The L
           p -dual mixed geominimal surface area for multiple star bodies

Yanan Li; Weidong Wang

doi:10.1186/1029-242X-2014-456

Research
Open Access

The $L_{p}$ -dual mixed geominimal surface area for multiple star bodies

Yanan Li¹ and
Weidong Wang¹Email author

Journal of Inequalities and Applications20142014:456

https://doi.org/10.1186/1029-242X-2014-456

Received: 27 August 2014
Accepted: 4 November 2014
Published: 13 November 2014

Abstract

According to the notion of the $L_{p}$ -mixed geominimal surface area of multiple convex bodies which were introduced by Ye et al., we define the concept of the $L_{p}$ -dual mixed geominimal surface area for multiple star bodies, and we establish several inequalities related to this concept.

MSC:52A20, 52A40.

Keywords

$L_{p}$ -mixed geominimal surface area
multiple convex bodies
$L_{p}$ -dual mixed geominimal surface area
multiple star bodies

1 Introduction

Let $K^{n}$ denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space $R^{n}$ . For the set of convex bodies containing the origin in their interiors and the set of convex bodies whose centroids lie at the origin in $R^{n}$ , we write $K_{o}^{n}$ and $K_{c}^{n}$ , respectively. $S_{o}^{n}$ and $S_{c}^{n}$ , respectively, denote the set of star bodies (about the origin) and the set of star bodies whose centroids lie at the origin in $R^{n}$ . Let $F_{o}^{n}$ denote the set of $K_{o}^{n}$ that have a positive continuous curvate function. Let $S^{n - 1}$ denote the unit sphere in $R^{n}$ and $V (K)$ the n-dimensional volume of the body K. For the standard unit ball B in $R^{n}$ , its volume is written by $ω_{n} = V (B)$ .

The notion of

L_{p}

-geominimal surface area was given by Lutwak in [1]. For

K \in K_{o}^{n}

, and

p \geq 1

, the

L_{p}

-geominimal surface area,

G_{p} (K)

, of K is defined by

ω_{n}^{\frac{p}{n}} G_{p} (K) = inf {n V_{p} (K, L) V {(L^{*})}^{\frac{p}{n}} : L \in K_{o}^{n}} .

Here

V_{p} (K, L)

denotes

L_{p}

-mixed volume of

K, L \in K_{o}^{n}

(see [1, 2]) and

L^{*}

denotes the polar of L. For the case

p = 1

,

G_{p} (K)

is just the classical geominimal surface area which was introduced by Petty [3]. Some affine isoperimetric inequalities related to the classical and

L_{p}

-geominimal surface areas can be found in [3–10]. Recently, the

L_{p}

-geominimal surface area was successfully extended to any real p (

p \neq - n

) by Ye in [11]. Especially, Ye et al. [12] studied the

L_{p}

-mixed geominimal surface area for multiple convex bodies. For

p > 0

, they defined the

L_{p}

-mixed geominimal surface areas for

K_{1}, \dots, K_{n} \in F_{o}^{n}

as

\begin{aligned} G_{p}^{(1)} (K_{1}, \dots, K_{n}) = inf_{L \in K_{o}^{n}} {n V_{p} {(K_{1}, \dots, K_{n}; L, \dots, L)}^{\frac{n}{n + p}} V {(L^{*})}^{\frac{p}{n + p}}}; \\ G_{p}^{(2)} (K_{1}, \dots, K_{n}) = inf_{L_{i} \in K_{o}^{n}} {n V_{p} {(K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}^{\frac{n}{n + p}} \prod_{i = 1}^{n} V {(L_{i}^{*})}^{\frac{p}{(n + p) n}}} . \end{aligned}

Here $Q^{*}$ denotes the polar body of Q, and $V_{p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})$ denotes a type of $L_{p}$ -mixed volume of $K_{1}, \dots, K_{n} \in F_{o}^{n}$ , $L_{1}, \dots, L_{n} \in K_{o}^{n}$ (see [12]).

