Dual Orlicz geominimal surface area

Abstract

The $L_{p}$-geominimal surface area was introduced by Lutwak in 1996, which extended the important concept of the geominimal surface area. Recently, Wang and Qi defined the p-dual geominimal surface area, which belongs to the dual Brunn-Minkowski theory. In this paper, based on the concept of the dual Orlicz mixed volume, we extend the dual geominimal surface area to the Orlicz version and give its properties. In addition, the isoperimetric inequality, a Blaschke-Santaló type inequality, and the monotonicity inequality for the dual Orlicz geominimal surface areas are established.

1 Introduction

Let $\mathcal{K}^{n}$ denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space $\mathbb {R}^{n}$. For the set of convex bodies containing the origin in their interiors, the set of convex bodies whose centroids lie at the origin, and the set of convex bodies having their Santaló point at the origin in $\mathbb{R}^{n}$, we write $\mathcal{K}^{n}_{o}$, $\mathcal{K}^{n}_{c}$, and $\mathcal {K}^{n}_{s}$, respectively. $\mathcal{S}^{n}_{o}$ denotes the set of star bodies (about the origin) in $\mathbb{R}^{n}$. Let $V(K)$ denote the n-dimensional volume of a body K, and let B denote the standard Euclidean unit ball in $\mathbb{R}^{n}$ and write $\omega_{n}=V(B)$ for its volume, as well as let $S^{n-1}$ denote the unit sphere for B.

The notion of geominimal surface area was introduced by Petty [1]. For $K\in\mathcal{K}^{n}_{o}$, the geominimal surface area, $G(K)$, of K is defined by

$$\omega_{n}^{\frac{1}{n}}G(K)=\inf\bigl\{ nV_{1}(K,Q)V \bigl(Q^{*}\bigr)^{\frac {1}{n}}:Q\in\mathcal{K}^{n}\bigr\} . $$

Here $Q^{*}$ denotes the polar of body Q, and $V_{1}(K,Q)$ denotes the mixed volume of $K, Q\in\mathcal{K}^{n}_{o}$ (see [2]). This concept has already attracted considerable interest; see, for example, [3–5].

The extension of the classical Brunn-Minkowski theory is the $L_{p}$-Brunn-Minkowski theory initialized by Lutwak [6], which has had an enormous impact, providing stronger affine isoperimetric inequalities than the classical counterparts (see, e.g., [7, 8]). In particular, Lutwak [6] introduced the p-geominimal surface area: For $K\in\mathcal{K}^{n}_{o}$, $p\geq1$, the p-geominimal surface area, $G_{p}(K)$, of K is defined by

$$\omega_{n}^{\frac{p}{n}}G_{p}(K)=\inf\bigl\{ nV_{p}(K,Q)V\bigl(Q^{*}\bigr)^{\frac {p}{n}}: Q\in \mathcal{K}^{n}_{o}\bigr\} . $$

Here $V_{p}(K,Q)$ denotes the p-mixed volume of $K,Q\in\mathcal{K}^{n}_{o}$ (see [6]).

Combining the homogeneousness of the volume with the p-mixed volume, the p-geominimal surface area could also be defined by

$$G_{p}(K)=\inf\bigl\{ nV_{p}(K,Q): Q\in\mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*}\bigr)=\omega_{n}\bigr\} . $$

The recent extension of the Brunn-Minkowski theory is the Orlicz-Brunn-Minkowski theory launched by Lutwak et al. [9, 10]. This theory is far more general than the $L_{p}$-Brunn-Minkowski theory, and has already attracted considerable interest; see, for example, [9–14]. In particular, Yuan et al. [15] introduced the Orlicz geominimal surface area $G_{\phi}(K)$ of $K\in\mathcal{K}^{n}_{o}$. Let Φ denote the set of convex functions $\phi: [0,\infty )\rightarrow[0,\infty)$ that satisfy $\phi(0)=0$, $\phi(1)=1$, and $\lim_{t\rightarrow\infty}\phi(t)=+\infty$. For $K\in\mathcal{K}^{n}_{o}$ and $\phi\in\Phi$,

$$G_{\phi}(K)=\inf\bigl\{ nV_{\phi}(K,Q): Q\in\mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*}\bigr)=\omega_{n}\bigr\} , $$

where $V_{\phi}(K,Q)$ denotes Orlicz mixed volume of K and Q (see [16]).

A dual theory to the $L_{p}$-Brunn-Minkowski theory was also developed by Lutwak (see [17, 18]). More recently, Wang and Qi [19] gave a definition of the p-dual geominimal surface area which is a dual concept for $L_{p}$-geominimal surface area and belongs to the dual $L_{p}$-Brunn-Minkowski theory: For $K\in\mathcal{S}^{n}_{o}$, the p-dual geominimal surface area, $\widetilde{G}_{-p}(K)$, of K is defined by

$$\omega_{n}^{-\frac{p}{n}}\widetilde{G}_{-p}(K)=\inf\bigl\{ n \widetilde {V}_{-p}(K,Q)V\bigl(Q^{*}\bigr)^{-\frac{p}{n}}: Q \in\mathcal{K}^{n}_{o}\bigr\} . $$

Here $\widetilde{V}_{-p}(M,N)$ denotes the p-dual mixed volume of $M,N\in\mathcal{S}^{n}_{o}$ (see [6]).

Further, they [19] established the affine isoperimetric inequality, the Blaschke-Santaló inequality and the monotone inequality for the p-dual geominimal surface area.

For $K\in\mathcal{S}^{n}_{o}$ and $p\geq1$,

$$ \widetilde{G}_{-p}(K)\geq n\omega_{n}^{-\frac{p}{n}}V(K)^{\frac{n+p}{n}}, $$

(1.1)

with equality if and only if K is an ellipsoid centered at the origin.

