2015, Vol.20 No.4, 277-282 Article ID 1007-1202(2015)04-0277-06 DOI 10.1007/s11859-015-1093-x

Some Inequalities for Lp-Dual Mixed Quermassintegrals □ ZHANG Ping, WANG Weidong†

0

Introduction

Department of Mathematics, China Three Gorges University, Yichang 443002, Hubei, China © Wuhan University and Springer-Verlag Berlin Heidelberg 2015

Abstract: Wang and Zhang defined a type of Lp-dual mixed quermassintegrals based on the Lp-radial combinations and dual quermassintegrals of star bodies. In the article, the product inequalities for this Lp-dual mixed quermassintegrals are established. As the applications, we obtain the lower bounds of dual quermassintegrals product. Further, the Brunn-Minkowski type inequality and the cycle inequality for the Lp-dual mixed quermassintegrals are given. Key words: dual quermassintegrals; Lp-radial combination; Lp-dual mixed quermassintegrals; product inequality; Brunn-Minkowski type inequality CLC number: O 178; O 18

Received date: 2015-02-28 Foundation item: Supported by the National Natural Science Foundation of China (11371224) Biography: ZHANG Ping, female, Lecturer, research direction: convex geometric analysis. E-mail: [email protected] † To whom correspondence should be addressed. E-mail: [email protected]

Let K n denote the set of convex bodies (compact, convex subsets with non-empty interiors) in Euclidean space R n . For the set of convex bodies containing the origin in their interiors in R n , we write K on . Let Son denote the set of star bodies (about the origin) in R n . Denote by V ( K ) the n-dimensional volume of body K , for the standard unit ball B in R n , write ωn = V ( B ) .

The Lp-Brunn-Minkowski theory is established by Lutwak. In Ref.[1] Lutwak showed that the Fiery Lp-combination of convex bodies led to the Brunn-Minkowski theory, and he introduced the notions of Lp-mixed surface area measure and Lp-mixed quermassintegrals (Lp-surface area measure and Lp-mixed volume are their special case, respectively), and established some inequalities for the Lp-mixed quermassintegrals. The Lp-harmonic radial combinations were instigated from convex bodies to star bodies by Firey (see Refs.[2-4]) and Lutwak (see Ref.[5]), respectively. Based on the Lp-harmonic radial combinations of star bodies, Lutwak [5] showed the notion of Lp-dual mixed volume. In 2005, Wang and Leng [6] put forward the notion of Lp-dual mixed quermassintegrals by the Lp-harmonic radial combination and dual quermassintegrals, they extended the Lp-dual mixed volume to the Lp-dual mixed quermassintegrals. Recently, based on the Lp-radial combinations and dual quermassintegrals of star bodies, Wang and Zhang in Ref.[7] (also see Ref.[8]) defined a type of Lp-dual mixed quermassintegrals. In this article, we establish some inequalities for this Lp-dual mixed quermassinte-

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Wuhan University Journal of Natural Sciences 2015, Vol.20 No.4

grals. First, we give the following product inequality. Theorem 1 If K , L ∈ Son ,∨ p 0 and real i ≠ n , ∨ then for i n − p ,  p ,i ( K , L)W  p ,i ( K * , L ) ≥ W  p , i ( B, L ) 2 W (1) with equality if and only if K is a ball centered at the  p ,i ( K , L ) origin; for i = n − p , (1) is identic. Here W denotes the Lp-dual mixed quermassintegrals of K and L , and K * denotes the polar of K . Let L = B in Theorem 1, we obtain a sharp lower bound of the dual quermassintegrals product as follows: Corollary 1 If K ∈ Son and i is any real, then for∨i n ,  i ( K )W  i (K * ) ≥ ω 2 W n

n

with equality if and only if K is a ball centered at the origin. Comparing inequality (2) with (3), we know that inequality (2) extend inequality (3) from K ∈ K on to K ∈ S n and i = n + p ( p ≥ 1) ≥ n + 1 to∨i n . o

Next, for Q ∈ K on , let (see Ref.[10]) r r ( x) = max (4) x ∈ Q R R( x) where r ( x) is the radius of the largest ball contained in Q and centered at x and R( x) is the radius of the smallest ball containing Q and centered at x . Then the following product inequality of the Lp-dual mixed quermassintegrals can be established. Theorem 2 For K , L ∈ Son ,∨ p 0, i is any real.