Wang and Qi in [13] introduced the

L_{p}

-dual geominimal surface area as follows: For

K \in S_{o}^{n}

, and

p \geq 1

, the

L_{p}

-dual geominimal surface area,

{\tilde{G}}_{- p} (K)

, of K is defined by

ω_{n}^{- \frac{p}{n}} {\tilde{G}}_{- p} (K) = inf {n {\tilde{V}}_{- p} (K, L) V {(L^{*})}^{- \frac{p}{n}} : L \in K_{c}^{n}} .

(1.1)

Here ${\tilde{V}}_{- p} (K, L)$ denotes the $L_{p}$ -dual mixed volume of $K, L \in S_{o}^{n}$ (see Section 2).

Note that we extend L from an origin-symmetric convex body to $L \in K_{c}^{n}$ in definition (1.1). Actually, we can prove that the results of [13] all are correct under this extension.

In this paper, we first define the $L_{p}$ -dual mixed geominimal surface area for multiple star bodies with the same idea in mind as [12].

Definition 1.1 For

K_{1}, \dots, K_{n} \in S_{o}^{n}

,

p \geq 1

, the

L_{p}

-dual mixed geominimal surface areas,

{\tilde{G}}_{- p}^{(j)} (K_{1}, \dots, K_{n})

(

j = 1, 2

), of

K_{1}, \dots, K_{n}

, are defined by

ω_{n}^{- \frac{p}{n}} {\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n}) = inf_{L \in K_{c}^{n}} {n {\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L, \dots, L) V {(L^{*})}^{- \frac{p}{n}}};

(1.2)

ω_{n}^{- \frac{p}{n}} {\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) = inf_{L_{i} \in K_{c}^{n}} {n {\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n}) \prod_{i = 1}^{n} V {(L_{i}^{*})}^{- \frac{p}{n^{2}}}} .

(1.3)

Here ${\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})$ denotes a type of $L_{p}$ -dual mixed volume of the star bodies $K_{1}, \dots, K_{n}$ , $L_{1}, \dots, L_{n}$ (see (2.5)).

Comparing the definitions (1.2) and (1.3), we easily obtain

{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) \leq {\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n}) .

(1.4)

When

K_{1} = \dots = K_{n} = K

in (1.2), then

{\tilde{G}}_{- p}^{(1)} (K, \dots, K) = {\tilde{G}}_{- p} (K) .

(1.5)

Further, we establish some inequalities for the $L_{p}$ -dual mixed geominimal surface area. Our results can be stated as follows.

Theorem 1.1 If

K_{1}, \dots, K_{n} \in S_{o}^{n}

,

p \geq 1

,

1 \leq m \leq n

, then

(1.6)

(1.7)

Equality holds in inequality (1.6) if and only if

K_{i}

(

i = n - m + 1, \dots, n

) all are dilates of each other. Equality holds in inequality (1.7) if and only if there exist constants

c_{1}, c_{2}, \dots, c_{m}

(not all zero) such that, for all

u \in S^{n - 1}

,

c_{1} ρ_{K_{n}}^{n + p} (u) ρ_{L_{n}}^{- p} (u) = c_{2} ρ_{K_{n - 1}}^{n + p} (u) ρ_{L_{n - 1}}^{- p} (u) = \dots = c_{m} ρ_{K_{n - m + 1}}^{n + p} (u) ρ_{L_{n - m + 1}}^{- p} (u) .

In particular, if $m = n$ , then we have the following.

Corollary 1.1 If

K_{1}, \dots, K_{n} \in S_{o}^{n}

,

p \geq 1

, then

{[{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n})]}^{n} \leq {[{\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n})]}^{n} \leq {\tilde{G}}_{- p} (K_{1}) \dots {\tilde{G}}_{- p} (K_{n}) .

(1.8)

Equality holds in the second inequality of (1.8) if and only if $K_{i}$ ( $i = 1, 2, \dots, n$ ) all are dilates of each other.

