Dual Orlicz Mixed Quermassintegral

Abstract

We study the dual Orlicz mixed Quermassintegral. For arbitrary monotone continuous function $\phi$, the dual Orlicz radial sum and dual Orlicz mixed Quermassintegral are introduced. Then the dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral are obtained. These inequalities are just the special cases of their $L_p$ analogues (including cases $-\infty<p<0$, $p=0$, $0<p<1$, $p=1$, and $1<p<+\infty$). These inequalities for $\phi =\log t$ are related to open problems including log-Minkowski problem and log-Brunn-Minkowski problem. Moreover, the equivalence of the dual Orlicz–Minkowski inequality for dual Orlicz mixed Quermassintegral and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral is shown.

1 Introduction

The classical Brunn–Minkowski theory for convex bodies (compact convex sets with nonempty interior) is known as consequences of the combination of Minkowski addition and volume, which constitutes the core of convex geometry. Significant results in this theory, for instance the Minkowski’s first inequality and the Brunn–Minkowski inequality, have important applications in analysis, geometry, random matrices, and many other fields (see [28]).

In 1960s, Firey extended Minkowski addition to $L_p$ addition in [2]. Since then, the Brunn–Minkowski theory has gained amazing developments. This extended theory is called $L_p$ Brunn–Minkowski theory, which connects volumes with $L_p$ addition (see e.g. [7, 9, 19,20,21,22,23, 31]). As a development of $L_p$ Brunn–Minkowski theory, Orlicz–Brunn–Minkowski theory is a new blossom in recent years, which is motivated by [8, 15, 16, 24, 25]. For more references, see [3, 11, 14, 34, 35, 38]. Specifically, Xiong and Zou studied Orlicz mixed Quermassintegral in [35].

In [17, 18], Lutwak introduced duality of the Brunn–Minkowski theory, in which the research object substitutes star bodies for convex bodies, obtained dual counterparts of the several wonderful results in the Brunn–Minkowski theory. Intersection body is a useful geometrical object in dual Brunn–Minkowski theory, introduced by Lutwak in [18]. The class of intersection bodies and mixed intersection bodies are valuable in geometry, especially in answering the known Busemann–Petty problem (see [12]). We refer the reader to [5, 6, 13, 26, 32, 33] for the extended intersection bodies and their applications.

In [37], a dual Orlicz–Brunn–Minkowski theory was presented and the dual Orlicz–Brunn–Minkowski inequality for volume was established. An Orlicz radial sum and dual Orlicz mixed volumes were introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality were established. The variational formula for the volume with respect to the Orlicz radial sum was proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality was demonstrated. Orlicz intersection bodies were introduced and the Orlicz–Busemann–Petty problem was posed. It should noted that analog theory was also discussed in [4]. Following ideas of [4, 37], the dual Orlicz–Brunn–Minkowski inequality for dual mixed Quermassintegral was also discussed in [35].

Motivated by works of [4, 37], we consider the dual Orlicz–Brunn–Minkowski inequality for dual mixed Quermassintegral in the n-dimensional Euclidean space $\mathbb {R}^n$. We denote by $\mathscr {C}^{in}$ the set of all increasing continuous functions $\phi :~(0,\infty )\rightarrow (-\infty ,\infty )$ and by $\mathscr {C}^{de}$ the set of all decreasing continuous functions $\phi :~(0,\infty )\rightarrow (-\infty ,\infty )$. Let $\mathscr {C}$ denote the union of $\mathscr {C}^{in}$ and $\mathscr {C}^{de}$. The n dimensional unit ball and the unit sphere are denoted by B and $S^{n-1}$ respectively.

A set K in $\mathbb {R}^n$ is star-shaped set with respect to $z\in K$ if the intersection of every line through z with K is a line segment. The radial function, $\rho _K:\ S^{n-1}\rightarrow [0,\infty )$, of a compact star-shaped set (about the origin) is defined by

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

(1)

If $\rho (K,\cdot )$ is positive and continuous, K is called a star body. Let ${\mathscr {S}^n}$ and ${\mathscr {S}_0^n}$ denote the set of start bodies and the set of start bodies about the origin in $\mathbb {R}^n$, respectively.

Definition 1

Let $K,\ L \in {\mathscr {S}}_0^n$, $a,b>0$.

If $\phi \in \mathscr {C}^{in}$, then Orlicz radial sum $a\cdot K\tilde{+}_\phi ~b\cdot L$ is defined by

$$\begin{aligned} \rho _{a\cdot K\tilde{+}_\phi ~b\cdot L}(u)=\inf \left\{ t>0:a\phi \left( \frac{\rho _K(u)}{t}\right) +b\phi \left( \frac{\rho _L(u)}{t}\right) \le \phi (1)\right\} ,\forall u\in S^{n-1}. \end{aligned}$$

(2)

If $\phi \in \mathscr {C}^{de}$, then Orlicz radial sum $a\cdot K\tilde{+}_\phi ~b\cdot L$ is defined by

$$\begin{aligned} \rho _{a\cdot K\tilde{+}_\phi ~b\cdot L}(u)=\sup \left\{ t>0:a\phi \left( \frac{\rho _K(u)}{t}\right) +b\phi \left( \frac{\rho _L(u)}{t}\right) \le \phi (1)\right\} ,\forall u\in S^{n-1}. \end{aligned}$$

(3)

The dual mixed Quermassintegral $\tilde{W}_i(K,L)$, defined in [17], is

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

(4)

Motivated by this, we define the following dual Orlicz mixed Quermassintegral.