If $K\in\mathcal{K}^{n}_{c}$ and $p\geq1$, then

$$ \widetilde{G}_{-p}(K)\widetilde{G}_{-p} \bigl(K^{*}\bigr)\leq(n\omega_{n})^{2}, $$

(1.2)

with equality if and only if K is an ellipsoid.

If $K\in\mathcal{S}^{n}_{o}$, $1\leq p< q$, then

$$ \biggl(\frac{\widetilde{G}_{-p}(K)^{n}}{n^{n}V(K)^{n+p}} \biggr)^{\frac {1}{p}}\leq \biggl( \frac{\widetilde{G}_{-q}(K)^{n}}{n^{n}V(K)^{n+q}} \biggr)^{\frac{1}{q}}. $$

(1.3)

Combining the homogeneousness of the volume and the p-dual mixed volume, the p-dual geominimal surface area could also be defined by

$$\widetilde{G}_{-p}(K)=\inf\bigl\{ n\widetilde{V}_{-p}(K,Q): Q\in\mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*} \bigr)=\omega_{n}\bigr\} . $$

A step toward a dual Orlicz-Brunn-Minkowski theory for star sets has already been made by Gardner et al. [20–22] and Zhu et al. [23]. In some respects, the dual Orlicz-Brunn-Minkowski theory is far more general than the Orlicz-Brunn-Minkowski theory for convex bodies.

The purpose of this article is to define a new operator $\widetilde {G}_{-\phi}(K)$ of $K\in\mathcal{S}^{n}_{o}$, called the dual Orlicz geominimal surface area, by

$$\widetilde{G}_{-\phi}(K)=\inf\bigl\{ n\widetilde{V}_{-\phi}(K,Q): Q\in \mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*} \bigr)=\omega_{n}\bigr\} , $$

where $\widetilde{V}_{-\phi}(K,Q)$ denotes the dual Orlicz mixed volume of K, Q (see Section 2).

Our main results are as follows.

Theorem 1.1

If $K\in\mathcal{S}^{n}_{o}$ and $\phi\in\Phi$, then there exists a unique body $\tilde{K}\in\mathcal{K}^{n}_{o}$ such that

$$ \widetilde{G}_{-\phi}(K)=n\widetilde{V}_{-\phi}(K, \tilde{K}) \quad\textit{and}\quad V\bigl(\tilde{K}^{*}\bigr)=\omega_{n}. $$

(1.4)

Theorem 1.2

Let $K\in\mathcal{S}^{n}_{o}$ and $\phi\in\Phi$, then

$$ \widetilde{G}_{-\phi}(K)\geq nV(K)\phi \biggl( \biggl( \frac{V(K)}{\omega _{n}} \biggr)^{\frac{1}{n}} \biggr), $$

(1.5)

with equality if and only if K is an ellipsoid.

When $\phi(t)=t^{p}$, with $p\geq1$, the above affine isoperimetric inequality for the dual Orlicz geominimal surface area reduces to affine isoperimetric inequality (1.1) for the p-dual geominimal surface area.

Theorem 1.3

Let $K\in\mathcal{K}^{n}_{c}$ and $\phi\in\Phi$, then

$$ \widetilde{G}_{-\phi}(K)\widetilde{G}_{-\phi} \bigl(K^{*}\bigr)\leq(n\omega_{n})^{2}, $$

with equality if and only if K is an ellipsoid.

When $\phi(t)=t^{p}$, with $p\geq1$, the above Blaschke-Santaló type inequality for the dual Orlicz geominimal surface area reduces to the Blaschke-Santaló type inequality (1.2) for the p-dual geominimal surface area.

Theorem 1.4

Let $\phi_{1},\phi_{2}\in\Phi$ and $\phi_{1}\leq\phi_{2}$. If $K\in \mathcal{S}^{n}_{o}$, then

$$\phi_{2}^{-1} \biggl(\frac{(nV(K))^{n}}{\widetilde{G}_{-\phi _{2}}(K)^{n}\phi_{2}(V(K)^{-1})} \biggr)\leq\phi_{1}^{-1} \biggl(\frac {(nV(K))^{n}}{\widetilde{G}_{-\phi_{1}}(K)^{n}\phi_{1}(V(K)^{-1})} \biggr). $$

When $\phi_{1}(t)=t^{p}$, $\phi_{2}(t)=t^{q}$, with $1\leq p\leq q<\infty$, the above inequality reduces to Wang’s monotonicity inequality (1.3) for the p-dual geominimal surface area ratio.

2 Background material

The support function $h_{K}=h(K,\cdot): S^{n-1}\rightarrow\mathbb{R}$ of a compact convex set $K\subset\mathbb{R}^{n}$ is defined, for $u\in S^{n-1}$, by

$$h_{K}(u)=\max\{u\cdot x: x\in K\}, $$

and it uniquely determines the compact convex set. Here $u\cdot x$ denotes the standard inner product of u and x in $\mathbb{R}^{n}$.