Using Corollary 1.1, we may get the following Blaschke-Santalö type inequality.

Corollary 1.2 If

K_{1}, \dots, K_{n} \in K_{c}^{n}

,

n \geq p \geq 1

, then

{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) {\tilde{G}}_{- p}^{(2)} (K_{1}^{*}, \dots, K_{n}^{*}) \leq {\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n}) {\tilde{G}}_{- p}^{(1)} (K_{1}^{*}, \dots, K_{n}^{*}) \leq n^{2} ω_{n}^{2} .

(1.9)

Equality holds in the second inequality of (1.9) if and only if $K_{i}$ ( $i = 1, 2, \dots, n$ ) all are balls centered at the origin.

Theorem 1.2 If

K_{1}, \dots, K_{n} \in K_{c}^{n}

,

p \geq 1

, then

{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) \leq n ω_{n}^{\frac{2 n + p}{n}} \prod_{i = 1}^{n} V {(K_{i}^{*})}^{\frac{- (n + p)}{n^{2}}} .

Theorem 1.3 If

K_{1}, \dots, K_{n} \in S_{o}^{n}

,

1 \leq p < q

, then

{(\frac{{\tilde{G}}_{- p}^{(1)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + p}})}^{\frac{1}{p}} \leq {(\frac{{\tilde{G}}_{- q}^{(1)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + q}})}^{\frac{1}{q}};

(1.10)

{(\frac{{\tilde{G}}_{- p}^{(2)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + p}})}^{\frac{1}{p}} \leq {(\frac{{\tilde{G}}_{- q}^{(2)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + q}})}^{\frac{1}{q}} .

(1.11)

Equality holds in (1.10) and (1.11) if and only if each $K_{i} \in K_{c}^{n}$ ( $i = 1, 2, \dots, n$ ).

2 Notations and background materials

2.1 Radial function and polar set

If K is a compact star-shaped (with respect to the origin) in

R^{n}

, then its radial function,

ρ_{K} = ρ (K, \cdot) : R^{n} ∖ {0} \to [0, \infty)

, is defined by (see [6, 14])

ρ (K, u) = max {λ \geq 0 : λ u \in K}, u \in S^{n - 1} .

If $ρ_{K}$ is positive and continuous, K will be called a star body (with respect to the origin). Two star bodies K and L are said to be dilates (of one another) if $ρ_{K} (u) / ρ_{L} (u)$ is independent of $u \in S^{n - 1}$ .

If E is a nonempty subset in

R^{n}

, the polar set,

E^{*}

, of E is defined by (see [6, 14])

E^{*} = {x \in R^{n} : x \cdot y \leq 1, y \in E} .

For

K \in K_{o}^{n}

and its polar body, the well-known Blaschke-Stantalö inequality can be stated (see [14]): If

K \in K_{c}^{n}

, then

V (K) V (K^{*}) \leq ω_{n}^{2},

(2.1)

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

2.2 Dual mixed volume

The dual mixed volume of star bodies was introduced by Lutwak (see [15]). For

K_{1}, \dots, K_{n} \in S_{o}^{n}

, the dual mixed volume,

\tilde{V} (K_{1}, \dots, K_{n})

, of

K_{1}, \dots, K_{n}

is given by

\tilde{V} (K_{1}, K_{2}, \dots, K_{n}) = \frac{1}{n} \int_{S^{n - 1}} ρ_{K_{1}} (u) ρ_{K_{2}} (u) \dots ρ_{K_{n}} (u) d u .

(2.2)

The classical Alexander-Fenchel inequality for the dual mixed volume (see [6, 14]) asserts that the integer m satisfies

1 \leq m \leq n

such that

with equality if and only if $K_{n - m + 1}, \dots, K_{n}$ are all dilations of each other.