Definition 2

Let $K,L\in {\mathscr {S}}_0^n$, $i\in \mathbb {R}$, $\phi \in \mathscr {C}$. The dual Orlicz mixed Quermassintegral $\tilde{W}_{\phi ,i}(K,L)$ is defined by

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,L)=\frac{1}{n}\int _{S^{n-1}}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) \rho _K^{n-i}(u)dS(u). \end{aligned}$$

(5)

When $\phi (t)=t^p$, with $p\ne 0$, the dual Orlicz mixed volume reduces to $L_p$ dual mixed Quermassintegral (see [20] for the case $p\ge 1$ and $i=0$)

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

When $\phi (t)=\log t$, one has

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

In Sect. 2, we introduce some basic concepts. In Sect. 3, the Orlicz radial sum and some related properties are discussed. Some important properties of dual Orlicz mixed Quermassintegral are investigated in Sect. 4.

In Sect. 5, dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality are established for dual Orlicz mixed Quermassintegral. As special cases, these inequalities are just the $L_p$ counterparts, including the cases $-\infty<p<0$, $p=0$, $0<p<1$, $p=1$ and $1<p<+\infty$. These inequalities for $\phi =\log t$ are related to open problems, such as, the log-Brunn–Minkowski problem and the log-Minkowski problem. Moreover, we prove the equivalence of dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality.

2 Preliminaries

Let $K,L\in {\mathscr {S}_0^n}$. By (1), one has

$$\begin{aligned} K\subset L\quad \text {if and only if}\quad \rho _K(u)\le \rho _L(u). \end{aligned}$$

(6)

Two star bodies K and L are dilates (of each other) if ${\rho _K(u)}/{\rho _L(u)}$ is independent of $u\in S^{n-1}$. If $t>0$, we have

$$\begin{aligned} \rho (tK,u)=t\rho (K,u),~~~~~~\text {for all}~u\in S^{n-1}. \end{aligned}$$

We write $A^{-1}$ for the inverse matrix of A where $A\in GL(n)$. So associated with the definition of the radial function, for $A\in GL(n)$, the radial function of the image $AK=\{Ay: y\in K\}$ of K is shown by

$$\begin{aligned} \rho (AK,u)=\rho (K,A^{-1}u),~~~~~\text {for all}~u\in S^{n-1}. \end{aligned}$$

(7)

The radial Hausdorff metric between the star bodies K and L is

$$\begin{aligned} \tilde{\delta }(K,L)=\max \limits _{u\in S^{n-1}}\mid \rho _K(u)-\rho _L(u)\mid . \end{aligned}$$

A sequence $\{K_i\}$ of star bodies is said to be convergent to K if

$$\begin{aligned} \tilde{\delta }(K_i,K)\rightarrow 0,~~~~~\text {as}~i\rightarrow \infty . \end{aligned}$$

Therefore, a sequence of star bodies $K_i$ converges to K if and only if the sequence of the radial function $\rho (K_i,\cdot )$ converges uniformly to $\rho (K,\cdot )$ [27].

Let $K,L\in {\mathscr {S}}^n$. We have

$$\begin{aligned} K\tilde{+}_\phi \varepsilon L\rightarrow K \end{aligned}$$

in the radial Hausdorff metric as $\varepsilon \rightarrow 0^+$ [36].

The radial Minkowski linear combination of sets $K_1,\ldots ,K_r$ in $\mathbb {R}^n$ is defined by

$$\begin{aligned} \lambda _1 K_1\tilde{+}\cdots \tilde{+}\lambda _rK_r= \{\lambda _1 x_1\tilde{+}\cdots \tilde{+}\lambda _rx_r\},~~~~~\text {for all}~\lambda _i\in \mathbb {R},~~i=1,\ldots , r. \end{aligned}$$

If $K,L\in {\mathscr {S}}_0^n$ and $a,b>0,~aK\tilde{+}bL$ can be defined as a star body with satisfying that

$$\begin{aligned} \rho _{aK\tilde{+}bL}(u)=a\rho _K(u)+b\rho _L(u),~~~~~\text {for all}~u\in S^{n-1}. \end{aligned}$$

(8)

Write V(K) for the volume of the compact set K in $\mathbb {R}^n$. In fact, the volume of the radial Minkowski linear combination $\lambda _1 x_1\tilde{+}\cdots \tilde{+}\lambda _rx_r$ is a homogeneous n-th polynomial in $\lambda _i$ (see [17, 18]).

$$\begin{aligned} V(\lambda _1 K_1\tilde{+}\cdots \tilde{+}\lambda _rK_r)=\sum \limits _{r\le n} \tilde{V}({K_{i_1},\ldots ,K_{i_n}})\lambda _{i_1}\cdots \lambda _{i_n}. \end{aligned}$$

The coefficient $\tilde{V}({K_{i_1},\ldots ,K_{i_n}})$ is called the dual mixed volume of $K_{i_1},\ldots ,K_{i_n}$, it is nonnegative and only depends on the sets $K_{i_1},\ldots ,K_{i_n}$. Or write $\tilde{V}_i(K,L)=\tilde{V}(\underbrace{K,\ldots ,K}_{n-i},\underbrace{L,\ldots ,L}_i)$. If $L=B$, the dual mixed volume $\tilde{V}_i(K,B)$ is written as $\tilde{W}_i(K)$ which is called the dual Quermassintegral of K.

