ,a_n\}$ and $b=\{b_1,b_2,\cdots ,b_n\}$$(n\ge 1)$ be two sequences of positive real numbers and $p>1$ such that $a_1^p-{\Sigma }_{i=2}^n{a}_i^p>0$ and $b_1^p-{\Sigma }_{i=2}^n{b}_i^p>0$. Then

$${\left(a_1^p-{\Sigma }_{i=2}^n{a}_i^p\right)}^{\frac{1}{p}}+{\left(b_1^p-{\Sigma }_{i=2}^n{b}_i^p\right)}^{\frac{1}{p}}\le {\left({(a_1+b_1)}^p-{\Sigma }_{i=2}^n{(a_i+b_i)}^p\right)}^{\frac{1}{p}},$$

(2f)

If $p<0$ or $0<p<1$, then

$${\left({(a_1^p-{\Sigma }_{i=2}^n{a}_i^p)}^{\frac{1}{p}}+{(b_1^p-{\Sigma }_{i=2}^n{b}_i^p)}^{\frac{1}{p}})\right)}^p\ge {(a_1+b_1)}^p-{\Sigma }_{i=2}^n{(a_i+b_i)}^p,$$

with equality if and only if $a=vb$, where v is a constant.

Lemma 2

([28], p.26). If $x_i>0,y_i>0,i=1,2,\cdots ,n$, then

$${\left(\prod _{i=1}^n(x_i+y_i)\right)}^{\frac{1}{n}}\ge {(\prod _{i=1}^n{x}_i)}^{\frac{1}{n}}+{(\prod _{i=1}^n{y}_i)}^{\frac{1}{n}},$$

(2g)

with equality if and only if $\frac{x_1}{y_1}=\frac{x_2}{y_2}=\cdots =\frac{x_n}{y_n}.$

Lemma 3

([5]). Suppose that $f_i,g_i$$(i=1,2)$ are non-negative continuous functions on $S^{n-1}$ such that

$${\int }_{S^{n-1}}{f}_1^s(u)du\ge {\int }_{S^{n-1}}{f}_2^s(u)du,$$

$${\int }_{S^{n-1}}{g}_1^t(u)du\ge {\int }_{S^{n-1}}{g}_2^t(u)du,$$

for $s>1,\frac{1}{s}+\frac{1}{t}=1$, and

$$f_1^s(u)=\lambda g_1^t(u),\forall u\in S^{n-1},$$

where λ is a constant. Then

$${\left({\int }_{S^{n-1}}\left(f_1^s-f_2^s\right)du\right)}^{\frac{1}{s}}{\left({\int }_{S^{n-1}}\left(g_1^t-g_2^t\right)du\right)}^{\frac{1}{t}}\le {\int }_{S^{n-1}}\left(f_1{g}_1-f_2{g}_2\right)du,$$

(2h)

with equality if and only if $f_2^s(u)=\lambda g_2^t(u)$ for any $u\in S^{n-1}.$

4. Proofs of Main Results

In this section, we prove Theorems 5–7.

Proof of Theorem 5.

We only prove Equation (1f). The proof of Equation (1g) is similar to Equation (1f). Let $i\le n-p$. Since K and L have similar general $L_p$-chord, by Equation (1b),

$$C_{p,i}^{(\tau )}{(K{\widetilde{+}}_{p}L)}^{\frac{p}{n-i}}=C_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+C_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}},$$

(3a)

for M and $M^{\prime }$,

$$C_{p,i}^{(\tau )}{(M{\widetilde{+}}_p{M}^{\prime })}^{\frac{p}{n-i}}\le C_{p,i}^{(\tau )}{(M)}^{\frac{p}{n-i}}+C_{p,i}^{(\tau )}{(M^{\prime })}^{\frac{p}{n-i}}.$$

(3b)

Let $a_1=C_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}},a_2=C_{p,i}^{(\tau )}{(M)}^{\frac{p}{n-i}}$ and $b_1=C_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}},b_2=C_{p,i}^{(\tau )}{(M^{\prime })}^{\frac{p}{n-i}}$, then from Equations (3a) and (3b) and Lemma 1, we have

$$\begin{array}{cc}& {\left(C_{p,i}^{(\tau )}(K{\widetilde{+}}_{p}L)-C_{p,i}^{(\tau )}(M{\widetilde{+}}_p{M}^{\prime })\right)}^{\frac{p}{n-i}}\hfill \\ & \ge {\left({\left(C_{p,i}^{(\tau )}{(K)}^{\frac{p}{n-i}}+C_{p,i}^{(\tau )}{(L)}^{\frac{p}{n-i}}\right)}^{\frac{n-i}{p}}+{\left(C_{p,i}^{(\tau )}{(M)}^{\frac{p}{n-i}}+C_{p,i}^{(\tau )}{(M^{\prime })}^{\frac{p}{n-i}}\right)}^{\frac{n-i}{p}}\right)}^{\frac{p}{n-i}}\hfill \\ & \ge {\left(C_{p,i}^{(\tau )}(K)-C_{p,i}^{(\tau )}(M)\right)}^{\frac{p}{n-i}}+{\left(C_{p,i}^{(\tau )}(L)-C_{p,i}^{(\tau )}(M^{\prime })\right)}^{\frac{p}{n-i}}\hfill \end{array}$$

This gives the desired inequality of Equation (1f) and according to the equality condition of Lemma 1, we obtain that equality holds if and only if M and $M^{\prime }$ have a similar general $L_p$-chord. □

Notice that from the notion of $L_p$-radial Minkowski homomorphism and Equation (2e), we have the following direct Corollary 1.