In particular, if

m = n

, one has the Minkowski inequality

\tilde{V} {(K_{1}, K_{2}, \dots, K_{n})}^{n} \leq V (K_{1}) V (K_{2}) \dots V (K_{n}),

(2.3)

with equality if and only if $K_{1}, \dots, K_{n}$ are all dilations of each other.

2.3 $L_{p}$ -Dual mixed volume

Lutwak in [1] introduced the

L_{p}

-dual mixed volume. For

K, L \in S_{o}^{n}

,

p \geq 1

, the

L_{p}

-dual mixed volume,

{\tilde{V}}_{- p} (K, L)

, of K and L is defined by

{\tilde{V}}_{- p} (K, L) = \frac{1}{n} \int_{S^{n - 1}} ρ_{K}^{n + p} (u) ρ_{L}^{- p} (u) d u .

(2.4)

Associated with (2.4), for all

K_{1}, \dots, K_{n} \in S_{o}^{n}

,

L_{1}, \dots, L_{n} \in S_{o}^{n}

, and

p \geq 1

, we define

{\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n}) = \frac{1}{n} \int_{S^{n - 1}} \prod_{i = 1}^{n} {[ρ_{K_{i}}^{n + p} (u) ρ_{L_{i}}^{- p} (u)]}^{\frac{1}{n}} d u .

(2.5)

From (2.2) and (2.5), we easily get

{\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; K_{1}, \dots, K_{n}) = \tilde{V} (K_{1}, \dots, K_{n}) .

(2.6)

If

K_{1} = \dots = K_{n} = K

and

L_{1} = \dots = L_{n} = L

in (2.5), then (2.4) and (2.5) yield

{\tilde{V}}_{- p} (K, \dots, K; L, \dots, L) = {\tilde{V}}_{- p} (K, L) .

3 Results and proofs

In this section, we will prove Theorems 1.1-1.3 and Corollaries 1.1-1.2.

Proof of Theorem 1.1 We first prove inequality (1.7) is true.

Let

ρ_{0}, ρ_{1}, \dots, ρ_{m}

be nonnegative bounded Borel functions on

S^{n - 1}

. By the Hölder inequality (see [16]), we have (see [14])

{(\frac{1}{n} \int_{S^{n - 1}} ρ_{0} (u) ρ_{1} (u) \dots ρ_{m} (u) d u)}^{m} \leq \prod_{i = 0}^{m - 1} (\frac{1}{n} \int_{S^{n - 1}} ρ_{0} (u) {[ρ_{i + 1} (u)]}^{m} d u),

(3.1)

with equality if and only if there exist constants $b_{1}, \dots, b_{m} \geq 0$ (not all zero) such that $b_{1} ρ_{1}^{m} (u) = \dots = b_{m} ρ_{m}^{m} (u)$ for all $u \in S^{n - 1}$ .

For

i = 0, \dots, m - 1

, we let

\begin{matrix} ρ_{0} (u) = {[ρ_{K_{1}}^{n + p} (u) ρ_{L_{1}}^{- p} (u) \dots ρ_{K_{n - m}}^{n + p} (u) ρ_{L_{n - m}}^{- p} (u)]}^{\frac{1}{n}}, \\ ρ_{i + 1} (u) = {[ρ_{K_{n - i}}^{n + p} (u) ρ_{L_{n - i}}^{- p} (u)]}^{\frac{1}{n}} . \end{matrix}

In association with (2.5), we get

(3.2)

Combining with (1.3), (3.2), we get

i.e.,

This gives (1.7).

According to the equality condition of inequality (3.1), we see that equality holds in inequality (1.7) if and only if

c_{1} ρ_{K_{n}}^{n + p} (u) ρ_{L_{n}}^{- p} (u) = c_{2} ρ_{K_{n - 1}}^{n + p} (u) ρ_{L_{n - 1}}^{- p} (u) = \dots = c_{m} ρ_{K_{n - m + 1}}^{n + p} (u) ρ_{L_{n - m + 1}}^{- p} (u)

for all $u \in S^{n - 1}$ , where $c_{i} = b_{i}^{n / m}$ ( $i = 1, 2, \dots, m$ ).