If r and R satisfy (4) in L* (notice L* ∈ K on ), then for i ≥ n − p , p

 p ,i ( K , L)W  p ,i ( K * , L* ) ≥  r  ω 2 W (5)   n R Taking K = L in Theorem 2, we may obtain another lower bound of the dual quermassintegrals product p 0) as follows: of any index of i ≥ n − p (∨ Corollary 2 For K ∈ S n ,∨ p 0, i is any real. If o

r and R satisfy (4) in K , then for i ≥ n − p , *

 i ( K )W  i (K * ) ≥  r  ω 2 W (6)   n R In particular, let i = 0 in Corollary 2, we have Corollary 3.

For K ∈ Son ,∨ p

0. If r and R

*

satisfy (4) in K , then for n ≤ p , p

r V ( K )V ( K * ) ≥   ωn2 R If Q is a centered convex body, let rin and Rout denote the inradius and outradius of Q , respectively.

Then r / R = rin / Rout . Hence, as a special case of Corollary 3, we obtain the following result. Corollary 4 For each centered star body K containing the origin in their interiors and ∨ p 0 . If n ≤ p , then p

(2)

with equality if and only if K is a ball centered at the origin; For i = n , (2) is identic. Remark 1 Recently, Wei and Wang in Ref.[9] gave a sharp lower bound of the dual quermassintegrals product: If K ∈ K on , p ≥ 1, then  n + p ( K )W  n+ p ( K * ) ≥ ω 2 W (3)

p

Corollary 3

where rin

 r  (7) V ( K )V ( K * ) ≥  in  ωn2  Rout  and Rout are the inradius and outradius of

K * , respectively. Remark 2 Recall that the Bourgain-Milman inequality can be stated as follows [11]: For each centered convex body K , there exits an absolute constant ∨ c 0 (independent of the dimension n ) such that V ( K )V ( K * ) ≥ c nωn2 (8)

Inequality (7) is interesting. It may be regarded as an analogy form of the Bourgain-Milman inequality (8). Further, based on the Lp -radial combinations of star bodies, we obtain the Brunn-Minkowski type inequality for the Lp -dual mixed quermassintegrals as follows: p 0, λ , μ ≥ 0 (not Theorem 3 Let K , L ∈ Son ,∨  p μ ⋅ L denote the Lp-radial combinaboth zero), λ ⋅ K + tion of K and L , and real i ≠ n . If i ≤ n − 2 p , then for any Q ∈ Son , p

 p , i (λ ⋅ K +  p μ ⋅ L, Q) n− p−i W p

p

 p ,i ( K , Q) n− p−i + μW  p , i ( L, Q ) n − p −i ≤ λW (9) with equality if and only if K and L are dilates. If n − 2 p ≤∧i n − p , or∨i n − p, then inequality (9) is reversed. Besides, we also establish the following cycle inequality for the Lp-dual mixed quermassintegrals. Theorem 4 If K , L ∈ Son ,∨ p 0, real i, j , k ≠ n ∧ ∧ and i j k , then  p ,i ( K,L) k − j W  p , k ( K , L) j −i ≥W  p , j ( K , L) k −i W (10)

with equality if and only if K is a ball centered at the origin. In this article, we will prove Theorems 1,2 in Section 2. The proofs of Theorems 3,4 be completed in Section 3. Actually, we give more general result than Theorem 3 in Section 3.