Corollary 1.

Let $K,L,M,M^{\prime }\in S_o^n$ and $\tau \in (-1,1),p>0$. ${\mathsf{\Psi }}_p$ is a radial Blaschke–Minkowski homomorphism. K and L have a similar general $L_p$-chord and $M\subseteq K$, $M^{\prime }\subseteq L$, then for $i\le n-p,$

$${[C_{p,i}^{(\tau )}({\mathsf{\Psi }}_p(K{\check{+}}_{p}L))-C_{p,i}^{(\tau )}({\mathsf{\Psi }}_p(M{\check{+}}_p{M}^{\prime }))]}^{\frac{p}{n-i}}\ge {[C_{p,i}^{(\tau )}({\mathsf{\Psi }}_{p}K)-C_{p,i}^{(\tau )}({\mathsf{\Psi }}_{p}M)]}^{\frac{p}{n-i}}+{[C_{p,i}^{(\tau )}({\mathsf{\Psi }}_{p}L)-C_{p,i}^{(\tau )}({\mathsf{\Psi }}_p{M}^{\prime })]}^{\frac{p}{n-i}},$$

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

$${[C_{p,i}^{(\tau )}({\mathsf{\Psi }}_p(K{\check{+}}_{p}L))-C_{p,i}^{(\tau )}({\mathsf{\Psi }}_p(M{\check{+}}_p{M}^{\prime }))]}^{\frac{p}{n-i}}\le {[C_{p,i}^{(\tau )}({\mathsf{\Psi }}_{p}K)-C_{p,i}^{(\tau )}({\mathsf{\Psi }}_{p}M)]}^{\frac{p}{n-i}}+{[C_{p,i}^{(\tau )}({\mathsf{\Psi }}_{p}L)-C_{p,i}^{(\tau )}({\mathsf{\Psi }}_p{M}^{\prime })]}^{\frac{p}{n-i}},$$

with equality in each inequality if and only if M and $M^{\prime }$ have a similar general $L_p$-chord.

Further, since the $L_p$ intersection map is a special $L_p$-radial Minkowski homomorphism, we have the following corollary

Corollary 2.

Let $K,L,M,M^{\prime }\in S_o^n$ and $\tau \in (-1,1),p>0$. If K and L have a similar general $L_p$-chord and $M\subseteq K$, $M^{\prime }\subseteq L$, then for $i\le n-p,$

$${[C_{p,i}^{(\tau )}(I_p(K{\check{+}}_{p}L))-C_{p,i}^{(\tau )}(I_p(M{\check{+}}_p{M}^{\prime }))]}^{\frac{p}{n-i}}\ge {[C_{p,i}^{(\tau )}(I_{p}K)-C_{p,i}^{(\tau )}(I_{p}M)]}^{\frac{p}{n-i}}+\mu {[C_{p,i}^{(\tau )}(I_{p}L)-C_{p,i}^{(\tau )}(I_p{M}^{\prime })]}^{\frac{p}{n-i}},$$

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

$${[C_{p,i}^{(\tau )}(I_p(K{\check{+}}_{p}L))-C_{p,i}^{(\tau )}(I_p(M{\check{+}}_p{M}^{\prime }))]}^{\frac{p}{n-i}}\le {[C_{p,i}^{(\tau )}(I_{p}K)-C_{p,i}^{(\tau )}(I_{p}M)]}^{\frac{p}{n-i}}+{[C_{p,i}^{(\tau )}(I_{p}L)-C_{p,i}^{(\tau )}(I_p{M}^{\prime })]}^{\frac{p}{n-i}},$$

with equality in each inequality if and only if M and $M^{\prime }$ have a similar general $L_p$-chord.

Proof of Theorem 6.