Now we complete the proof of (1.6). For

i = 0, \dots, m - 1

, we let

\begin{matrix} ρ_{0} (u) = {[ρ_{K_{1}}^{n + p} (u) ρ_{L}^{- p} (u) \dots ρ_{K_{n - m}}^{n + p} (u) ρ_{L}^{- p} (u)]}^{\frac{1}{n}}, \\ ρ_{i + 1} (u) = {[ρ_{K_{n - i}}^{n + p} (u) ρ_{L}^{- p} (u)]}^{\frac{1}{n}} . \end{matrix}

In association with (2.5) and (3.1), we get

(3.3)

Similar to the proof of (1.7), combining with (1.2) and (3.3), we obtain

According to the equality condition of inequality (3.1), we see that equality holds in inequality (1.6) if and only if

c_{1} ρ_{K_{n}}^{n + p} (u) ρ_{L}^{- p} (u) = c_{2} ρ_{K_{n - 1}}^{n + p} (u) ρ_{L}^{- p} (u) = \dots = c_{m} ρ_{K_{n - m + 1}}^{n + p} (u) ρ_{L}^{- p} (u)

for all

u \in S^{n - 1}

, where

c_{i} = b_{i}^{n / m}

(

i = 1, 2, \dots, m

). This means that

c_{1} ρ_{K_{n}}^{n + p} (u) = c_{2} ρ_{K_{n - 1}}^{n + p} (u) = \dots = c_{m} ρ_{K_{n - m + 1}}^{n + p} (u)

for all $u \in S^{n - 1}$ , i.e., $K_{i}$ ( $i = n - m + 1, \dots, n$ ) all are dilates of each other. □

Proof of Corollary 1.1 Let

m = n

in (1.6), and together with (1.4) and (1.5), we easily obtain

\begin{aligned} {[{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n})]}^{n} & \leq {[{\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n})]}^{n} \\ \leq \prod_{i = 0}^{n - 1} {\tilde{G}}_{- p}^{(1)} (K_{n - i}, \dots, K_{n - i}) \\ \leq {\tilde{G}}_{- p} (K_{1}) {\tilde{G}}_{- p} (K_{2}) \dots {\tilde{G}}_{- p} (K_{n}) . \end{aligned}

This gives (1.8).

From the equality condition of (1.6), we easily find that equality holds in the second inequality of (1.8) if and only if there exist constants $c_{1}, c_{2}, \dots, c_{n}$ (not all zero) such that, for all $u \in S^{n - 1}$ , $c_{1} ρ_{K_{n}}^{n + p} (u) ρ_{L}^{- p} (u) = c_{2} ρ_{K_{n - 1}}^{n + p} (u) ρ_{L}^{- p} (u) = \dots = c_{n} ρ_{K_{1}}^{n + p} (u) ρ_{L}^{- p} (u)$ . This means all $K_{i}$ ( $i = 1, 2, \dots, n$ ) are dilates of each other. □

In order to prove Corollary 1.2, we give the following lemma.

Lemma 3.1 ([13])

If

K \in K_{c}^{n}

,

n \geq p \geq 1

, then

{\tilde{G}}_{- p} (K) {\tilde{G}}_{- p} (K^{*}) \leq n^{2} ω_{n}^{2},

(3.4)

with equality if and only if K is a ball centered at the origin.

Proof of Corollary 1.2 Corollary 1.1, for K and

K^{*}

, immediately yields

{[{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n})]}^{n} \leq {[{\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n})]}^{n} \leq {\tilde{G}}_{- p} (K_{1}) \dots {\tilde{G}}_{- p} (K_{n}),

(3.5)

{[{\tilde{G}}_{- p}^{(2)} (K_{1}^{*}, \dots, K_{n}^{*})]}^{n} \leq {[{\tilde{G}}_{- p}^{(1)} (K_{1}^{*}, \dots, K_{n}^{*})]}^{n} \leq {\tilde{G}}_{- p} (K_{1}^{*}) \dots {\tilde{G}}_{- p} (K_{n}^{*}) .