For $K, L\in\mathcal{K}^{n}$, and $\lambda,\mu\geq0$ (not both zero), the Minkowski linear combination $\lambda\cdot K+ \mu\cdot L\in \mathcal{K}^{n}$ is defined by

$$h(\lambda\cdot K+ \mu\cdot L, \cdot)=\lambda h(K,\cdot)+\mu h(L,\cdot). $$

The classical Brunn-Minkowski inequality (see [24]) states that for convex bodies $K,L\in\mathcal{K}^{n}$ and real $\lambda,\mu \geq0$ (not both zero), the volume of the bodies and of their Minkowski linear combination $\lambda\cdot K+ \mu\cdot L\in\mathcal{K}^{n}$ are related by

$$ V(\lambda\cdot K+ \mu\cdot L)^{\frac{1}{n}}\geq\lambda V(K)^{\frac {1}{n}}+\mu V(L)^{\frac{1}{n}}, $$

(2.1)

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

For real $p\geq1$, $K,L\in\mathcal{K}^{n}_{o}$, and $\lambda,\mu\geq0$ (not both zero), the Firey linear combination $\lambda\cdot K+_{p}\mu\cdot L$, is defined by (see [25])

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

Firey [25] also established the following $L_{p}$ Brunn-Minkowski inequality. If $p>1$, $\lambda,\mu\geq0$ (not both zero), and $K,L\in\mathcal {K}^{n}_{o}$, then

$$V(\lambda\cdot K+_{p} \mu\cdot L)^{\frac{p}{n}}\geq \lambda V(K)^{\frac{p}{n}}+\mu V(L)^{\frac{p}{n}}, $$

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

The radial function of $K, \rho_{K}: S^{n-1}\rightarrow[0,\infty)$, is defined by

$$\rho_{K}(u)=\max\{\lambda: \lambda u\in K\}. $$

A set $K\subset\mathbb{R}^{n}$ is said to be a star body about the origin, if the line segment from the origin to any point $x\in K$ is contained in K and K has continuous and positive radial function $\rho _{K}(\cdot)$.

Note that $K\in\mathcal{S}^{n}_{o}$ can be uniquely determined by its radial function $\rho_{K}(\cdot)$ and vice versa. If $\lambda>0$, we have

$$\rho_{K}(\lambda x)=\lambda^{-1} \rho_{K}(x)\quad \mbox{and} \quad\rho _{\lambda K}(x)=\lambda \rho_{K}(x). $$

More generally, from the definition of the radial function it follows immediately that for $\Lambda\in\operatorname{GL}(n)$ the radial function of the image $\Lambda K=\{\Lambda y: y\in K\}$ of $K\in\mathcal {S}^{n}_{o}$ is given by (see [26])

$$ \rho(\Lambda K,x)=\rho\bigl(K,\Lambda^{-1}x\bigr), \quad\mbox{for all } x\in \mathbb{R}^{n}. $$

(2.2)

Two star bodies $K,L\in\mathcal{S}^{n}_{o}$ are said to be dilates of each other if there is a constant $\lambda>0$ such that $L=\lambda K$, and the equation $\rho_{L}(u)=\lambda\rho_{K}(u)$ for all $u\in S^{n-1}$. Clearly, for $K,L\in\mathcal{S}^{n}_{o}$,

$$K\subseteq L \quad\mbox{if and only if}\quad \rho_{K}(u)\leq\rho _{L}(u),\quad \mbox{for all } u\in S^{n-1}. $$

The natural metric on $\mathcal{S}^{n}_{o}$ is the radial metric $\widetilde{\delta}(\cdot,\cdot): \mathcal{S}^{n}_{o}\times\mathcal {S}^{n}_{o}\rightarrow\mathbb{R}$ defined as

$$\widetilde{\delta}(K,L)=\|\rho_{K}-\rho_{L}\|_{\infty}= \sup_{u\in S^{n-1}}\bigl|\rho_{K}(u)-\rho_{L}(u)\bigr|, \quad\mbox{for } K,L\in\mathcal{S}^{n}_{o}. $$

A sequence of star bodies $\{K_{j}\}\subset\mathcal{S}^{n}_{o}$ is said to be convergent to $K\in\mathcal{S}^{n}_{o}$ in δ̃ if $\widetilde{\delta}(K_{j},K)\rightarrow0$ as $j\rightarrow\infty$, and equivalently, $\rho_{K_{j}}$ is uniformly convergent to $\rho_{K}$ on $S^{n-1}$.

If $K\in\mathcal{K}_{o}^{n}$, the polar body, $K^{\ast}$, of K is defined by

$$\begin{aligned} K^{*}=\bigl\{ x\in\mathbb{R}^{n}: x\cdot y \leq1, \mbox{ for all } y\in K\bigr\} . \end{aligned}$$

Obviously, we have $(K^{*})^{*}=K$.

The Blaschke-Santaló inequality [27] is one of the fundamental affine isoperimetric inequalities. It states that if $Q\in \mathcal{K}^{n}_{c}$ then

$$ V(Q)V\bigl(Q^{*}\bigr)\leq\omega_{n}^{2}, $$

(2.3)

with equality if and only if Q is an ellipsoid.

For $\phi\in\Phi$ and $\lambda, \mu\geq0$ (not both zero), we define the Orlicz radial sum $\lambda\circ K\widetilde{+}_{-\phi}\mu\circ L$ of two star bodies $K, L\in\mathcal{S}^{n}_{o}$, by

$$ \rho(\lambda\circ K\widetilde{+}_{-\phi}\mu\circ L,u)=\sup \biggl\{ t>0: \lambda\phi \biggl(\frac{t}{\rho_{K}(u)} \biggr)+\mu\phi \biggl( \frac {t}{\rho_{L}(u)} \biggr)\leq1 \biggr\} , $$

(2.4)

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

When $\phi(t)=t^{p}$, with $p\geq1$, it is easy to show that the Orlicz radial sum reduces to Lutwak’s p-harmonic radial combination (see [6]):

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

If $K,L\in\mathcal{K}^{n}_{o}$, then

$$\lambda\circ K\widetilde{+}_{-p} \mu\circ L=\bigl(\lambda\cdot K^{*}+_{p}\mu\cdot L^{*}\bigr)^{*}. $$

We denote the right derivative of a real-valued function f by $f_{r}'$. For $\phi\in\Phi$, there is $\phi_{r}'(1)>0$ because ϕ is convex and strictly increasing.

