 # Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas

• Juan Zhang , Weidong Wang and
From the journal Open Mathematics

## Abstract

Motivated by the celebrated work of Lutwak, Yang and Zhang  on ( p , q ) -mixed volumes and that of Feng and He  on ( p , q ) -mixed geominimal surface areas, we in the present paper establish and confirm the affine isoperimetric and Brunn-Minkowski-type inequalities with respect to ( p , q ) -mixed geominimal surface areas.

MSC 2010: 52A20; 52A39; 52A40

## 1 Introduction

Let K n denote the set of convex bodies (compact, convex subsets with nonempty interiors) in the Euclidean n -space R n . For the set of convex bodies containing the origin in their interiors and the set of convex bodies whose centroid lie at the origin, we write K o n and K c n , respectively. For the set of star bodies (about the origin) in R n , we write S o n . Let S n 1 denote the unit sphere in R n and V ( K ) denote the n -dimensional volume of a body K . For the standard unit ball B in R n , its volume is written by ω n = V ( B ) .

If K K n , then the support function of K , h K = h ( K , ) : R n R , is defined by 

h ( K , x ) = max { x y : y K } , x R n ,

where x y denotes the standard inner product of x and y in R n .

Let K be a compact star shaped (about the origin) in R n , the radial function of K , ρ K = ρ ( K , ) : R n { 0 } [ 0 , + ) , is defined by 

ρ ( K , x ) = max { λ 0 : λ x K } , x R n { 0 } .

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 dilated (of one another) if ρ K ( u ) / ρ L ( u ) is independent of u S n 1 .

For p 1 , the L p mixed volume, V p ( K , L ) , of K , L K o n is defined by 

(1.1) V p ( K , L ) = 1 n S n 1 h L p ( v ) d S p ( K , v ) ,

where S p ( K , ) is the L p surface area measure.

Based on the L p mixed volume (1.1), Lutwak  in 1996 introduced the notion of L p geominimal surface areas: For p 1 and K K o n , the L p geominimal surface area, G p ( K ) , is defined by

ω n p n G p ( K ) = inf { n V p ( K , Q ) V ( Q ) p n : Q K o n } ,

where Q denotes the polar of Q . When p = 1 , G 1 ( K ) is just Petty’s classical geominimal surface area . In particular, Lutwak  proved the following inequalities.

## Theorem 1.A

If K K o n and 1 p < q , then

G p ( K ) n n n V ( K ) n p 1 p G q ( K ) n n n V ( K ) n q 1 q ,

with equality if and only if K is p-self-minimal.

## Theorem 1.B

If K K o n and p 1 , then

ω n G p ( K ) n n n V ( K ) n p 1 p V ( K ) V ( K ) ,

with equality if and only if K is p-self-minimal.

Very recently, Lutwak et al.  introduced the L p dual curvature measures as follows: Suppose p , q R . If K K o n while L S o n , then the L p dual curvature measures, C ˜ p , q ( K , L , ) , on S n 1 are defined by

S n 1 g ( v ) d C ˜ p , q ( K , L , v ) = 1 n S n 1 g ( α K ( u ) ) h K ( α K ( u ) ) p ρ K ( u ) q ρ L ( u ) n q d u ,

for each continuous g : S n 1 R , where α K is the radial Gauss map (see , Section 3).

Using the L p dual curvature measures, Lutwak et al.  defined the ( p , q ) -mixed volumes as follows: For p , q R , K , Q K o n and L S o n , the ( p , q ) -mixed volume, V ˜ p , q ( K , Q , L ) , is defined by

V ˜ p , q ( K , Q , L ) = S n 1 h Q p ( v ) d C ˜ p , q ( K , L , v ) .

Meanwhile, Lutwak et al.  gave the following formula of ( p , q ) -mixed volume:

(1.2) V ˜ p , q ( K , Q , L ) = 1 n S n 1 h Q h K ( α K ( u ) ) p ρ K ( u ) q ρ L ( u ) n q d u .

In , Lutwak et al. introduced the L p mixed volume V p ( K , Q ) of K , Q K o n for all p R by

V p ( K , Q ) = 1 n S n 1 h Q p ( v ) d S p ( K , v ) .

The case p 1 is just Lutwak’s L p mixed volume (1.1). Moreover, for q R and K , Q S o n , they in  also defined the qth dual mixed volume, V ˜ q ( K , Q ) , by

V ˜ q ( K , Q ) = 1 n S n 1 ρ K q ( v ) ρ Q n q ( v ) d v .

By (1.2), the following special cases are showed:

(1.3) V ˜ p , q ( K , Q , K ) = V p ( K , Q ) ,

(1.4) V ˜ p , q ( K , K , L ) = V ˜ q ( K , L ) ,

(1.5) V ˜ p , n ( K , Q , L ) = V p ( K , Q ) .