(3.6)

Combining with (3.5), (3.6), and (3.4), we obtain

\begin{matrix} {[{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n})]}^{n} {[{\tilde{G}}_{- p}^{(2)} (K_{1}^{*}, \dots, K_{n}^{*})]}^{n} \\ \leq {[{\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n})]}^{n} {[{\tilde{G}}_{- p}^{(1)} (K_{1}^{*}, \dots, K_{n}^{*})]}^{n} \\ \leq {\tilde{G}}_{- p} (K_{1}) {\tilde{G}}_{- p} (K_{1}^{*}) \dots {\tilde{G}}_{- p} (K_{n}) {\tilde{G}}_{- p} (K_{n}^{*}) \leq {[n^{2} ω_{n}^{2}]}^{n}, \end{matrix}

i.e.,

{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) {\tilde{G}}_{- p}^{(2)} (K_{1}^{*}, \dots, K_{n}^{*}) \leq {\tilde{G}}_{- p}^{(1)} (K_{1}, \dots, K_{n}) {\tilde{G}}_{- p}^{(1)} (K_{1}^{*}, \dots, K_{n}^{*}) \leq n^{2} ω_{n}^{2} .

This yields (1.9).

By the equality conditions of inequality (3.4) and the second inequality of (1.8), we know that equality holds in the second inequality of (1.9) if and only if $K_{1}, \dots, K_{n}$ all are balls centered at the origin. □

Proof of Theorem 1.2 From (1.3), it follows that, for any

L_{i} \in K_{c}^{n}

,

{\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) \leq n ω_{n}^{\frac{p}{n}} {\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n}) \prod_{i = 1}^{n} V {(L_{i}^{*})}^{\frac{- p}{n^{2}}} .

Since

K_{1}, \dots, K_{n} \in K_{c}^{n}

, taking

K_{1}, \dots, K_{n}

for

L_{1}, \dots, L_{n}

, and using (2.6), (2.3), and (2.1), we get

\begin{aligned} {\tilde{G}}_{- p}^{(2)} (K_{1}, \dots, K_{n}) & \leq n ω_{n}^{\frac{p}{n}} {\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; K_{1}, \dots, K_{n}) \prod_{i = 1}^{n} V {(K_{i}^{*})}^{- \frac{p}{n^{2}}} \\ = n ω_{n}^{\frac{p}{n}} \tilde{V} (K_{1}, \dots, K_{n}) \prod_{i = 1}^{n} V {(K_{i}^{*})}^{- \frac{p}{n^{2}}} \\ \leq n ω_{n}^{\frac{p}{n}} {[V (K_{1}) \dots V (K_{n})]}^{\frac{1}{n}} \prod_{i = 1}^{n} V {(K_{i}^{*})}^{- \frac{p}{n^{2}}} \\ = n ω_{n}^{\frac{p}{n}} {[V (K_{1}) \dots V (K_{n}) V (K_{1}^{*}) \dots V (K_{n}^{*})]}^{\frac{1}{n}} \prod_{i = 1}^{n} V {(K_{i}^{*})}^{- \frac{(n + p)}{n^{2}}} \\ \leq n ω_{n}^{\frac{2 n + p}{n}} \prod_{i = 1}^{n} V {(K_{i}^{*})}^{- \frac{(n + p)}{n^{2}}} . \end{aligned}