Let $\phi\in\Phi$. By the Orlicz radial sum (2.4), we define the dual Orlicz mixed volume $\widetilde{V}_{-\phi}(K,L)$ of convex bodies $K,L\in\mathcal{S}^{n}_{o}$ by

$$ \frac{n}{-\phi'_{r}(1)}\widetilde{V}_{-\phi}(K,L)=\lim _{\varepsilon \rightarrow0^{+}}\frac{V(K\widetilde{+}_{-\phi}\varepsilon\circ L)-V(K)}{\varepsilon}. $$

(2.5)

From (2.5) we easily obtain the following integral formula of the dual Orlicz mixed volume:

$$ \widetilde{V}_{-\phi}(K,L)=\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho _{K}(u)}{\rho_{L}(u)} \biggr) \rho_{K}^{n}(u)\,\mathrm{d}S(u). $$

(2.6)

Apparently, we have

$$ \widetilde{V}_{-\phi}(K,K)=V(K). $$

(2.7)

For $\phi(t)=t^{p}$ with $p\geq1$, the dual Orlicz mixed volume $\widetilde{V}_{\phi}(K,L)$ reduces to Lutwak’s p-dual mixed volume formula (see [6]):

$$\widetilde{V}_{-p}(K,L)=\frac{1}{n} \int_{S^{n-1}}\rho_{K}^{n+p}(u)\rho _{L}^{-p}(u)\,\mathrm{d}S(u), $$

for all $K,L\in\mathcal{S}^{n}_{o}$.

Using the same argument as in [23] we establish the following dual Orlicz-Minkowski inequality: Suppose $\phi\in\Phi$. If $K,L\in \mathcal{S}^{n}_{o}$, then

$$ \widetilde{V}_{-\phi}(K,L)\geq V(K)\phi \biggl( \biggl( \frac {V(K)}{V(L)} \biggr)^{\frac{1}{n}} \biggr), $$

(2.8)

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

When $\phi(t)=t^{p}$, with $p\geq1$. Then (2.8) reduces to the following $L_{p}$-dual Minkowski inequality (see [6]):

$$\widetilde{V}_{-p}(K,L)^{n}\geq V(K)^{n+p}V(L)^{-p}, $$

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

We also establish the dual Orlicz-Brunn-Minkowski inequality as follows: Suppose $K,L\in\mathcal{S}^{n}_{o}$, and $\lambda,\mu>0$. If $\phi\in\Phi$, then

$$\lambda\phi \biggl( \biggl(\frac{V(\lambda\circ K\widetilde{+}_{\phi}\mu \circ L)}{V(K)} \biggr)^{\frac{1}{n}} \biggr)+\mu\phi \biggl( \biggl(\frac {V(\lambda\circ K\widetilde{+}_{\phi}\mu\circ L)}{V(L)} \biggr)^{\frac {1}{n}} \biggr)\leq1, $$

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

When $\phi(t)=t^{p}$, with $p\geq1$, the above inequality reduces to Lutwak’s $L_{p}$-dual Brunn-Minkowski inequality (see [6]):

$$V(\lambda\circ K\widetilde{+}_{-p}\mu\circ L)^{-\frac{p}{n}}\geq \lambda V(K)^{-\frac{p}{n}}+\mu V(L)^{-\frac{p}{n}}, $$

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

The following results are required in the proofs of our main results.

Lemma 2.1

If $\phi\in\Phi$, and $K,L\in\mathcal{S}^{n}_{o}$, then for $\Lambda \in\operatorname{SL}(n)$,

$$\widetilde{V}_{-\phi}(\Lambda K,\Lambda L)=\widetilde{V}_{\phi}(K,L). $$

Proof

For $x\in\mathbb{R}^{n}\backslash\{0\}$, let $\langle x\rangle=x/|x|$. By (2.6) and (2.2), we have

$$\begin{aligned} \widetilde{V}_{-\phi}(\Lambda K,L) =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{\Lambda K}(u)}{\rho_{L}(u)} \biggr) \rho_{\Lambda K}^{n}(u)\,\mathrm{d}S(u)\\ =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(\Lambda^{-1} u)}{\rho_{L}(\Lambda\Lambda^{-1}u)} \biggr) \rho_{K}^{n}\bigl(\Lambda ^{-1}u\bigr)\,\mathrm{d}S \bigl(\Lambda\Lambda^{-1}u\bigr)\\ =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(\langle \Lambda ^{-1} u\rangle)}{\rho_{\Lambda^{-1} L}(\langle\Lambda^{-1}u\rangle )} \biggr) \rho_{K}^{n}\bigl(\bigl\langle \Lambda^{-1}u\bigr\rangle \bigr)|\det\Lambda|\,\mathrm{d}S\bigl(\bigl\langle \Lambda^{-1}u\bigr\rangle \bigr)\\ =&\widetilde{V}_{-\phi}\bigl(K,\Lambda^{-1}L\bigr), \end{aligned}$$

where $\Lambda^{-1}$ denotes the inverse of Λ. □

It is easy to check that $\widetilde{V}_{-\phi}(\lambda K,\lambda L)=\lambda^{n}\widetilde{V}_{-\phi}(K,L)$, for $\lambda>0$. Therefore, we have the following.