The ( p , q ) -mixed volumes unify the mixed volumes of convex bodies in the L p Brunn-Minkowski theory and the dual mixed volumes of star bodies in the L p dual Brunn-Minkowski theory. In the last 20 years, the L p Brunn-Minkowski theory and its dual theory have been developed very rapidly, see e.g., [3, 4,5,6,8, 9,10,11, 12,13,14, 15,16,17, 18,19,20, 21,22,23].

Based on the ( p , q ) -mixed volumes, Feng and He in  introduced the concept of ( p , q ) -mixed geominimal surface areas as follows:

## Definition 1.1

For p , q R , K K o n and L S o n , the ( p , q ) -mixed geominimal surface areas, G ˜ p , q ( K , L ) , of K and L are defined by

(1.6) ω n p n G ˜ p , q ( K , L ) = inf { n V ˜ p , q ( K , Q , L ) V ( Q ) p n : Q K o n } .

If L = K or q = n in (1.6), then from (1.3) or (1.5) we see that for p 1 the definition is just Lutwak’s L p geominimal surface area in . For the studies of L p geominimal surface areas, some results have been obtained in these articles (see, e.g., [24,25,26, 27,28,29, 30,31,32, 33,34,35]).

According to the definition of ( p , q ) -mixed geominimal surface areas, Feng and He  extended Theorem 1.A to the following result:

## Theorem 1.C

Suppose q R . If K K o n and L S o n , then for 0 < r < s ,

G ˜ r , q ( K , L ) n n n V ˜ q ( K , L ) n r 1 r G ˜ s , q ( K , L ) n n n V ˜ q ( K , L ) n s 1 s ,

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

Obviously, Theorem 1.C for L = K and 1 r < s implies Theorem 1.A.

In addition, they  gave a Brunn-Minkowski-type inequality for the ( p , q ) -mixed geominimal surface areas as follows:

## Theorem 1.D

Suppose p , q R such that 0 < n q q < 1 and q n , and let λ , μ R . If K K o n , and L 1 , L 2 S o n , then

G ˜ p , q ( K , λ L 1 + ˜ q μ L 2 ) q n q λ G ˜ p , q ( K , L 1 ) q n q + μ G ˜ p , q ( K , L 2 ) q n q ,

with equality if and only if L 1 and L 2 are dilates.

## 2 Main results

Here, λ L 1 + ˜ q μ L 2 is the L q -radial combination of L 1 and L 2 (see (2.1)).

In this article, associated with the ( p , q ) -mixed geominimal surface areas, we first establish two monotonic inequalities as follows:

## Theorem 1.1

Suppose p , q R . If K K o n and L S o n , then for 1 p < q n 1 ,

(1.7) G ˜ p , n p ( K , L ) n n n V ( K ) n p 1 p G ˜ q , n q ( K , L ) n n n V ( K ) n q 1 q ,

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

## Theorem 1.2

Suppose p , q R such that 1 p < q . If K K o n and L S o n , then

(1.8) G ˜ p , p ( K , L ) n n n V ( L ) n p 1 p G ˜ q , q ( K , L ) n n n V ( L ) n q 1 q ,

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

## Remark 1.1

When L = K and K is p -self-minimal, Theorems 1.1 and 1.2 both become Theorem 1.A.

Next, we obtain an affine isoperimetric inequality for the ( p , q ) -mixed geominimal surface areas.

## Theorem 1.3

Suppose p , q R such that p > 0 and 0 < q n . If K K o n and L S o n , then

(1.9) ω n G ˜ p , q ( K , L ) n n n V ( K ) q p V ( L ) n q 1 p V ( K ) V ( K ) ,

for 0 < q < n equality holds when K and L are dilates.

## Remark 1.2

If q = n and p 1 , Theorem 1.3 only contains Lutwak’s inequality.

Finally, together with the L q harmonic Blaschke combination, we give the following Brunn-Minkowski-type inequality for the ( p , q ) -mixed geominimal surface areas.

## Theorem 1.4

Suppose p , q R such that 0 < q < n , and let λ , μ 0 (not both zero). If K K o n and L 1 , L 2 S o n , then

(1.10) G ˜ p , q ( K , λ L 1 + ^ q μ L 2 ) n + q n q V ( λ L 1 + ^ q μ L 2 ) λ G ˜ p , q ( K , L 1 ) n + q n q V ( L 1 ) + μ G ˜ p , q ( K , L 2 ) n + q n q V ( L 2 ) ,

with equality if and only if L 1 and L 2 are dilates.

Here, λ L 1 + ^ q μ L 2 is the L q harmonic Blaschke combination of L 1 , L 2 S o n (see (2.2)).

## 3 Background

In order to complete the proofs of Theorems 1.1–1.4, we collect some basic facts about convex bodies and star bodies.

If E is a nonempty subset in R n , then the polar set, E , of E is defined by [1,2]

E = { x R n : x y 1 , y E } .

Meanwhile, it is easy to get that ( K ) = K for all K K o n .