If $K_1,\ldots ,K_n\in S_0^n$, the dual mixed volume $\tilde{V}(K_{1},\ldots ,K_{n})$ is defined [17]

$$\begin{aligned} \tilde{V}(K_{1},\ldots ,K_{n})=\frac{1}{n}\int _{S^{n-1}}\rho _{K_1}(u)\cdots \rho _{K_n}(u)dS(u), \end{aligned}$$

where S is the Lebesgue measure on $S^{n-1}$ (i.e., the (n-1)-dimensional Hausdorff measure). Let $K\in {\mathscr {S}_0^n}$ and $i\in \mathbb {R}$. A slight extension (see [29]) of the notation $\tilde{W}_i(K)$ is

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

(9)

In (4), let $i=0$, we immediately get the following integral representation for the first dual mixed volume proved by Lutwak in [17]: if $K,L\in {\mathscr {S}_0^n}$, then

$$\begin{aligned} \tilde{V}_1(K,L)=\frac{1}{n}\int _{S^{n-1}}\rho _K(u)^{n-1}\rho _L(u)dS(u). \end{aligned}$$

The integral representation (4), together with the H$\ddot{\text {o}}$lder inequality and (9), immediately lead to the following dual Minkowski inequality about the dual mixed Quermassintegral $\tilde{W}_i(K,L)$.

Lemma 1

If $K,L\in {\mathscr {S}}_0^n$ and $i<n-1$, then

$$\begin{aligned} \tilde{W}_i(K,L)^{n-i}\le \tilde{W}_i(K)^{n-i-1}\tilde{W}_i(L), \end{aligned}$$

(10)

with equality if and only if K and L are dilates of each other.

If $i>n-1$ and $i\ne n$, (10) is reversed, with equality if and only if K and L are dilates.

We shall obtain the dual Brunn–Minkowski inequality for the dual Quermassintegral $\tilde{W}_i(aK\tilde{+}bL)$.

Lemma 2

If $K,L\in {\mathscr {S}}_0^n$, $i<n-1$ and $a,b>0$, then

$$\begin{aligned} \tilde{W}_i(aK\tilde{+}bL)^\frac{1}{n-i}\le a\tilde{W}_i(K)^{\frac{1}{n-i}}+b\tilde{W}_i(L)^{\frac{1}{n-i}}, \end{aligned}$$

(11)

with equality if and only if K and L are dilates of each other.

If $i>n-1$ and $i\ne n$, (11) is reversed, with equality if and only if K and L are dilates.

Upon the definition of the function $\phi$, suppose that $\mu$ is a probability measure on a space X and $g:X\rightarrow I\subset \mathbb {R}$ is a $\mu$-integrable function, where I is a possibly infinite interval. Jensen’s inequality (see [10]) shows that if $\phi :I\rightarrow \mathbb {R}$ is a concave function, then

$$\begin{aligned} \int _X\phi (g(x))d\mu (x)\le \phi \bigg (\int _Xg(x)d\mu (x)\bigg ). \end{aligned}$$

(12)

If $\phi \in \mathscr {C}_2$, the inequality is reverse, that is

$$\begin{aligned} \int _X\phi (g(x))d\mu (x)\ge \phi \bigg (\int _Xg(x)d\mu (x)\bigg ). \end{aligned}$$

(13)

If $\phi$ is strictly concave or convex, each equality in (12) and (13) holds if and only if g(x) is constant for $\mu$-almost all $x\in X$.

3 Orlicz Radial Sum

From (7) and the definition of the Orlicz radial sum, we have

Proposition 1

Let $K,L\in {\mathscr {S}}_0^n$, and $a,b>0$. If $\phi \in \mathscr {C}$, then for $A\in GL(n)$,

$$\begin{aligned} A(a\cdot K\tilde{+}_\phi ~b\cdot L)=a\cdot AK\tilde{+}_\phi ~b\cdot AL. \end{aligned}$$

(14)

Proof

For $\phi \in \mathscr {C}^{in}, u\in S^{n-1}$, by (7)

$$\begin{aligned} \rho _{a\cdot AK\tilde{+}_\phi ~b\cdot AL}(u)&=\inf \left\{ t>0:a\phi \left( \frac{\rho _{AK}(u)}{t}\right) +b\phi \left( \frac{\rho _{AL}(u)}{t}\right) \le \phi (1)\right\} \\&=\inf \left\{ t>0:a\phi \left( \frac{\rho _K(A^{-1}u)}{t}\right) +b\phi \left( \frac{\rho _L(A^{-1}u)}{t}\right) \le \phi (1)\right\} \\&=\rho _{a\cdot K\tilde{+}_\phi ~b\cdot L}(A^{-1}u) \\&=\rho _{A(a\cdot K\tilde{+}_\phi ~b\cdot L)}(u). \end{aligned}$$

If $\phi \in \mathscr {C}^{de}$, in the same way, we also have (14).

Since $K,L\in {\mathscr {S}}_0^n$ and $u\in S^{n-1}$, $0<\rho _K(u)<\infty$ and $0<\rho _L(u)<\infty$, hence $\frac{\rho _K(u)}{t}\rightarrow 0$ and $\frac{\rho _L(u)}{t}\rightarrow 0$ as $t\rightarrow \infty$. By the assumption that $\phi$ is monotone increasing (or decreasing) in $(0,\infty )$, so the function

$$\begin{aligned} t\mapsto a\phi \left( \frac{\rho _K(u)}{t}\right) +b\phi \left( \frac{\rho _L(u)}{t}\right) , \end{aligned}$$

is monotone decreasing (or increasing) in $(0,\infty )$. Since it is also continuous, we have

Lemma 3

Let $K,L\in {\mathscr {S}}_0^n$, $a,b>0$, and $u\in S^{n-1}$. If $\phi \in \mathscr {C}$, then

$$\begin{aligned} a\phi \left( \frac{\rho _K(u)}{t}\right) +b\phi \left( \frac{\rho _L(u)}{t}\right) =\phi (1), \end{aligned}$$

if and only if

$$\begin{aligned} \rho _{a\cdot K\tilde{+}_\phi b\cdot L}(u)=t. \end{aligned}$$

Remark 1

We shall provide several special examples of the Orlicz radial sum. Let $K,L\in {\mathscr {S}_0^n}$, $a,b>0$.