Proposition 2.2

Suppose $K, L\in\mathcal{S}^{n}_{o}$. If $\phi\in\Phi$ and $\Lambda \in\operatorname{GL}(n)$, then

$$\widetilde{V}_{-\phi}(\Lambda K,\Lambda L)=|\det\Lambda|\widetilde {V}_{-\phi}(K,L). $$

Lemma 2.3

Suppose $f_{i}$, f are strictly positive and continuous functions on $S^{n-1}$; $\phi_{j},\phi\in\Phi$; $\mu_{k}$, μ are Borel probability measures on $S^{n-1}$; $i,j,k\in\mathbb{N}$. If $f_{i}\rightarrow f$ pointwise, $\phi _{j}\rightarrow\phi$ uniformly, and $\mu_{k}\rightarrow\mu$ weakly, then

$$ \int_{S^{n-1}}\phi_{j}(f_{i})\,\mathrm{d} \mu_{k}\rightarrow \int _{S^{n-1}}\phi(f)\,\mathrm{d}\mu, \quad\textit{as } i,j,k\rightarrow \infty. $$

(2.9)

Proof

The continuity of $f_{i}$ and f, and $f_{i}\rightarrow f$ pointwise guarantee that $f_{i}\rightarrow f$ uniformly. Thus, there exists an $N_{0}\in\mathbb{N}$, such that

$$\frac{1}{2}\min_{u\in S^{n-1}}f(u)\leq f_{i}(u)\leq2 \max_{u\in S^{n-1}}f(u), \quad\mbox{for } i>N_{0}. $$

Let

$$c_{m}=\min \biggl\{ \frac{1}{2}\min_{u\in S^{n-1}}f(u), \min_{u\in S^{n-1}}f_{i}(u), \mbox{ with } i\leq N_{0} \biggr\} $$

and

$$c_{M}=\min \Bigl\{ 2\max_{u\in S^{n-1}}f(u),\max _{u\in S^{n-1}}f_{i}(u), \mbox{ with } i\leq N_{0} \Bigr\} . $$

The strict positivity and the continuity of $f_{i}$ and f imply that $0< c_{m}\leq c_{M}<\infty$. Thus,

$$ c_{m}\leq f(u)\leq c_{M}\quad \mbox{and}\quad c_{m}\leq f_{i}(u)\leq c_{M}, \quad\mbox{for } u\in S^{n-1} \mbox{ and } i\in\mathbb{N}. $$

(2.10)

Since $\phi_{j}\rightarrow\phi$ uniformly on $[c_{m}, c_{M}]$, by (2.10) and $f_{i}\rightarrow f$ uniformly, it follows that as $i,j\rightarrow\infty$, $\phi_{j}(f_{i})\rightarrow\phi(f)$, uniformly on $S^{n-1}$. Combined with $\mu_{k}\rightarrow\mu$ weakly, one concludes that, as $i,j,k\rightarrow\infty$,

$$\int_{S^{n-1}}\phi_{j}(f_{i})\,\mathrm{d} \mu_{k}\rightarrow \int _{S^{n-1}}\phi(f)\,\mathrm{d}\mu, $$

as for (2.9) as desired. □

Using Lemma 2.3, we immediately obtain the following result.

Lemma 2.4

Suppose $K, K_{i}, L, L_{j}\in\mathcal{S}^{n}_{o}$ and $\phi,\phi _{k}\in\Phi$. If $K_{i}\rightarrow K$, $L_{j}\rightarrow L$ and $\phi _{k}\rightarrow\phi$, then $\lim_{i,j,k\rightarrow\infty}\widetilde{V}_{-\phi _{k}}(K_{i},L_{j})=\widetilde{V}_{-\phi}(K,L)$.

3 Dual Orlicz geominimal surface areas

In this section we first propose the following problem.

Problem 3.1

For $K\in\mathcal{S}^{n}_{o}$, find a convex body Q, amongst all convex bodies containing the origin in their interiors, which solves the constrained infimum problem

$$\widetilde{G}_{-\phi}(K):=\inf \bigl(n\widetilde{V}_{-\phi}(K,Q) \bigr)\quad \mbox{subject to}\quad V\bigl(Q^{*}\bigr)=\omega_{n}. $$

In order to demonstrate the existence and uniqueness of Problem 3.1, the following simple fact will be needed.

Lemma 3.2

([6])

Let $\mathcal{C}^{n}$ denote the set of compact convex subsets of Euclidean n-space $\mathbb{R}^{n}$, and suppose $K_{i}\in\mathcal {K}^{n}_{o}$ such that $K_{i}\rightarrow L\in\mathcal{C}^{n}$. If the sequence $V(K_{i}^{*})$ is bounded, then $L\in\mathcal{K}^{n}_{o}$.

Now we answer Problem 3.1, namely, the proof of Theorem 1.1 is given.

Proof of Theorem 1.1

Problem 3.1 can be equivalently restated as: For a fixed star body $K\in\mathcal{S}^{n}_{o}$, there exists a sequence $M_{i}\in\mathcal{K}^{n}_{o}$ such that $V(M_{i}^{*})=\omega_{n}$, with $\widetilde{V}_{-\phi }(K,B)\geq\widetilde{V}_{-\phi}(K,M_{i})$, for all i, and $n\widetilde{V}_{-\phi}(K,M_{i})\rightarrow\widetilde{G}_{-\phi}(K)$. To see that the $M_{i}\in\mathcal{K}^{n}_{o}$ are uniformly bounded, let

$$R_{i}=R(M_{i})=\rho(M_{i},u_{i})= \max\bigl\{ \rho(M_{i},u): u\in S^{n-1}\bigr\} , $$

where $u_{i}$ is any of the points in $S^{n-1}$ at which this maximum is attained.

