A convex body in ℝ^{n} is a compact convex subset with non-empty interior. Let 𝒦^{n} denote the set of all convex bodies in ℝ^{n}. For the set of convex bodies containing the origin in their interiors, the set of convex bodies whose centroids are at the origin and the set of origin-symmetric convex bodies in ℝ^{n}, we write ${\mathcal{K}}_{o}^{n}$, ${\mathcal{K}}_{c}^{n}$ and ${\mathcal{K}}_{os}^{n}$, respectively. Let *B* denote the standard Euclidean unit ball in ℝ^{n} and write *S*_{n−1} for its surface. In addition, we write *V*(*K*) for the *n*-dimensional volume of a body *K*. Especially, *V*(*B*)=*ω*_{n}.

If *K*∈𝒦^{n}, then its support function *h*_{K}=*h*(*K*,⋅) :ℝ^{n}→(−∞,+∞) is defined by (see [1,2])

$$h(K,x)=max\{x\cdot y:y\in K\},x\in {\mathbb{R}}^{n},$$(1.1)

where *x*⋅*y* denotes the standard inner product of *x* and *y*.

From (1.1), we easily know that: If *A*∈*GL*(*n*), then (see [1])

$$h(AK,u)=h(K,{A}^{\mathrm{\prime}}u),\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thickmathspace}{0ex}}\mathrm{a}\mathrm{l}\mathrm{l}u\in {S}^{n-1}.$$(1.2)Here, *GL*(*n*) denotes the set of all general (non-singular) affine transformations and *A*^{′} denotes the transpose of *A*. Further, from (1.2), it is easy to get that *h*(−*K*,*u*)=*h*(*K*,−*u*), for any *u*∈*S*^{n−1}.

For *K*,*L*∈𝒦^{n}, and *λ*,*μ*≥0 (not both zero), the Minkowski linear combination, *λK*+*μL*∈𝒦^{n}, of *K* and *L* is defined by (see [1,2])

$$h(\lambda K+\mu L,\cdot )=\lambda h(K,\cdot )+\mu h(L,\cdot ),$$(1.3)where *λK*={*λx*:*x*∈*K*}.

Taking *λ*=*μ*=1/2,*L*=−*K* in (1.3), then the difference body, *Δ**K*, of *K*∈𝒦^{n} is given by (see [1])

$$\mathrm{\Delta}K=\frac{1}{2}K+\frac{1}{2}(-K).$$(1.4)Obviously, *Δ**K* is an origin-symmetric convex body.

For the difference body, we know that (see [1]): *If K*∈𝒦^{n}, then

$$V(\mathrm{\Delta}K)\ge V(K),$$*with equality if and only if K is centrally symmetric*.

A recent extension of the Brunn-Minkowski theory is the Orlicz-Brunn-Minkowski theory, which was first launched by Lutwak, Yang and Zhang ([4, 3]) with affine isoperimetric inequalities for Orlicz centroid and projection bodies. In addition, Gardner, Hug and Weil ([5]) built the foundation and provided a general framework for the Orlicz-Brunn-Minkowski theory. In [6], Xi, Jin and Leng obtained the beautiful Orlicz-Brunn-Minkowski inequality, which can be viewed as a generalization of the classical Brunn-Minkowski inequality. Corresponding to Orlicz-Brunn-Minkowski theory, Zhu, Zhou and Xu ([7]) also established the dual Orlicz-Brunn-Minkowski theory. For the recent development of the Orlicz theory, see also [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 21, 22, 23].