This gives the proof of Theorem 1.2. □

Proof of Theorem 1.3 Using the Hölder inequality, (2.5), and (2.2), we get

\begin{aligned} {\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n}) & = \frac{1}{n} \int_{S^{n - 1}} \prod_{i = 1}^{n} {[ρ_{K_{i}}^{n + p} (u) ρ_{L_{i}}^{- p} (u)]}^{\frac{1}{n}} d u \\ = \frac{1}{n} \int_{S^{n - 1}} {(\prod_{i = 1}^{n} {[ρ_{K_{i}}^{n + q} (u) ρ_{L_{i}}^{- q} (u)]}^{\frac{1}{n}})}^{\frac{p}{q}} {(\prod_{i = 1}^{n} {[ρ_{K_{i}}^{n} (u)]}^{\frac{1}{n}})}^{\frac{q - p}{q}} d u \\ \leq {\tilde{V}}_{- p} {(K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}^{\frac{p}{q}} \tilde{V} {(K_{1}, \dots, K_{n})}^{\frac{q - p}{q}}, \end{aligned}

that is,

{(\frac{{\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{1}{p}} \leq {(\frac{{\tilde{V}}_{- q} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{1}{q}} .

(3.7)

According to the equality condition in the Hölder inequality, we know that equality holds in (3.7) if and only if there exist constants $c_{i} > 0$ ( $i = 1, 2, \dots, n$ ) such that $ρ (K_{i}, u) = c_{i} ρ (L_{i}, u)$ for any $u \in S^{n - 1}$ , i.e., for each $i = 1, 2, \dots, n$ , $K_{i}$ and $L_{i}$ both are dilates.

From definition (1.3) and inequality (3.7), we have

\begin{matrix} ω_{n}^{- 1} {(\frac{{\tilde{G}}_{- p}^{(2)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + p}})}^{\frac{1}{p}} \\ = inf_{L_{i} \in K_{c}^{n}} {{(\frac{{\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{n}{p}} \frac{1}{\tilde{V} (K_{1}, \dots, K_{n})} {(\prod_{i = 1}^{n} V {(L_{i}^{*})}^{- \frac{p}{n^{2}}})}^{\frac{n}{p}}} \\ = inf_{L_{i} \in K_{c}^{n}} {{(\frac{{\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{n}{p}} \frac{1}{\tilde{V} (K_{1}, \dots, K_{n})} \prod_{i = 1}^{n} V {(L_{i}^{*})}^{- \frac{1}{n}}} \\ \leq inf_{L_{i} \in K_{c}^{n}} {{(\frac{{\tilde{V}}_{- q} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{n}{q}} \frac{1}{\tilde{V} (K_{1}, \dots, K_{n})} \prod_{i = 1}^{n} V {(L_{i}^{*})}^{- \frac{1}{n}}} \\ = inf_{L_{i} \in K_{c}^{n}} {{(\frac{{\tilde{V}}_{- q} (K_{1}, \dots, K_{n}; L_{1}, \dots, L_{n})}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{n}{q}} \frac{1}{\tilde{V} (K_{1}, \dots, K_{n})} {(\prod_{i = 1}^{n} V {(L_{i})}^{- \frac{q}{n^{2}}})}^{\frac{n}{q}}} \\ = ω_{n}^{- 1} {(\frac{{\tilde{G}}_{- q}^{(2)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + q}})}^{\frac{1}{q}}, \end{matrix}

i.e.,

{(\frac{{\tilde{G}}_{- p}^{(2)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + p}})}^{\frac{1}{p}} \leq {(\frac{{\tilde{G}}_{- q}^{(2)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + q}})}^{\frac{1}{q}} .

This is just (1.11). Because of each $L_{i} \in K_{c}^{n}$ in inequality (3.7), together with the equality condition of (3.7), we see that equality holds in (1.11) if and only if each $K_{i} \in K_{c}^{n}$ .

In order to prove (1.10), (3.7) can be written

{(\frac{{\tilde{V}}_{- p} (K_{1}, \dots, K_{n}; L, \dots, L)}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{1}{p}} \leq {(\frac{{\tilde{V}}_{- q} (K_{1}, \dots, K_{n}; L, \dots, L)}{\tilde{V} (K_{1}, \dots, K_{n})})}^{\frac{1}{q}} .