(1) When $\phi (t)=t^p$, with $p\ne 0$, it is easy to show that the Orlicz radial sum reduces to an analogue form of Lutwak’s $L_p$ radial combination ($p\ge 1$, see [20])

$$\rho (a\cdot K\tilde{+}_\phi b\cdot L,u)^p = a\rho (K,u)^p+b\rho (L,u)^p.$$

(2) When $\phi (t)=\log t$, we obtain

$$ (a+b)\log \rho _{a\cdot K\tilde{+}_\phi b\cdot L}(u)=a\log \rho _K(u)+b\log \rho _L(u). $$

This sum is dual of the logarithm sum which is an important notion (see [1, 30]).

(3) When $\phi (t)=\log (t+1)$, we have

$$ \left( \dfrac{\rho _K(u)}{\rho _{a\cdot K\tilde{+}_\phi b\cdot L}(u)}+1\right) ^a\left( \dfrac{\rho _L(u)}{\rho _{a\cdot K\tilde{+}_\phi b\cdot L}(u)}+1\right) ^b=2, $$

and $\phi (0)=0$.

Lemma 4

Let $K,L\in {\mathscr {S}}_0^n$, for $0<\lambda <1$,

(1) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$ or $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$, then

$$\begin{aligned} (1-\lambda )\cdot K\tilde{+}_\phi \lambda \cdot L\subseteq (1-\lambda )K\tilde{+}\lambda L. \end{aligned}$$

(15)

When $\phi$ is strictly concave or convex, the equality holds if and only if $K=L$.

(2) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$ or $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, then

$$\begin{aligned} (1-\lambda )\cdot K\tilde{+}_\phi \lambda \cdot L\supseteq (1-\lambda )K\tilde{+}\lambda L. \end{aligned}$$

(16)

When $\phi$ is strictly concave or convex, the equality holds if and only if $K=L$.

Proof

Let $K_\lambda =(1-\lambda )\cdot K\tilde{+}_\phi \lambda \cdot L$.

(1) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$, by Lemma 3 and concavity of $\phi$, we have

$$\begin{aligned} \phi (1)&=(1-\lambda )\phi \left( \frac{\rho _K(u)}{\rho _{K_\lambda }(u)}\right) +\lambda \phi \left( \frac{\rho _L(u)}{\rho _{K_\lambda }(u)}\right) \\&\le \phi \left( \frac{(1-\lambda )\rho _K(u)+\lambda \rho _L(u)}{\rho _{K_\lambda }(u)}\right) . \end{aligned}$$

Since $\phi$ is monotone increasing on $(0,\infty )$, hence we have

$$\begin{aligned} (1-\lambda )\rho _K(u)+\lambda \rho _L(u)\ge \rho _{K_\lambda }(u), \end{aligned}$$

that is,

$$\begin{aligned} \rho _{(1-\lambda )K\tilde{+}\lambda L}(u)\ge \rho _{K_\lambda }(u). \end{aligned}$$

(17)

If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$, by Lemma 3 convexity of $\phi$ and $\phi$ is monotone decreasing on $(0,\infty )$, in the same way, we can obtain (17). Then by (6), (17) deduces the helpful conclusion (15).

(2) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$, by Lemma 3 and convexity of $\phi$, we have

$$\begin{aligned} \phi (1)&=(1-\lambda )\phi \left( \frac{\rho _K(u)}{\rho _{K_\lambda }(u)}\right) +\lambda \phi \left( \frac{\rho _L(u)}{\rho _{K_\lambda }(u)}\right) \\&\ge \phi \left( \frac{(1-\lambda )\rho _K(u)+\lambda \rho _L(u)}{\rho _{K_\lambda }(u)}\right) . \end{aligned}$$

Since $\phi$ is monotone increasing on $(0,\infty )$, hence we also have

$$\begin{aligned} (1-\lambda )\rho _K(u)+\lambda \rho _L(u)\le \rho _{K_\lambda }(u), \end{aligned}$$

that is,

$$\begin{aligned} \rho _{(1-\lambda )K\tilde{+}\lambda L}(u)\le \rho _{K_\lambda }(u). \end{aligned}$$

(18)

If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, by Lemma 3, concavity of $\phi$ and $\phi$ is monotone decreasing on $(0,\infty )$, in the same way, we can obtain (18). Then by (6), (18) deduce the helpful conclusion (16).

From the equality condition in the concavity (or convexity) of $\phi$, then each equality in (15) and (16) holds if and only if $K=L$.

Corollary 1

Let $K,L\in {\mathscr {S}}_0^n$, $0<\lambda <1$ and $\tilde{W}_i(K)=\tilde{W}_i(L)$.

(1) If $i<n-1$, $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$ or $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$, then

$$\begin{aligned} \tilde{W}_i((1-\lambda )\cdot K\tilde{+}_\phi \lambda \cdot L)\le \tilde{W}_i(K), \end{aligned}$$

(19)

with equality if and only if $K=L$.

(2) If $i>n-1$ and $i\ne n$, $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$ or $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, then

$$\begin{aligned} \tilde{W}_i((1-\lambda )\cdot K\tilde{+}_\phi \lambda \cdot L)\ge \tilde{W}_i(K), \end{aligned}$$

(20)

with equality if and only if $K=L$.