Let Φ denote the set of convex and strictly increasing functions *φ*:[0,∞)→[0,+∞) and *φ*(0)=0. In 2014, Gardner, Hug and Weil (see [5], also see [6]) defined the Orlicz-Minkowski combination: Let *φ*∈Φ satisfy *φ*(1)=1, for *K*, $K,L\in {\mathcal{K}}_{o}^{n}$, *α*,*β*≥0 (not both zero), define the Orlicz-Minkowski combination +*φ*(*K*,*L*,*α*,*β*) by

$${h}_{{+}_{\phi}(K,L,\alpha ,\beta )}(u)=inf\{\lambda >0:\alpha \phi (\frac{{h}_{K}(u)}{\lambda})+\beta \phi (\frac{{h}_{L}(u)}{\lambda})\le \phi (1)\}.$$(1.5)Notice that since the function $z\to \alpha \phi (\frac{{h}_{K}(u)}{z})+\beta \phi (\frac{{h}_{L}(u)}{z})$ is strictly decreasing, we have

$${h}_{{+}_{\phi}(K,L,\alpha ,\beta )}(u)={\lambda}_{u},$$if and only if

$$\alpha \phi (\frac{{h}_{K}(u)}{{\lambda}_{u}})+\beta \phi (\frac{{h}_{L}(u)}{{\lambda}_{u}})=\phi (1).$$(1.6)From definition (1.5), we first give the concept of Orlicz difference body as follows:

For $K\in {\mathcal{K}}_{o}^{n}$, define the Orlicz difference body *Δ**φ**K* of *K* by

$${h}_{{\mathrm{\Delta}}_{\phi}K}(u)=inf\{\lambda >0:\frac{1}{2}\phi (\frac{{h}_{K}(u)}{\lambda})+\frac{1}{2}\phi (\frac{{h}_{-K}(u)}{\lambda})\le \phi (1)\}.$$(1.7)From (1.7), it is easy to conclude that ${\mathrm{\Delta}}_{\phi}K\in {\mathcal{K}}_{os}^{n}$, and $K\in {\mathcal{K}}_{os}^{n}$ implies *Δ**φ**K*=*K*. If *φ*(*t*)=*t* in (1.7), then by (1.4), we have *Δ**φ**K*=*Δ**K*.

By (1.7), for $K\in {\mathcal{K}}_{o}^{n}$ and real *τ*∈[−1,1], we also define the general Orlicz difference body, ${\mathrm{\Delta}}_{\phi}^{\tau}K$, of *K* by

$${h}_{{\mathrm{\Delta}}_{\phi}^{\tau}K}(u)=inf\{\lambda >0:{f}_{1}(\tau )\phi (\frac{{h}_{K}(u)}{\lambda})+{f}_{2}(\tau )\phi (\frac{{h}_{-K}(u)}{\lambda})\le \phi (1)\}.$$(1.8)Here,

$${f}_{1}(\tau )=\frac{(1+\tau {)}^{2}}{2(1+{\tau}^{2})},{f}_{2}(\tau )=\frac{(1-\tau {)}^{2}}{2(1+{\tau}^{2})}$$(1.9)for *τ*∈[−1,1]. Obviously, by (1.9), functions *f*_{1}(*τ*) and *f*_{2}(*τ*) satisfy

$${f}_{1}(\tau )+{f}_{2}(\tau )=1,$$(1.10)$${f}_{1}(-\tau )={f}_{2}(\tau ).$$(1.11)From (1.7), (1.8) and (1.9), we easily know that if *τ*=0, then ${\mathrm{\Delta}}_{\phi}^{0}K={\mathrm{\Delta}}_{\phi}K$, and if *τ*=±1, then ${\mathrm{\Delta}}_{\phi}^{+1}K=K,{\mathrm{\Delta}}_{\phi}^{-1}K=-K$.

The aim of this paper is to study the general Orlicz difference bodies and their polars. First, we obtain the following inclusion relationship between general difference bodies *Δ*^{τ}*K* and general Orlicz difference bodies ${\mathrm{\Delta}}_{\phi}^{\tau}K$.

#### Theorem 1.2

*If $K\in {\mathcal{K}}_{o}^{n}$, then for any **τ*∈[−1,1], we have

$${\mathrm{\Delta}}^{\tau}K\subseteq {\mathrm{\Delta}}_{\phi}^{\tau}K,$$(1.12)

*with equality either if and only if $K\in {\mathcal{K}}_{os}^{n}$ or **φ* is a linear function.