279

ZHANG Ping et al : Some Inequalities for Lp-Dual Mixed Quermassintegrals

1

Preliminaries

Remark 3 Let i = 0 in definition (16) and notice

that

1.1 Support Function, Radial Function and Polar Body If K ∈ K n , then its support function, hK = h( K , ⋅) :

 0 (K ) = 1 W ρ ( K , u )n dS (u ) = V ( K ) n −1 n S then

 p ε ⋅ L) − V ( K ) V (K + n W p ,0 ( K , L) = lim+ ε →0 p ε n = V p ( K , L) p

R n → (−∞, + ∞), is defined by (see Refs.[12,13])

h( K , x) = max{x ⋅ y : y ∈ K }, x ∈ R n (11) where x ⋅ y denotes the standard inner product of x and y . If K is a compact star-shaped (about the origin) in n R , its radial function, ρ K = ρ ( K , ⋅) : R n \ {0} → [0, + ∞) , is defined by (see Refs. [12,13])

ρ ( K , x) = max{λ ≥ 0 : λ x ∈ K }, x ∈ R n \ {0} (12) If ρ K is positive and continuous, K will be called a star body (about 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 ∈ S n −1 . From definitions (11) and (12), it is obvious that for K ∈ Kn, ρ ( K , ⋅) ≤ h( K , ⋅) (13) If E is a nonempty set in R n , E * (the polar set of E ) is a convex set containing the origin in their interior whose definition is given by (see Refs.[12, 14]) E * = {x ∈ R n : x ⋅ y ≤ 1, y ∈ E}. From this, if K ∈ Son , then K * ∈ K on and for all u ∈ S n −1 ,

ρ ( K , u ) −1 = h( K * , u ) (14) 1.2 Lp-dual Mixed Quermassintegrals If K , L ∈ Son ,∨ p 0, λ , μ ≥ 0 (not both zero), the  p μ ⋅ L ∈ Son of K and Lp-radial combination λ ⋅ K + L is defined by (see Refs.[15-17], for p ≥ 1 also see Ref.[18]) (15) ρ (λ ⋅ K + p μ ⋅ L, ⋅) p = λρ ( K , ⋅) p + μρ ( L, ⋅) p

Here λ ⋅ K = λ 1/ p K . From (15), Ref.[7] (also see Ref.[8]) defined a type of Lp-dual mixed quermassintegrals as follows: For K , L ∈ Son , ∨ P 0, ∨ ε 0 and real i ≠ n, the Lp-dual  p ,i ( K , L), of K and L mixed quermassintegrals, W is defined by  i (K +  i (K )  p ε ⋅ L) − W W n−i  W p ,i ( K , L) = lim+ (16) ε →0 p ε  i ( K ) denotes the dual quermassintegrals of Here W K ∈ Son which is given by (see Refs.[19, 20])  i (K ) = 1 W ρ ( K , u ) n −i dS (u ) n −1  S n

Here V p ( K , L) denotes a type of Lp-dual mixed volume of K and L which is defined in Refs.[15, 17] (for p ≥ 1 also see Ref.[18]). For the Lp-dual mixed quermassintegrals, Ref.[7] (also see Ref.[8]) gave its integral representation as follows: Theorem 5 If K , L ∈ Son ,∨ p 0 and real i ≠ n , then  p ,i ( K , L) = 1 W ρ Kn − p −i ( u ) ρ Lp (u )dS (u ) (18) n−1 n S Together with (17) and (18), we easily see that if K , L ∈ Son ,∨ p 0 and i ≠ n , then  p ,i ( K , K ) = W  i (K ) W (19)  p , n − p ( K , L) = W  n − p ( L) W