Proof

(1) By Lemmas 4 and 2, we have

$$\begin{aligned} \tilde{W}_i((1-\lambda )\cdot K\tilde{+}_\phi \lambda \cdot L)^{\frac{1}{n-i}}&\le \tilde{W}_i((1-\lambda )\cdot K\tilde{+}\lambda \cdot L)^{\frac{1}{n-i}}\\&\le (1-\lambda )\tilde{W}_i(K)^{\frac{1}{n-i}}+\lambda \tilde{W}_i(L)^{\frac{1}{n-i}}\\&=\tilde{W}_i(K)^{\frac{1}{n-i}}. \end{aligned}$$

The equality condition in (19) can be obtained from the equality condition of (11).

(2) Similarly, from Lemmas 3.4 and 2.2, we can obtain (20).

4 Dual Orlicz Mixed Quermassintegral

We denote the right derivative of a real-valued function f by $f_r^{\prime }$. In the following Lemma 5 the function $\phi$ is different from $\phi$ in Lemma 4.1 of [37]. However, we can use the similar argument to prove Lemma 5, so we omit the details.

Lemma 5

Let $\phi \in \mathscr {C}$ and $K,L\in {\mathscr {S}}_0^n$. Then

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\rho _{K\tilde{+}_\phi \varepsilon \cdot L}(u)-\rho _K(u)}{\varepsilon }=\frac{\rho _K(u)}{\phi _r^{\prime }(1)}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) , \end{aligned}$$

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

Theorem 1

Let $\phi \in \mathscr {C}$, $K,L\in {\mathscr {S}}_0^n$ and $i\ne n$. Then

$$\begin{aligned} \frac{n}{n-i}\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(K\tilde{+}_\phi \varepsilon \cdot L)-\tilde{W}_i(K)}{\varepsilon }= \frac{1}{\phi _r^{\prime }(1)}\int _{S^{n-1}}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) \rho _K^{n-i}(u)dS(u). \end{aligned}$$

Proof

Let $\varepsilon >0$, $K,L\in {\mathscr { S}_0^n}$, $i\ne n$ and $u\in S^{n-1}$. By Lemma 5, it follows that

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\rho _{K\tilde{+}_\phi \varepsilon \cdot L}^{n-i}(u)-\rho _K^{n-i}(u)}{\varepsilon }&=(n-i)\rho _K^{n-i-1}(u)\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\rho _{K\tilde{+}_\phi \varepsilon \cdot L}(u)-\rho _K(u)}{\varepsilon }\\&=\frac{(n-i)\rho _K^{n-i}(u)}{\phi _r^{\prime }(1)}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) , \end{aligned}$$

uniformly on $S^{n-1}$. Hence

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(K\tilde{+}_\phi \varepsilon \cdot L)-\tilde{W}_i(K)}{\varepsilon }&=\lim \limits _{\varepsilon \rightarrow 0^+}\bigg (\frac{1}{n}\int _{S^{n-1}} \frac{\rho _{K\tilde{+}_\phi \varepsilon \cdot L}^{n-i}(u)-\rho _K^{n-i}(u)}{\varepsilon }dS(u)\bigg )\\&=\frac{1}{n}\int _{S^{n-1}}\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\rho _{K\tilde{+}_\phi \varepsilon \cdot L}^{n-i}(u)-\rho _K^{n-i}(u)}{\varepsilon }dS(u)\\&=\frac{n-i}{n\phi _r^\prime (1)}\int _{S^{n-1}}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) \rho _K^{n-i}(u)dS(u), \end{aligned}$$

we complete the proof of Theorem 1.

From Definition 2 and Theorem 1, we have

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(K\tilde{+}_\phi \varepsilon \cdot L)-\tilde{W}_i(K)}{\varepsilon }=\frac{n-i}{\phi _r^\prime (1)}\tilde{W}_{\phi ,i}(K,L). \end{aligned}$$

(21)

An immediate consequence of Proposition 1 and (21) is contained in:

Proposition 2

If $\phi \in \mathscr {C}$, $K,L\in {\mathscr {S}}_0^n$ and $i\ne n$, then for $A\in SL(n)$,

$$\begin{aligned} \tilde{W}_{\phi ,i}(AK,AL)=\tilde{W}_{\phi ,i}(K,L). \end{aligned}$$

Proof

From Proposition 1 and (21), for $A\in SL(n)$, we have

$$\begin{aligned} \tilde{W}_{\phi ,i}(AK,AL)&=\frac{\phi _r^\prime (1)}{n-i}\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(AK\tilde{+}_\phi \varepsilon \cdot AL)-\tilde{W}_i(AK)}{\varepsilon }\\&=\frac{\phi _r^\prime (1)}{n-i}\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(A(K\tilde{+}_\phi \varepsilon \cdot L))-\tilde{W}_i(K)}{\varepsilon }\\&=\frac{\phi _r^\prime (1)}{n-i}\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(K\tilde{+}_\phi \varepsilon \cdot L)-\tilde{W}_i(K)}{\varepsilon }\\&=\tilde{W}_{\phi ,i}(K,L). \end{aligned}$$

5 Geometric Inequalities

For $K\in {\mathscr {S}}_0^n$ and $i\in \mathbb {R}$, it will be rather good to use the volume-normalized dual conical measure $\tilde{W}_i^*(K)$ defined by

$$\begin{aligned} d\tilde{W}_i^*(K)=\frac{1}{n\tilde{W}_i(K)}\rho _K^{n-i}dS, \end{aligned}$$

(22)

where S is the Lebesgue measure on $S^{n-1}$ and $\tilde{W}_i^*(K)$ is a probability measure on $S^{n-1}$. When i=0, this is same as the definition in [4].