Let $r_{K}=\min_{S^{n-1}}\rho_{K}$. Then $r_{K}B\subseteq K$. From the definition (2.6) of the dual Orlicz mixed volume and the Jensen inequality, it follows that

$$\begin{aligned} \frac{\widetilde{V}_{-\phi}(K,B)}{V(K)} \geq& \frac{\widetilde{V}_{-\phi }(K,M_{i})}{V(K)} \\ =& \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(u)}{\rho_{M_{i}}(u)} \biggr)\,\mathrm{d} \widetilde{V}_{K}^{*} \\ \geq&\phi \biggl( \int_{S^{n-1}}\frac{\rho_{K}(u)}{\rho_{M_{i}}(u)}\,\mathrm{d}\widetilde{V}_{K}^{*} \biggr) \\ \geq&\phi \biggl( \int_{S^{n-1}}\frac{\rho_{K}(u)}{R_{i}}\,\mathrm{d}\widetilde {V}_{K}^{*} \biggr) \\ =&\phi \biggl(\frac{1}{nV(K)R_{i}} \int_{S^{n-1}}\rho_{K}(u)^{n+1}\,\mathrm{d}S(u) \biggr) \\ \geq&\phi \biggl(\frac{r_{K}}{nV(K)R_{i}} \int_{S^{n-1}}\rho _{K}(u)^{n}\,\mathrm{d}S(u) \biggr) \\ =&\phi \biggl(\frac{r_{K}}{R_{i}} \biggr). \end{aligned}$$

Namely,

$$\omega_{n}r_{K}^{n}\phi \biggl(\frac{r_{K}}{R_{i}} \biggr)\leq\widetilde {V}_{-\phi}(K,M_{i})\leq \widetilde{V}_{-\phi}(K,B)< \infty. $$

Since the $M_{i}$ are uniformly bounded, the Blaschke selection theorem guarantees the existence of a subsequence of the $M_{i}$, which will also be denoted by $M_{i}$, and a compact convex $L\in\mathcal{C}^{n}$, such that $M_{i}\rightarrow L$. Since $V(M_{i}^{*})=\omega_{n}$, Lemma 3.2 gives $L\in\mathcal{K}^{n}_{o}$. Now, $M_{i}\rightarrow L$ implies that $M_{i}^{*}\rightarrow L^{*}$, and since $V(M_{i}^{*})=\omega_{n}$, it follows that $V(L^{*})=\omega_{n}$. Lemma 2.4 can now be used to conclude that L will serve as the desired body K̃.

The uniqueness of the minimizing body is easily demonstrated as follows. Suppose $L_{1},L_{2}\in\mathcal{K}^{n}_{o}$ and $L_{1}\neq L_{2}$, such that $V(L_{1}^{*})=\omega_{n}=V(L_{2}^{*})$, and

$$\widetilde{V}_{-\phi}(K,L_{1})=\widetilde{V}_{-\phi}(K,L_{2}). $$

Define $L\in\mathcal{K}^{n}_{o}$, by

$$L=\frac{1}{2}\circ L_{1}\widetilde{+}_{-1} \frac{1}{2}\circ L_{2}. $$

Since obviously

$$L^{*}=\frac{1}{2} \cdot L_{1}^{*}+ \frac{1}{2} \cdot L_{2}^{*}, $$

and $V(L_{1}^{*})=\omega_{n}=V(L_{2}^{*})$, it follows from the Brunn-Minkowski inequality (2.1) that

$$V\bigl(L^{*}\bigr)\geq\omega_{n}, $$

with equality if and only if $L_{1}=L_{2}$.

By the definition (2.6) of the dual Orlicz mixed volume, together with the convexity of ϕ, we have

$$\begin{aligned} \widetilde{V}_{-\phi}(K,L) =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac {\rho_{K}(u)}{\rho_{\frac{1}{2}\circ L_{1}\widetilde{+}_{-1}\frac {1}{2}\circ L_{2}}(u)} \biggr) \rho_{K}(u)^{n}\,\mathrm{d}S(u)\\ =&\frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(u)}{ (\frac {1}{2\rho_{L_{1}}(u)}+\frac{1}{2\rho_{L_{2}}(u)} )^{-1}} \biggr)\rho _{K}(u)^{n}\,\mathrm{d}S(u)\\ =& \frac{1}{n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(u)}{2\rho _{L_{1}}(u)}+\frac{\rho_{K}(u)}{2\rho_{L_{2}}(u)} \biggr)\rho _{K}(u)^{n}\,\mathrm{d}S(u)\\ \leq&\frac{1}{2n} \int_{S^{n-1}}\phi \biggl(\frac{\rho_{K}(u)}{\rho _{L_{1}}(u)} \biggr) \rho_{K}(u)^{n}\,\mathrm{d}S(u)+\frac{1}{2n} \int _{S^{n-1}}\phi \biggl(\frac{\rho_{K}(u)}{\rho_{L_{2}}(u)} \biggr)\rho _{K}(u)^{n}\,\mathrm{d}S(u)\\ =&\frac{1}{2}\widetilde{V}_{-\phi}(K,L_{1})+ \frac{1}{2}\widetilde {V}_{-\phi}(K,L_{2}) \\ =&\widetilde{V}_{-\phi}(K,L_{1}) \\ =&\widetilde{V}_{-\phi}(K,L_{2}), \end{aligned}$$

with equality if and only if $L_{1}=L_{2}$. Thus,

$$\widetilde{V}_{-\phi}(K,L)< \widetilde{V}_{-\phi}(K,L_{1})= \widetilde {V}_{-\phi}(K,L_{2}), $$

is the contradiction that would arise if it were the case that $L_{1}\neq L_{2}$. This completes the proof. □

In view of Theorem 1.1, naturally, we introduce the following definition.

Definition 3.3

For $K\in\mathcal{S}^{n}_{o}$, and $\phi\in\Phi$, we define the dual Orlicz geominimal surface area, $\widetilde{G}_{-\phi}(K)$, of K by

$$\widetilde{G}_{-\phi}(K)=\inf\bigl\{ n\widetilde{V}_{-\phi}(K,Q): Q\in \mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*} \bigr)=\omega_{n}\bigr\} . $$

When $\phi(t)=t^{p}$, with $p\geq1$, the dual Orlicz geominimal surface area reduces to Wang’s p-geominimal surface area.

We will show that the dual Orlicz geominimal surface area of a stat body is invariant under unimodular centro-transformations of the body.