2

Product Inequalities

Theorems 1,2 show that the product inequalities for the Lp-dual mixed quermassintegrals. In this section, we will complete the proofs of Theorems 1,2. Proof of Theorem 1 Since K * ∈ K on , thus from (18), (13)-(14) and the Cauchy-Schwarz integral inequality (see Ref.[21]), and notice that ρ ( B, ⋅) = 1 , then we get that for∨i n − p ,  p ,i ( K , L)W  p ,i ( K * , L ) W 1 ρ Kn − p −i (u ) ρ Lp (u )dS (u )] n −1 n S 1 ×[  n−1 ρ Kn −* p −i (u ) ρ Lp (u )dS (u )] n S 1 = [  n−1 hKi +* p − n (u ) ρ Lp (u )dS (u )] n S 1 ×[  n−1 ρ Kn −* p −i (u ) ρ Lp (u )dS (u )] n S h * (u ) i+ p2−n p 1 ) ρ L (u )dS (u )]2 ≥ [  n−1 ( K n S ρ K * (u ) =[

1  p , i ( B, L ) 2 ρ Lp (u )dS (u )]2 = W n −1 n S This yields inequality (1). ≥[

(17)

(20)

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Wuhan University Journal of Natural Sciences 2015, Vol.20 No.4

By the equality conditions of Cauchy-Schwarz integral inequality [21], we see that equality holds in above first inequality if and only if there exits a constant ∨ c 0 such that ρ ( K * , u ) = ch( K * , u ) for all u ∈ S n −1 . Notice that equality holds in the above second inequality if and only if ρ ( K * , u ) = h( K * , u ) for all u ∈ S n −1 , which implies K * is a ball centered at the origin. Hence, equality holds in (1) if and only if K is a ball centered at the origin. For i = n − p , by (20) we know that (1) is identic. Proof of Corollary 1 Since∨i n − p , let L = B in Theorem 1, then by (17) and (18) we obtain  p ,i ( B, B ) = W  i ( B ) = ω and W n  p ,i ( K , B) = 1  p +i ( K ) W ρ Kn − p −i (u )dS (u ) = W n −1  S n Hence, from inequality (1) we get  p + i ( K )W  p+i ( K * ) ≥ ω 2 W n ∨ ∨ Let j = i + p , thus j n when i n − p . Now replace j by i , then inequality (2) is obtained.

According to the equality condition of inequality (1), we know that equality holds in (2) if and only if K is a ball centered at the origin. Obviously, for i = n , (2) is identic by (17). Proof of Theorem 2 From equalities (18) and (14), we have  p ,i ( K , L ) = 1 W ρ Kn − p −i (u ) ρ Lp (u )dS (u ) n −1  S n 1 1 =  n−1 hKp*+ i − n (u ) p (21) dS (u ) S n hL* (u )  p ,i ( K * , L* ) = 1 W ρ Kn −* (pu−)i (u ) ρ Lp* (u )dS (u ) n −1  S n 1 1 =  n−1 p + i − n ρ Lp* (u )dS (u ) S n ρ K * (u )

(22)

Multiplying both sides of equalities (21) and (22), and using the Cauchy-Schwarz inequality and (13), we obtain that for i ≥ n − p (i ≠ n) ,  p ,i ( K , L)W  p ,i ( K * , L* ) W

p

2

2

p

 r  1  r ≥     n−1 dS (u )  =   ωn2 S  R  n  R The last inequality is valid since ρ ( L* , ⋅) ≥ r and h( L* , ⋅) ≤ R.

Proof of Corollary 2 Taking L = K in (5) and using (19), we immediately get inequality (6).

3 Brunn-Minkowski Type Inequality and Cycle Inequality In this section, we shall establish the BrunnMinkowski type inequality and the cycle inequality for the Lp-dual mixed quermassintegrals. First, based on the Lp-radial combinations, we prove a more general result than Theorem 3, i.e., a quotient form of the Brunn-Minkowski type inequality for the Lp-dual mixed quermassintegrals as follows: p 0 and λ , μ ≥ 0 Theorem 6 Let K , L ∈ Son , ∨ (not both zero), real i ≠ j . If i ≤ n − 2 p ≤ j ≤ n − p, then for any Q ∈ Son , p

 p , i (λ ⋅ K + W  p μ ⋅ L, Q)  j −i    p μ ⋅ L, Q)  W p , j (λ ⋅ K + p

p

 p ,i ( K , Q)  j −i  p ,i ( L, Q)  j −i W W + μ ≤λ  (23)     W p , j ( K , Q)   W p , j ( L, Q )  with equality if and only if K and L are dilates. If n − 2 p ≤ i ≤ n − p ≤ j , then inequality (23) is reversed. Proof of Theorem 6 For K , L ∈ S n ,∨ p 0 and o