Obviously, by (1.12) we immediately get the following corollary.

#### Corollary 1.3

*If $K\in {\mathcal{K}}_{o}^{n}$, **τ*∈[−1,1], and *φ*∈Φ, then

$$V({\mathrm{\Delta}}_{\phi}^{\tau}K)\ge V({\mathrm{\Delta}}^{\tau}K),$$

*with equality either if and only if $K\in {\mathcal{K}}_{os}^{n}$ or **φ* is a linear function.

Next, the extreme value of the general Orlicz difference bodies is also obtained:

#### Theorem 1.4

*If $K\notin {\mathcal{K}}_{os}^{n}$, **τ*∈[−1,1], and *φ*∈Φ, then

$$V({\mathrm{\Delta}}_{\phi}^{\tau}K)\ge V(K).$$(1.13)

*If $K\notin {\mathcal{K}}_{os}^{n}$, equality holds in (1.13) if and only if **τ*=±1; if $K\in {\mathcal{K}}_{os}^{n}$, then (1.13) becomes an equality.

Let ${\mathrm{\Delta}}_{\phi}^{\tau ,c}K=({\mathrm{\Delta}}_{\phi}^{\tau}K{)}^{c}$ denote the polar body of general Orlicz difference body ${\mathrm{\Delta}}_{\phi}^{\tau}K$, the following result gives the extreme value of the volume, $V({\mathrm{\Delta}}_{\phi}^{\tau ,\ast}K)$, of ${\mathrm{\Delta}}_{\phi}^{\tau ,\ast}K$.

#### Theorem 1.5

*If $K\notin {\mathcal{K}}_{os}^{n}$, **τ*∈[−1,1], and *φ*∈Φ, then

$$V({\mathrm{\Delta}}_{\phi}^{\tau ,\ast}K)\le V({K}^{\ast}).$$(1.14)

*If K\hspace*{1.6ex}{\not}\hspace*{-1.3ex}{\in}\,\mathcal{K}_{os}^n, equality holds in (1.14) if and only if **τ*=±1; When $K\in {\mathcal{K}}_{os}^{n}$, inequality (1.14) becomes an identity.

According to the fact that the Orlicz-Minkowski combination +*φ*(*K*,*L*,*α*,*β*)∈K_{no} (see [6]), we easily deduce that ${\mathrm{\Delta}}_{\phi}^{\tau}K\in {\mathcal{K}}_{o}^{n}$. Using inequality (1.13) and the Blaschke-Santal*o*´ inequality for convex bodies, we have

#### Theorem 1.6

*If $K\in {\mathcal{K}}_{os}^{n}$, **φ*∈Φ, then for any *τ*∈[−1,1], we have

$$V(K)V({\mathrm{\Delta}}_{\phi}^{\tau ,c}K)\le {\omega}_{n}^{2},$$(1.15)

*with equality if and only if **K* is an ellipsoid centered at the origin. Here ${\mathrm{\Delta}}_{\phi}^{\tau ,c}K=({\mathrm{\Delta}}_{\phi}^{\tau}K{)}^{c}$, and for $Q\in {\mathcal{K}}_{o}^{n}$, *Q*_{c}=(*Q*−cent*Q*)_{∗} and cent*Q* denotes the centroid of *Q*.

Obviously, when *φ*(*t*)=*t*^{p} (*p*≥1), it is easily checked that Theorems 1.2-1.6 reduce to the results of general *L*_{p}-difference body *Δ*_{τp}*K* (see [30]).

This paper is organized as follows. In Section 2, we list some basic notions that will be indispensable to the proofs of our results. In Section 3, several elementary properties of the general Orlicz difference bodies are listed. In Section 4, the proofs of Theorems 1.2-1.6 are completed.

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