(3.8)

In the same way as (1.11), from definition (1.2) and (3.8), we get

{(\frac{{\tilde{G}}_{- p}^{(1)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + p}})}^{\frac{1}{p}} \leq {(\frac{{\tilde{G}}_{- q}^{(1)} {(K_{1}, \dots, K_{n})}^{n}}{n^{n} \tilde{V} {(K_{1}, \dots, K_{n})}^{n + q}})}^{\frac{1}{q}} .

From the equality condition in (1.11), we see that equality holds in (1.10) if and only if each $K_{i} \in K_{c}^{n}$ . □

Declarations

Acknowledgements

The authors would like to sincerely thank the referees for very valuable and helpful comments and suggestions which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No. 2014PY067).

Authors’ Affiliations

(1)

Department of Mathematics, China Three Gorges University, Yichang, 443002, P.R. China

References

Lutwak E: The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas. Adv. Math. 1996, 118: 244-294. 10.1006/aima.1996.0022MathSciNetView ArticleMATHGoogle Scholar
Lutwak E: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 1993, 38: 131-150.MathSciNetMATHGoogle Scholar
Petty CM: Geominimal surface area. Geom. Dedic. 1974, 3: 77-97.MathSciNetView ArticleMATHGoogle Scholar
Petty CM: Affine isoperimetric problems. Annals of the New York Academy of Sciences 440. Discrete Geometry and Convexity 1985, 113-127.Google Scholar
Schneider R: Affine surface area and convex bodies of elliptic type. Period. Math. Hung. 2014. 10.1007/s10998-014-0050-3Google Scholar
Schneider R: Convex Bodies: The Brunn-Minkowski Theory. 2nd edition. Cambridge University Press, Cambridge; 2014.MATHGoogle Scholar
Wang WD, Feng YB:A general $L_{p}$ -version of Petty’s affine projection inequality. Taiwan. J. Math. 2013,17(2):517-528.View ArticleMATHGoogle Scholar
Zhu BC, Li N, Zhou JZ:Isoperimetric inequalities for $L_{p}$ geominimal surface area. Glasg. Math. J. 2011, 53: 717-726. 10.1017/S0017089511000292MathSciNetView ArticleMATHGoogle Scholar
Zhu BC, Zhou JZ, Xu WX: $L_{p}$ Mixed geominimal surface area. J. Math. Anal. Appl. 2015, 422: 1247-1263. 10.1016/j.jmaa.2014.09.035MathSciNetView ArticleMATHGoogle Scholar
Zhu, BC, Zhou, JZ, Xu, WX: Affine isoperimetric inequalities for L p geominimal surface area. Springer Proceedings in Mathematics and Statistics (in press)Google Scholar
Ye, DP: On the L_p-geominimal surface area and related inequalities (submitted). arXiv: 1308.4196
Ye, DP, Zhu, BC, Zhou, JZ: The mixed L_p-geominimal surface areas for multiple convex bodies. arXiv: 1311.5180v1
Wang WD, Qi C: $L_{p}$ -Dual geominimal surface area. J. Inequal. Appl. 2011., 2011: Article ID 6Google Scholar
Gardner RJ: Geometric Tomography. 2nd edition. Cambridge University Press, Cambridge; 2006.View ArticleMATHGoogle Scholar
Lutwak E: Dual mixed volumes. Pac. J. Math. 1975, 58: 531-538. 10.2140/pjm.1975.58.531MathSciNetView ArticleMATHGoogle Scholar
Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1959.MATHGoogle Scholar

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

The L p -dual mixed geominimal surface area for multiple star bodies

2.1 Radial function and polar set

2.2 Dual mixed volume

2.3 L p -Dual mixed volume

Acknowledgements

The $L_{p}$ -dual mixed geominimal surface area for multiple star bodies

2.3 $L_{p}$ -Dual mixed volume