Since $K_1,\cdots ,K_n$ have a similar general $L_p$-chord, from (1d) we have for $1<m\le n$,

$$C_p^{(\tau )}{(K_1,\cdots ,K_n)}^m=\prod _{i=1}^m{C}_p^{(\tau )}(K_1,\cdots ,K_{n-m},K_{n-i+1},K_{n-i+1},\cdots ,K_{n-i+1}).$$

(3c)

For $M_1,\cdots ,M_n$,

$$C_p^{(\tau )}{(M_1,\cdots ,M_n)}^m\le \prod _{i=1}^m{C}_p^{(\tau )}(M_1,\cdots ,M_{n-m},M_{n-i+1},M_{n-i+1},\cdots ,M_{n-i+1}).$$

(3d)

The condition $M_i\subseteq K_i,i=1,2,\cdots ,n$ means that $C_p^{(\tau )}{(K_1,\cdots ,K_n)}^m\ge C_p^{(\tau )}{(M_1,\cdots ,M_n)}^m$. From Equations (3c) and (3d) and Lemma 2, we obtain

$$\begin{array}{cc}& C_p^{(\tau )}(K_1,\cdots ,K_n)-C_p^{(\tau )}(M_1,\cdots ,M_n)\hfill \\ & \ge {\left(\prod _{i=1}^m{C}_p^{(\tau )}(K_1,\cdots ,K_{n-m},K_{n-i+1},K_{n-i+1},\cdots ,K_{n-i+1})\right)}^{\frac{1}{m}}\hfill \\ & -{\left(\prod _{i=1}^m{C}_p^{(\tau )}(M_1,\cdots ,M_{n-m},M_{n-i+1},M_{n-i+1},\cdots ,M_{n-i+1})\right)}^{\frac{1}{m}}.\hfill \end{array}$$

Let $x_i+y_i=C_p^{(\tau )}(K_1,\cdots ,K_{n-m},K_{n-i+1},K_{n-i+1},\cdots ,K_{n-i+1})$ and $y_i=C_p^{(\tau )}(M_1,\cdots ,M_{n-m},M_{n-i+1},M_{n-i+1},\cdots ,M_{n-i+1})$ in Lemma 2. Then by Equation (2g)

$$\begin{array}{cc}& C_p^{(\tau )}(K_1,\cdots ,K_n)-C_p^{(\tau )}(M_1,\cdots ,M_n)\hfill \\ & \ge {\left(\prod _{i=1}^m\left[C_p^{(\tau )}(K_1,\cdots ,K_{n-m},K_{n-i+1},\cdots ,K_{n-i+1})-C_p^{(\tau )}(M_1,\cdots ,M_{n-m},M_{n-i+1},\cdots ,M_{n-i+1})\right]\right)}^{\frac{1}{m}},\hfill \end{array}$$

which implies that Equation (1h) is proved. According to the equality condition of Lemma 2, we know that equality holds in Equation (1h) if and only if $M_1,\cdots M_n$ all have a similar general $L_p$-chord. □

Proof of Theorem 7.

For $i<j<k$, let $s=\frac{k-i}{k-j},t=\frac{k-i}{j-i}$. Then, $s>1$ and $\frac{1}{s}+\frac{1}{t}=1$. Let

$$f_1^s=c_p^{(\tau )}{(K,u)}^{n-i}{c}_p^{(\tau )}{(L,u)}^i,f_2^s=c_p^{(\tau )}{(M,u)}^{n-i}{c}_p^{(\tau )}{(M^{\prime },u)}^i$$

and

$$g_1^t=c_p^{(\tau )}{(K,u)}^{n-k}{c}_p^{(\tau )}{(L,u)}^k,g_2^t=c_p^{(\tau )}{(M,u)}^{n-k}{c}_p^{(\tau )}{(M^{\prime },u)}^k.$$

After a simple calculation, we obtain

$$\begin{array}{cc}\hfill {\int }_{S^{n-1}}\left(f_1{g}_1-f_2{g}_2\right)du& ={\int }_{S^{n-1}}\left(c_p^{(\tau )}{(K,u)}^{n-j}{c}_p^{(\tau )}{(L,u)}^j-c_p^{(\tau )}{(M,u)}^{n-j}{c}_p^{(\tau )}{(M^{\prime },u)}^j\right)du\hfill \\ & =C_{p,j}^{(\tau )}(K,L)-C_{p,j}^{(\tau )}(M,M^{\prime }).\hfill \end{array}$$

The left-hand side of Equation (2h) leads to ${[C_{p,i}^{(\tau )}(K,L)-C_{p,i}^{(\tau )}(M,M^{\prime })]}^{\frac{1}{s}}{[C_{p,k}^{(\tau )}(K,L)-C_{p,k}^{(\tau )}(M,M^{\prime })]}^{\frac{1}{t}}.$

By Lemma 3, Equation (1i) immediately holds.

The equality condition of Equation (2h) means that $\frac{f_1^s}{g_1^t}={\left(\frac{c_p^{(\tau )}(K,u)}{c_p^{(\tau )}(L,u)}\right)}^{k-i}$ is a constant, that is, K and L have a similar general $L_p$-chord. This completes the proof. □

5. Conclusions

The asymmetric operators belong to a new and rapidly evolving asymmetric $L_p$-Brunn–Minkowski theory that has its origins in the work of Ludwig, Haberl and Schuster (see [9,11,12,16,18,19,20]). The general $L_p$-mixed chord integral difference of star bodies was motivated by the notion of mixed width-integrals of convex bodies. We hope that besides the inequalities mentioned in this article, we can deduce some other inequalities in the future.