λ , μ ≥ 0 (not both zero), real i ≠ j , according to (15) and (18), then  p , i (λ ⋅ K +  p μ ⋅ L, Q ) W 1 ρλn⋅−Kp+−pi μ ⋅ L (u ) ρQp (u )dS (u ) n −1  S n n − p −i 1 =  n−1 (λρ Kp (u ) + μρ Lp (u )) p ρQp (u )dS (u ) n S

=

1  1 =   n−1 hKp*+ i − n (u ) p dS (u )  S hL* (u )  n 

p2

1  1 ×  n−1 p + i − n ρ Lp* (u )dS (u )  S ρ K * (u )  n  1  hKp*+ i − n (u ) ρ Lp* (u )  dS (u )  ≥  n−1 p i n p + − ρ K * (u )hL* (u ) n S   

1  ρ Lp* (u )  ≥   n−1 d S ( u ) hLp* (u ) n S   

2

p2

n − p −i 1 =  n−1 (λρ Kp (u ) ρQn − p −i (u ) + μρ Lp (u ) ρQn − p −i (u )) p dS (u ) n S And  p , j (λ ⋅ K +  p μ ⋅ L, Q ) W

p2

p2

n− p− j 1 =  n−1 (λρ Kp (u ) ρQn − p − j (u ) + μρ Lp (u ) ρQn − p − j (u )) p dS (u ) n S

281

ZHANG Ping et al : Some Inequalities for Lp-Dual Mixed Quermassintegrals

For i ≤ n − 2 p ≤ j ≤ n − p , we know n− p− j n− p−i 0≤ ≤1≤ . p p According to the Minkowski inequality, we have [

(λρ Kp (u ) ρ

n −1

S

≤ [

S n−1

+ [

S

p2 n − p −i Q

(λρ Kp (u ) ρ n −1

(u ) + μρ Lp (u ) ρ

p2 n − p −i Q

( μρ Lp (u ) ρ

(u ))

p2 n − p −i Q

n − p −i p

p2 n − p −i Q

dS (u )]

(u ))

n − p −i p

dS (u )]

S n−1

S n−1

p2 n − p −i Q

(λρ Kp (u ) ρ

+ (

S n −1

S n−1

(u ))

dS (u )]

(u ) + μρ Lp (u ) ρ

p2 n − p −i Q

( μρ Lp (u ) ρ

(λρ Kp (u ) ρ

S

p2 n− p− j Q

(λρ Kp (u ) ρ n −1

+ (

S n −1

p n − p −i

.

(u ))

p2 n − p −i Q

n − p −i p

(u ))

p2 n − p −i Q

dS (u ))

n − p −i p

(u ))

n − p −i p

dS (u )

p n − p −i

dS (u ))

]

(u ) + μρ Lp (u ) ρ

p2 n− p− j Q

p2 n− p− j Q

( μρ Lp (u ) ρ

(u ))

p2 n− p− j Q

n− p − j p

(u ))

dS (u ))

n− p− j p

dS (u )

+ (( 

( μρ (u ) ρ

p2 n− p− j Q

p

× [

dS (u ))

]

p

p

2

(u ))

n − p −i p

p

dS (u )) n − p −i ]

2

(λρ Kp (u ) ρQn − p − j (u )) n −1

n− p − j p

n − p −i p

/

p

dS (u )) n − p − j p

p2 n− p − j Q

+( 

( μρ Lp (u ) ρ

= [( 

(λρ (u ) ρ n −1

S n −1

S

p K

(u ))

p2 n − p −i Q

n− p− j p

(u ))

p

n − p −i p

(u ))