We now set up the dual Orlicz–Minkowski inequality for the dual Quermassintegral as follows:

Theorem 2

Suppose $K,L\in {\mathscr {S}}_0^n$.

(1) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$ and $i<n-1$, then

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,L)\le \tilde{W}_i(K)\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(23)

(2) If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$ and $i<n-1$, then

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,L)\ge \tilde{W}_i(K)\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(24)

(3) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$, $i>n-1$ and $i\ne n$, then

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,L)\ge \tilde{W}_i(K)\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(25)

(4) If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, $i>n-1$ and $i\ne n$, then

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,L)\le \tilde{W}_i(K)\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(26)

Each equality in (23)–(26) holds if and only if K and L are dilates of each other.

Proof

(1) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$, then by dual Orlicz mixed Quermassintegral (5), and $\tilde{W}_i^*(K)$ defined by (22) is a probability measure on $S^{n-1}$, Jensen’s inequality (12), the integral formulas of dual mixed Quermassintegral (4), dual Minkowski inequality (10), and the fact that $\phi$ is increasing on $(0,\infty )$, we have

$$\begin{aligned} \frac{\tilde{W}_{\phi ,i}(K,L)}{\tilde{W}_i(K)}&=\frac{1}{n\tilde{W}_i(K)}\int _{S^{n-1}}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) \rho _K^{n-i}(u)dS(u)\\&\le \phi \bigg (\frac{1}{n\tilde{W}_i(K)}\int _{S^{n-1}}\frac{\rho _L(u)}{\rho _K(u)}\rho _K^{n-i}(u)dS(u)\bigg )\\&=\phi \left( \frac{\tilde{W}_{i}(K,L)}{\tilde{W}_i(K)}\right) \\&\le \phi \left( \frac{\tilde{W}_{i}(K)^{\frac{n-i-1}{n-i}}\tilde{W}_{i}(L)^{\frac{1}{n-i}}}{\tilde{W}_i(K)}\right) \\&=\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) , \end{aligned}$$

(2) If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$, from (5), (22), Jensen’s inequality (13), (4), (10), and $\phi$ is decreasing on $(0,\infty )$, we have

$$\begin{aligned} \frac{\tilde{W}_{\phi ,i}(K,L)}{\tilde{W}_i(K)}&=\frac{1}{n\tilde{W}_i(K)}\int _{S^{n-1}}\phi \left( \frac{\rho _L(u)}{\rho _K(u)}\right) \rho _K^{n-i}(u)dS(u)\\&\ge \phi \bigg (\frac{1}{n\tilde{W}_i(K)}\int _{S^{n-1}}\frac{\rho _L(u)}{\rho _K(u)}\rho _K^{n-i}(u)dS(u)\bigg )\\&=\phi \left( \frac{\tilde{W}_{i}(K,L)}{\tilde{W}_i(K)}\right) \\&\ge \phi \left( \frac{\tilde{W}_{i}(K)^{\frac{n-i-1}{n-i}}\tilde{W}_{i}(L)^{\frac{1}{n-i}}}{\tilde{W}_i(K)}\right) \\&=\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) , \end{aligned}$$

(3) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$, $i>n-1$ and $i\ne n$, proof as similar above, we can immediately obtain (25) which have the same form with (24).

(4) If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, $i>n-1$ and $i\ne n$, similarly, we can immediately obtain (26) which have the same form with (23).

Each equality in (23)–(26) holds if and only if K and L are dilates of each other. Thus we get the significant dual Orlicz–Minkowski inequality.

Remark 2

It immediately follows a few cases for all $K,L\in {\mathscr {S}}_0^n$.

(1) Let $\phi (t)=t^p$ with $p<0$. Equation (24) is just a similar result of Lutwak’s $L_p$ dual Minkowski inequality for the $L_p$ dual mixed volume (see [20]): for $i<n-1$,

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

(2) Let $\phi (t)=\log t$, we have

$$\tilde{W}_{p,i}(K,L)\le \frac{\tilde{W}_i(K)}{n-i}\log \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)},$$

it is a very meaningful result, see [1, 30].

(3) Let $\phi (t)=t^p$ with $0<p<1$. For $i<n-1$, (23) is just

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

(4) Let $\phi (t)=t$. From (23) and (25), we have for $i<n-1$,

$$\tilde{W}_{i}(K,L)^{n-i}\le \tilde{W}_i(K)^{n-i-1}\tilde{W}_i(L),$$

and for $i>n-1, i\ne n$, the above inequality is reversed.

(5) Let $\phi (t)=t^p$ with $p\ge 1$. It follows from (25) that for $i>n-1$, $i\ne n$,

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

Corollary 2

Let $K,L\in {\mathscr {S}}_0^n$, $i<n-1$, $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_{1}~(or~\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_{2})$. If

$$\begin{aligned} \tilde{W}_{\phi ,i}(M,K)=\tilde{W}_{\phi ,i}(M,L),~~~{ for}~{ all}~M\in S_0^n, \end{aligned}$$

(27)

or

$$\begin{aligned} \frac{\tilde{W}_{\phi ,i}(K,M)}{\tilde{W}_i(K)}=\frac{\tilde{W}_{\phi ,i}(L,M)}{\tilde{W}_i(L)},~~~{ for}~{ all}~ M\in S_0^n, \end{aligned}$$

(28)

then $K=L$.

Proof

Whatever $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_{1}, or ~\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_{2}$, the process of proof is almost identical, so we next just prove the situation that $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_{1}$.