Proposition 3.4

Suppose $K\in\mathcal{S}^{n}_{o}$. If $\phi\in\Phi$ and $\Lambda\in \operatorname{SL}(n)$, then

$$\widetilde{G}_{-\phi}(\Lambda K)=\widetilde{G}_{-\phi}(K). $$

Proof

From Definition 3.3 of the dual Orlicz geominimal surface area, and Lemma 2.1, we have for $\Lambda\in \operatorname{SL}(n)$

$$\begin{aligned} \widetilde{G}_{-\phi}(\Lambda K) =&\inf\bigl\{ n\widetilde{V}_{-\phi}( \Lambda K,Q): Q\in\mathcal{K}^{n}_{o} \mbox{ and } V \bigl(Q^{*}\bigr)=\omega _{n}\bigr\} \\ =&\inf\bigl\{ n\widetilde{V}_{-\phi}\bigl( K,\Lambda^{-1}Q \bigr): \Lambda^{-1}Q\in \mathcal{K}^{n}_{o} \mbox{ and } V\bigl(\bigl(\Lambda^{-1}Q\bigr)^{*}\bigr)= V\bigl( \Lambda ^{t}Q^{*}\bigr)=\omega_{n}\bigr\} \\ =&\widetilde{G}_{-\phi}(K). \end{aligned}$$

 □

The unique body whose existence is guaranteed by Theorem 1.1 will be denoted by $\widetilde{T}_{\phi}K$, and it is called the dual Orlicz-Petty body of K. The polar body of $\widetilde{T}_{\phi}K$ will be denoted by $\widetilde {T}^{*}_{\phi}K$, rather than $(\widetilde{T}_{\phi}K)^{*}$. Thus, for $K\in\mathcal{S}^{n}_{o}$, the body $\widetilde{T}_{\phi}K$ is defined by

$$\widetilde{G}_{-\phi}(K)=n\widetilde{V}_{-\phi}(K, \widetilde{T}_{\phi }K) \quad\mbox{and}\quad V\bigl(\widetilde{T}_{\phi}^{*}K \bigr)=\omega_{n}. $$

The next proposition shows that the mapping $\widetilde{T}_{\phi}: \mathcal{S}^{n}_{o}\rightarrow\mathcal{K}^{n}_{o}$ is a unimodular centro-affine invariant mapping.

Proposition 3.5

If $K\in\mathcal{S}^{n}_{o}$, then for $\Lambda\in\operatorname{SL}(n)$,

$$\widetilde{T}_{\phi}\Lambda K=\Lambda\widetilde{T}_{\phi}K. $$

Proof

From the definition of $\widetilde{T}_{\phi}$ and Proposition 3.4,

$$n\widetilde{V}_{-\phi}(K,\widetilde{T}_{\phi}K)= \widetilde{G}_{-\phi }(K)=\widetilde{G}_{-\phi}(\Lambda K)=n \widetilde{V}_{-\phi}(\Lambda K,\widetilde{T}_{\phi}\Lambda K). $$

By Lemma 2.1,

$$\widetilde{V}_{-\phi}(K,\widetilde{T}_{\phi}K)= \widetilde{V}_{-\phi }(\Lambda K,\widetilde{T}_{\phi}\Lambda K)= \widetilde{V}_{-\phi }\bigl(K,\Lambda^{-1}\widetilde{T}_{\phi} \Lambda K\bigr). $$

The uniqueness part of Theorem 1.1 shows that $\widetilde {T}_{\phi}K=\Lambda^{-1}\widetilde{T}_{\phi}\Lambda K$, which is the desired result. □

An immediate consequence of the definition of $\widetilde{G}_{-\phi}$ and Lemma 2.4 is the following.

Proposition 3.6

If $\phi\in\Phi$, then the functional $\widetilde{G}_{-\phi}: \mathcal{S}^{n}_{o}\rightarrow(0,\infty)$, is continuous.

For $Q\in\mathcal{K}^{n}$, let $\operatorname{Cen}(Q)\in\operatorname{int}Q$ denote the centroid of Q. Associated with each $Q\in\mathcal{K}^{n}$ is a point $s=\operatorname{San}(Q)\in\operatorname {int}K$, called the Santaló point of Q, the Santaló point also may be defined as the unique $s\in Q$, such that

$$V\bigl((-s+Q)^{*}\bigr)=\min\bigl\{ V\bigl((-x+Q)^{*}\bigr): x\in\operatorname{int}Q\bigr\} , $$

or equivalently, as the unique $s\in Q$, such that

$$\int_{S^{n-1}}u h(-s+Q,u)^{-(n+1)}dS(u)=0. $$

We recall that $\mathcal{K}^{n}_{s}$ denotes the set of convex bodies having their Santaló point at the origin. Thus (see [6]),

$$Q\in\mathcal{K}^{n}_{s} \quad\mbox{if and only if}\quad Q^{*}\in\mathcal {K}^{n}_{c}. $$

Now we give the proof of Theorem 1.2.

Proof of Theorem 1.2

Suppose $Q\in\mathcal {K}^{n}_{o}$, and s is the Santaló point of Q. Let $Q_{0}=-s+Q\in\mathcal{K}^{n}_{s}$. Since $V(Q^{*}_{0})\leq V(Q^{*})$ (see [6], p.263), then by the definition of the dual Orlicz geominimal surface area, the dual Orlicz-Minkowski inequality (2.8) and the Blaschke-Santaló inequality (2.3), it follows that