(u ))

j −i

p

dS (u ))

n− p− j p

n− p − j p

p j −i

)

j −i n − p −i

−p

]

n − p −i j −i

− ( j −i )

dS (u )) j −i ) n − p − j dS (u ))

−p j −i

)

− ( j −i ) n− p − j n − p − j − j −i

]

p

dS (u )] j −i

n− p − j p

dS (u )]

2

p2 n− p− j Q

]

dS (u )) j −i ) n − p −i

n − p −i p

(u ))

p n− p − j n − p − j − j −i

n − p −i p

n− p − j p

−p j −i p

dS (u )] j −i dS (u )]

−p j −i

p

j −i  λ p ρ Kn − p −i (u ) ρQp (u )dS (u )  n −1  S  = n− p − j   n− p− j p p λ ρ ( u ) ρ ( u )d S ( u )) K Q  S n−1  p

j −i  μ p ρ Ln − p −i (u ) ρQp (u )dS (u )   S n −1   + n− p − j   p μ ρ Ln − p − j (u ) ρQp (u )dS (u ))  n −  S 1  n − p −i

p

2

( μρ (u ) ρ n −1

p2 n− p− j Q

( μρ (u ) ρ

dS (u ))

2

( μρ Lp (u ) ρQn − p −i (u )) n −1 p L

dS (u )) n − p − j

n − p −i

n− p − j p

p p   j −i n − p −i   n−1 (λρ Kp (u ) ρQn − p −i (u ) + μρ Lp (u ) ρQn − p −i (u )) p dS (u )   = S p2 p2 n− p − j   p p n− p− j n− p− j p (u ) + μρ L (u ) ρQ (u )) dS (u )   S n−1 (λρ K (u ) ρQ  2 p p  n − p −i ≤ [(  n−1 (λρ Kp (u ) ρQn − p −i (u )) p dS (u )) n − p −i S 

p2 n − p −i Q

(λρ (u ) ρ p K

n − p −i p

(u ))

(λρ Kp (u ) ρQn − p −i (u )) n −1

S n−1

p n− p− j

p L

n− p − j p

n − p −i j −i

p

n− p − j p

(u ))

(λρ (u ) ρ n −1

+ [

p n− p− j

p2 n − p −i Q

× [(( 

S n−1

(u ))

( μρ (u ) ρ p L

p2 n− p − j Q

S

p

p L

n −1

× [ n− p − j p

(u ))

p

dS (u )) n − p −i ]

2

p K

≤ [

n − p −i p

 p , i (λ ⋅ K + W  p μ ⋅ L, Q)  j −i    p μ ⋅ L, Q)  W p , j (λ ⋅ K +

S

p

S n−1

p n − p −i

p L

n − p −i p

(u ))

(λρ Kp (u ) ρQn − p −i (u )) n −1

S

j −i ≤ 1, according to the Hölder inequality, n− p−i we have

[( 

( μρ (u ) ρ

+ (( 

For

S

+ (

p2 n− p− j Q

S

≥ [( 

+( 

(λρ (u ) ρ n −1

= [(( 

Similar to the above proof, we have



× [( 

S

n − p −i p

(u ))

p2 n− p− j Q

S n −1

p n − p −i

p2 n − p −i Q

p K

S

(λρ Kp (u ) ρ

≤ [( 

S

( μρ (u ) ρ n −1

S

p n − p −i

So



+ (

p L

dS (u ))

n − p −i p

p n− p− j

p

dS (u )) n − p −i

]

n− p − j p

 j −i  

 n−1 ρ Kn − p −i (u ) ρQp (u )dS (u )  j−i   =λ S   n−1 ρ Kn − p − j (u ) ρQp (u )dS (u )   S  p

 n−1 ρ Ln − p −i (u ) ρQp (u )dS (u )  j −i   +μ S   n−1 ρ Ln − p − j (u ) ρQp (u )dS (u )   S  p

p

 p , i ( K , Q )  j −i  p , i ( L , Q )  j −i W W =λ + μ     W p , j ( K , Q)   W p , j ( L, Q )  This yields inequality (23). According to the equality condition of the Minkowski inequality and the Hölder inequality, we see that equality holds in (23) if and only K and L are dilates. Similar to above proof, for n − 2 p ≤ i ≤ n − p ≤ j , we can prove that the reverse of (23) is true. From Theorem 6, we can complete the proof of