Suppose (27) holds, if we take K for M, then from Definition 2 and (9), we have

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,L)=\tilde{W}_{\phi ,i}(K,K)=\phi (1)\tilde{W}_i(K). \end{aligned}$$

However, from (23), we have

$$\begin{aligned} \tilde{W}_{\phi ,i}(K,K)=\tilde{W}_{\phi ,i}(K,L)\le \tilde{W}_i(K)\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) , \end{aligned}$$

then

$$\begin{aligned} \phi (1)\le \phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K)}\right) ^{\frac{1}{n-i}}\right) , \end{aligned}$$

with equality if and only if K and L are dilates of each other. Since $\phi$ is monotone increasing on $(0,\infty )$, we get

$$\begin{aligned} \tilde{W}_i(L)\ge \tilde{W}_i(K), \end{aligned}$$

with equality if and only if K and L are dilates of each other. If we take L for M, similarly we get $\tilde{W}_i(K)\ge \tilde{W}_i(L)$ which shows there is in fact equality in both inequalities and that $\tilde{W}_i(K)=\tilde{W}_i(L)$, hence the equality implies that $K=L$.

Next, assume (28) holds, if we take K for M, then from Definition 2 and (9), we have

$$\begin{aligned} \frac{\tilde{W}_{\phi ,i}(K,K)}{\tilde{W}_{i}(K)}=\phi (1)=\frac{\tilde{W}_{\phi ,i}(L,K)}{\tilde{W}_i(L)}. \end{aligned}$$

But from (23), we have

$$\begin{aligned} \frac{\tilde{W}_{\phi ,i}(L,K)}{\tilde{W}_i(L)}\le \frac{\tilde{W}_i(L)\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(L)}\right) ^{\frac{1}{n-i}}\right) }{\tilde{W}_i(L)}, \end{aligned}$$

then

$$\begin{aligned} \phi (1)\le \phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(L)}\right) ^{\frac{1}{n-i}}\right) , \end{aligned}$$

with equality if and only if K and L are dilates of each other. Since $\phi$ is strictly increasing on $(0,\infty )$, we have

$$\begin{aligned} \tilde{W}_i(K)\ge \tilde{W}_i(L), \end{aligned}$$

with equality if and only if K and L are dilates of each other.

On the other hand, taking L for M, similarly we have $\tilde{W}_i(L)\ge \tilde{W}_i(K)$, which shows that in fact equality holds in both inequalities and $\tilde{W}_i(K)=\tilde{W}_i(L)$. Hence the equality implies $K=L$.

We now establish the following dual Orlicz–Brunn–Minkowski inequality for dual Quermassintegral:

Theorem 3

Let $K,L\in {\mathscr {S}}_0^n$ and $a,b>0$.

(1) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$ and $i<n-1$ then

$$\begin{aligned} \phi (1)\le a\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) +b\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(29)

(2) If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$ and $i<n-1$, then

$$\begin{aligned} \phi (1)\ge a\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) +b\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(30)

(3) If $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$, $i>n-1$ and $i\ne n$, then

$$\begin{aligned} \phi (1)\ge a\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) +b\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(31)

(4) If $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, $i>n-1$ and $i\ne n$, then

$$\begin{aligned} \phi (1)\le a\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) +b\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(32)

Each equality in (29)–(32) holds if and only if K and L are dilates of each other.

Proof

Note $K_\phi =a\cdot K\tilde{+}_\phi b\cdot L$.

(1) When $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_1$ and $i<n-1$, by (9), Lemma 3, Definition 2 and (23), then

$$\begin{aligned} \phi (1)&=\frac{1}{n \tilde{W}_i(K_\phi )}\int _{S^{n-1}}\phi (1)\rho _{K_\phi }^{n-i}(u)dS(u)\\&=\frac{1}{n \tilde{W}_i(K_\phi )}\int _{S^{n-1}}\left[ a\phi \left( \frac{\rho _K(u)}{\rho _{K_\phi }(u)}\right) +b\phi \left( \frac{\rho _L(u)}{\rho _{K_\phi }(u)}\right) \right] \rho _{K_\phi }^{n-i}(u)dS(u)\\&=\frac{a}{n \tilde{W}_i(K_\phi )}\int _{S^{n-1}}\phi \left( \frac{\rho _K(u)}{\rho _{K_\phi }(u)}\right) \rho _{K_\phi }^{n-i}(u)dS(u)\\&~~~~~+\frac{b}{n \tilde{W}_i(K_\phi )}\int _{S^{n-1}}\phi \left( \frac{\rho _L(u)}{\rho _{K_\phi }(u)}\right) \rho _{K_\phi }^{n-i}(u)dS(u)\\&=\frac{a}{\tilde{W}_i(K_\phi )}\tilde{W}_{\phi ,i}(K_\phi ,K)+\frac{b}{\tilde{W}_i(K_\phi )}\tilde{W}_{\phi ,i}(K_\phi ,L)\\&\le a\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_\phi )}\right) ^{\frac{1}{n-i}}\right) +b\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K_\phi )}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(2) When $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_2$ and $i<n-1$, by (9), Lemma 3, Definition 2 and (24), we obtain (30).

(3) When $\phi \in \mathscr {C}^{in}\bigcap \mathscr {C}_2$, $i>n-1$ and $i\ne n$, by (9), Lemma 3, Definition 2 and (25), we obtain (31).

(4) When $\phi \in \mathscr {C}^{de}\bigcap \mathscr {C}_1$, $i>n-1$ and $i\ne n$, by (9), Lemma 3, Definition 2 and (26), we obtain (32).