$$\begin{aligned} \widetilde{G}_{-\phi}(K) =&\inf\bigl\{ n\widetilde{V}_{-\phi}(K,Q): Q\in \mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*} \bigr)=\omega_{n}\bigr\} \\ \geq&\inf \biggl\{ nV(K)\phi \biggl( \biggl(\frac{V(K)}{V(Q)} \biggr)^{\frac {1}{n}} \biggr): Q\in\mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*}\bigr)=\omega_{n} \biggr\} \\ =&\inf \biggl\{ nV(K)\phi \biggl( \biggl(\frac{V(K)}{V(s+Q_{0})} \biggr)^{\frac {1}{n}} \biggr): Q_{0}\in\mathcal{K}^{n}_{s} \mbox{ and } V\bigl(Q_{0}^{*}\bigr)\leq\omega_{n} \biggr\} \\ =&\inf \biggl\{ nV(K)\phi \biggl( \biggl(\frac{V(K)}{V(Q_{0})} \biggr)^{\frac {1}{n}} \biggr): Q_{0}\in\mathcal{K}^{n}_{s} \mbox{ and } V\bigl(Q_{0}^{*}\bigr)\leq\omega_{n} \biggr\} \\ \geq& nV(K)\phi \biggl( \biggl(\frac{V(K)}{\omega_{n}} \biggr)^{\frac {1}{n}} \biggr). \end{aligned}$$

By the equality conditions of the dual Orlicz-Minkowski inequality (2.8) and the Blaschke-Santaló inequality (2.3), we see that equality holds in (1.5) if and only if K is an ellipsoid centered at the origin. □

Proof of Theorem 1.3

By the dual Orlicz-Minkowski inequality (2.8), we get

$$\begin{aligned} &\omega_{n}\widetilde{G}_{-\phi}(K)\phi \biggl( \biggl( \frac {V(Q^{*})}{V(K^{*})} \biggr)^{\frac{1}{n}} \biggr) \\ &\quad =\inf \biggl\{ n\widetilde{V}_{-\phi}(K,Q)\phi \biggl( \biggl( \frac {V(Q^{*})}{V(K^{*})} \biggr)^{\frac{1}{n}} \biggr)V\bigl(Q^{*}\bigr): Q\in \mathcal {K}^{n}_{o} \mbox{ and } V\bigl(Q^{*}\bigr)= \omega_{n} \biggr\} \\ &\quad \leq\inf \bigl\{ n\widetilde{V}_{-\phi}(K,Q) \widetilde{V}_{\phi }\bigl(Q^{*},K^{*}\bigr): Q\in \mathcal{K}^{n}_{o} \mbox{ and } V\bigl(Q^{*}\bigr)= \omega _{n} \bigr\} . \end{aligned}$$

(3.1)

Since $K\in\mathcal{K}^{n}_{c}$, taking $Q=K$ in (3.1), and combining equation (2.7) with inequality (2.3), we get

$$\omega_{n}\widetilde{G}_{-\phi}(K)\leq nV(K)V \bigl(K^{*}\bigr)\leq n\omega_{n}^{2}, $$

i.e.,

$$ \widetilde{G}_{-\phi}(K)\leq n\omega_{n}. $$

(3.2)

Similarly,

$$ \widetilde{G}_{-\phi}\bigl(K^{*}\bigr)\leq n \omega_{n}. $$

(3.3)

Combining (3.2) and (3.3), we get

$$ \widetilde{G}_{-\phi}(K)\widetilde{G}_{-\phi} \bigl(K^{*}\bigr)\leq(n\omega_{n})^{2}. $$

(3.4)

By the equality conditions of the dual Orlicz-Minkowski inequality (2.8) and the Blaschke-Santaló inequality (2.3), we see that equality in (3.4) holds if and only if K is an ellipsoid centered at the origin. □

Proof of Theorem 1.4

By the definition of the dual Orlicz geominimal surface area and $\phi_{i}$ ($i=1,2$) being strictly increasing on $(0,\infty)$, we have

$$\begin{aligned} &\phi_{1} \biggl(\frac{n^{n}V(K)^{n}}{\widetilde{G}_{-\phi _{2}}(K)^{n}\phi_{2}(V(K)^{-1})} \biggr)\\ &\quad =\phi_{1} \biggl(\frac{n^{n}V(K)^{n}}{\inf\{n^{n}\widetilde{V}_{-\phi _{2}}(K,Q)^{n}: Q\in\mathcal{K}^{n}_{o} \mbox{ and } V(Q^{*})=\omega_{n}\}\phi_{2}(V(K)^{-1})} \biggr)\\ &\quad \leq\phi_{1} \biggl(\frac{n^{n}V(K)^{n}}{\inf\{n^{n}\widetilde {V}_{-\phi_{1}}(K,Q)^{n}: Q\in\mathcal{K}^{n}_{o} \mbox{ and } V(Q^{*})=\omega_{n}\}\phi_{1}(V(K)^{-1})} \biggr)\\ &\quad \leq\phi_{2} \biggl(\frac{n^{n}V(K)^{n}}{\inf\{n^{n}\widetilde {V}_{-\phi_{1}}(K,Q)^{n}: Q\in\mathcal{K}^{n}_{o} \mbox{ and } V(Q^{*})=\omega_{n}\}\phi_{1}(V(K)^{-1})} \biggr)\\ &\quad =\phi_{2} \biggl(\frac{n^{n}V(K)^{n}}{\widetilde{G}_{-\phi _{1}}(K)^{n}\phi_{1}(V(K)^{-1})} \biggr). \end{aligned}$$

According to the inverse $\phi_{i}^{-1}$ ($i=1,2$) of $\phi_{i}$ ($i=1,2$) being strictly increasing and having continuity on $(0,\infty)$, it follows that

$$\phi_{2}^{-1} \biggl(\frac{n^{n}V(K)^{n}}{\widetilde{G}_{-\phi _{2}}(K)^{n}\phi_{2}(V(K)^{-1})} \biggr)\leq \phi_{1}^{-1} \biggl(\frac {n^{n}V(K)^{n}}{\widetilde{G}_{-\phi_{1}}(K)^{n}\phi _{1}(V(K)^{-1})} \biggr). $$

This completes the proof. □