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Wuhan University Journal of Natural Sciences 2015, Vol.20 No.4

Theorem 3. Proof of Theorem 3 Let j = n − p in inequality  p ,n − p ( M , Q) = W  n − p (Q) by (20), (23) and notice that W then for i ≤ 2n − p and any Q ∈ Son , we have that p

 p , i (λ ⋅ K +  p μ ⋅ L, Q) n− p−i W p n − p −i

 p ,i ( K , Q )  p , i ( L, Q ) ≤ λW + μW This gives (9). From the equality condition of (23), we see that equality holds in (9) if and only if K and L are dilates. Similarly, let j = n − p in the reverse of inequality (23), we can obtain that for n − 2 p ≤∧i n − p , the reverse of inequality (9) is true. In addition, let i = n − p in the reverse of inequality (23), then for ∨j n − p , we have

Brunn-Minkowski theorem[J]. Canad J Math, 1961, 13: 444-453. [3] [4]

Firey W J. Mean cross-section measures of harmonic means Firey W J. p-means of convex bodies[J]. Math Scand, 1962, 10: 17-24.

[5]

Lutwak E. The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas[J]. Adv Math, 1996, 118: 244-294.

[6]

Wang W D, Leng G S. Lp-dual mixed quermassintegrals[J].

[7]

Wang W D , Zhang T. Inequalities for dual quermassin-

Indian J Pure Appl Mat, 2005, 36(4): 177-188. tegrals of the radial pth mean bodies[J]. J Inequal Appl, 2014, 2014: 252. [8]

p

−  p , j (λ ⋅ K +  p μ ⋅ L, Q ) j −( n − p ) W

Wang W D , Li Y P. Inequalities for dual quermassintegrals of the p-cross-section bodies[J]. J Math Inequal, 2015, 9(2):

p

 p , j ( K , Q ) − j − ( n − p ) + μW  p , j ( L, Q ) − j − ( n − p ) ≥ λW Replace j by i in above inequality, we easily see that for∨i n − p , the reverse of inequality (9) is also true. Proof of Theorem 4 Since ∧i ∧j k , thus ∧ 0 k −∧j k − i∧ , 0 j −∧i k − i. Therefore, together (18) with the Hölder integral inequality [21], we have that k− j j −i  p , i ( K , L ) k −i W  p ,k ( K , L) k −i W k− j 1 = [  n−1 ρ Kn − p −i (u ) ρ Lp (u )dS (u )] k −i S n j −i 1 ×[  n−1 ρ Kn − p − k (u ) ρ Lp (u )dS (u )] k −i S n ( n − p −i ) ( k − j ) p(k− j ) k −i k− j 1 = [  n−1 ( ρ K k −i (u ) ρ L k −i (u )) k − j dS (u )] k −i S n ( n − p − k ) ( j −i ) p ( j −i ) k −i j −i 1 × [  n−1 ( ρ K k −i (u ) ρ L k −i (u )) j −i dS (u )] k −i S n 1  p , j ( K , L) ≥  n−1 ρ Kn − p − j (u ) ρ Lp (u )dS (u ) = W n S So inequality (20) is obtained. According to equality condition of the Hölder integral inequality, we see that equality holds in (20) if and only if for any u ∈ S n −1 , ρ Kn − p −i (u ) ρ Lp (u ) / ( ρ Kn − p − k (u ) ρ Lp (u )) is a constant, i.e.

ρ K (u ) is a constant. This means that equality holds in (20) if and only if K is a dilate of standard ball B , that is, K is a ball centered at the origin.

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