Each equality in (29)–(32) holds as an equality if and only if K and L are dilates of each other. We obtain the desired dual Orlicz–Brunn–Minkowski inequality (29)–(32).

Remark 3

For $K,L\in {\mathscr {S}}_0^n$, $a,b>0$, some particular cases are as follows: each equality holds if and only if K and L are dilates of each other.

(1) Let $\phi (t)=t^p$ with $p<0$. From (30) we can deduces to the analogous form of Lutwak’s $L_p$ dual Brunn–Minkowski inequality (see [20]): for $i<n-1$,

$$\tilde{W}_{i}(a\cdot K\tilde{+}_\phi b\cdot L)^{\frac{p}{n-i}}\ge a\tilde{W}_i(K)^{\frac{p}{n-i}}+b\tilde{W}_i(L)^{\frac{p}{n-i}}.$$

(2) Let $\phi (t)=\log t$, from (29), we obtain

$$\frac{a}{n-i}\log \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) +\frac{b}{n-i}\log \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(a\cdot K\tilde{+}_\phi b\cdot L)}\right) \ge 0.$$

(3) Let $\phi (t)=t^p$ with $0<p<1$. For $i<n-1$, (29) is just

$$\tilde{W}_{i}(a\cdot K\tilde{+}_\phi b\cdot L)^{\frac{p}{n-i}}\le a\tilde{W}_i(K)^{\frac{p}{n-i}}+b\tilde{W}_i(L)^{\frac{p}{n-i}}.$$

(4) Let $\phi (t)=t$. From (29) and (31), we have for $i<n-1$,

$$\tilde{W}_{i}(a\cdot K\tilde{+}_\phi b\cdot L)^{\frac{1}{n-i}}\le a\tilde{W}_i(K)^{\frac{1}{n-i}}+b\tilde{W}_i(L)^{\frac{1}{n-i}},$$

and for $i>n-1, i\ne n$, the above inequality reversed.

(5) Let $\phi (t)=t^p$ with $p>1$. From (31), it follows that for $i>n-1, i\ne n$,

$$\tilde{W}_{i}(a\cdot K\tilde{+}_\phi b\cdot L)^{\frac{p}{n-i}}\ge a\tilde{W}_i(K)^{\frac{p}{n-i}}+b\tilde{W}_i(L)^{\frac{p}{n-i}}.$$

We derive the equivalence between the dual Orlicz–Minkowski inequalities (23)–(26) and the dual Orlicz–Brunn–Minkowski inequalities (29)–(32), respectively. Since we proved that (23)–(26) implies (29)–(32), respectively, so now we just need to prove that (29)–(32) can deduce (23)–(26), respectively. Since all the process are similar, so we just prove (23) by (29).

Proof of the implication (29) to (23). For $\varepsilon \ge 0$, let $K_\varepsilon =K\tilde{+}_\phi \varepsilon \cdot L$. By (29), the following function

$$\begin{aligned} G(\varepsilon )=\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) +\varepsilon \phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) -\phi (1), \end{aligned}$$

is non-negative and it easily get $G(0)=0$. Then,

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{G(\varepsilon )-G(0)}{\varepsilon }&=\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) +\varepsilon \phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) -\phi (1)}{\varepsilon }\nonumber \\&=\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}\right) -\phi (1)}{\varepsilon } +\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) \nonumber \\&=\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}\right) -\phi (1)}{\left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}-1} \cdot \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}-1}{\varepsilon }\nonumber \\&~~~~~~+\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) . \end{aligned}$$

(33)

Let $t=\left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}$ and note that $t\rightarrow 1^+$ as $\varepsilon \rightarrow 0^+$, consequently,

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\phi \left( \left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}\right) -\phi (1)}{\left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}-1} =\lim \limits _{t\rightarrow 1^+}\frac{\phi (t)-\phi (1)}{t-1}=\phi _r^\prime (1). \end{aligned}$$

(34)

By (21), we have

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\left( \frac{\tilde{W}_i(K)}{\tilde{W}_i(K_ \varepsilon )}\right) ^{\frac{1}{n-i}}-1}{\varepsilon }&=-\lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(K_ \varepsilon )^{\frac{1}{n-i}}-\tilde{W}_i(K)^{\frac{1}{n-i}}}{\varepsilon } \cdot \lim \limits _{\varepsilon \rightarrow 0^+}\tilde{W}_i(K_ \varepsilon )^{-\frac{1}{n-i}}\nonumber \\&=-\frac{1}{n-i}\tilde{W}_i(K)^{\frac{1-n+i}{n-i}}\cdot \lim \limits _{\varepsilon \rightarrow 0^+}\frac{\tilde{W}_i(K_ \varepsilon ) -\tilde{W}_i(K)}{\varepsilon }\cdot \tilde{W}_i(K)^{-\frac{1}{n-i}}\nonumber \\&=-\frac{\tilde{W}_{\phi ,i}(K,L)}{\phi _r^\prime (1)\tilde{W}_i(K)}. \end{aligned}$$

(35)

From (33), (34), (35) and since $G(\varepsilon )$ is non-negative, thus

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0^+}\frac{G(\varepsilon )-G(0)}{\varepsilon } =-\frac{\tilde{W}_{\phi ,i}(K,L)}{\tilde{W}_i(K)} +\phi \left( \left( \frac{\tilde{W}_i(L)}{\tilde{W}_i(K_\varepsilon )}\right) ^{\frac{1}{n-i}}\right) \ge 0. \end{aligned}$$

(36)

Therefore, we have the formula (23). The equality holds as an equality in (36) if and only if $G(\varepsilon )=G(0)=0$, and this means that the equality case in (23) can be obtained from the equality condition of (29).   $\square$