For K , L S o n , q 0 and λ , μ 0 (not both zero), the L q radial combination, λ K + ˜ q μ L S o n , of K and L is defined by 

(2.1) ρ ( λ K + ˜ q μ L , ) q = λ ρ ( K , ) q + μ ρ ( L , ) q ,

where the operation “ + ˜ q ” is called L q radial addition, λ K denotes L q radial scalar multiplication and λ K = λ 1 q K .

For K , L S o n , q > 0 and λ , μ 0 (not both zero), the L q harmonic Blaschke combination, λ K + ^ q μ L S o n , of K and L is defined by 

(2.2) ρ ( λ K + ^ q μ L , ) n + q V ( λ K + ^ q μ L ) = λ ρ ( K , ) n + q V ( K ) + μ ρ ( L , ) n + q V ( L ) ,

where the operation “ + ^ q ” is called L q harmonic Blaschke addition, λ K denotes L q harmonic Blaschke scalar multiplication and λ K = λ 1 q K . When λ = μ = 1 , K + ^ q L is called L q harmonic Blaschke sum.

## 4 Proofs of main theorems

In this section, we will prove Theorems 1.1–1.4. To complete the proof of Theorem 1.1, we require the following lemma.

## Lemma 3.1

 Suppose p , q R . If K , Q K o n and L S o n , then for 0 < p < q n 1 ,

(3.1) V ˜ p , n p ( K , Q , L ) V ( K ) 1 p V ˜ q , n q ( K , Q , L ) V ( K ) 1 q ,

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

## Proof of Theorem 1.1

Together (1.6) with inequality (3.1), we obtain that for 1 p < q n 1 ,

ω n G ˜ p , n p ( K , L ) n n n V ( K ) n p 1 p = inf V ˜ p , n p ( K , Q , L ) V ( K ) n p V ( K ) V ( Q ) : Q K o n inf V ˜ q , n q ( K , Q , L ) V ( K ) n q V ( K ) V ( Q ) : Q K o n = ω n G ˜ q , n q ( K , L ) n n n V ( K ) n q 1 q .

This is

G ˜ p , n p ( K , L ) n n n V ( K ) n p 1 p G ˜ q , n q ( K , L ) n n n V ( K ) n q 1 q .

This yields inequality (1.7). According to the equality condition of inequality (3.1), we see that the equality of the above inequality holds if and only if K and L are dilates.□

## Lemma 3.2

 Suppose p , q R satisfy 1 p < q . If K , Q K o n and L S o n , then

(3.2) V ˜ p , p ( K , Q , L ) V ( L ) 1 p V ˜ q , q ( K , Q , L ) V ( L ) 1 q ,

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

## Proof of Theorem 1.2

From Definition 1.1 and (3.2) we see that for 1 p < q ,

ω n G ˜ p , p ( K , L ) n n n V ( L ) n p 1 p = inf V ˜ p , p ( K , Q , L ) V ( L ) n p V ( L ) V ( Q ) : Q K o n inf V ˜ q , q ( K , Q , L ) V ( L ) n q V ( L ) V ( Q ) : Q K o n = ω n G ˜ q , q ( K , L ) n n n V ( L ) n q 1 q .

This is, for 1 p < q ,

G ˜ p , p ( K , L ) n n n V ( L ) n p 1 p G ˜ q , q ( K , L ) n n n V ( L ) n q 1 q .

This gives inequality (1.8). From the equality condition of inequality (3.2), we see that the equality holds in (1.8) if and only if K , L are dilates.□

## Lemma 3.3

 If K , L S o n , 0 < q n , then

(3.3) V ˜ q ( K , L ) V ( K ) q n V ( L ) n q n .

For 0 < q < n , equality holds in (3.3) if and only if K and L are dilates; for q = n , (3.3) becomes an equality.

## Proof of Theorem 1.3

Since p > 0 , it follows from (1.6) that

(3.4) ω n G ˜ p , q ( K , L ) n p = inf { n n p V ˜ p , q ( K , Q , L ) n p V ( Q ) : Q K o n } .

Taking Q = K in (3.4), it follows from (1.4) and (3.3) that for 0 < q n ,

ω n G ˜ p , q ( K , L ) n p n n p V ˜ p , q ( K , K , L ) n p V ( K ) = n n p V ˜ q ( K , L ) n p V ( K ) n n p V ( K ) q p V ( L ) n q p V ( K ) ,

i.e.,

ω n G ˜ p , q ( K , L ) n n n V ( K ) q p V ( L ) n q 1 p V ( K ) V ( K ) .

By the equality condition of inequality (3.3) and Definition 1.1, for 0 < q < n equality holds in (1.9) when K and L are dilates.□

## Lemma 3.4

Suppose p , q R such that 0 < q < n , and let λ , μ > 0 . If K , Q K o n and L 1 , L 2 S o n , then

(3.5) V ˜ p , q ( K , Q , λ L 1 + ^ q μ L 2 ) n + q n q V ( λ L 1 + ^ q μ L 2 ) λ V ˜ p , q ( K , Q , L 1 ) n + q n q V ( L 1 ) + μ V ˜ p , q ( K , Q , L 2 ) n + q n q V ( L 2 ) ,

with equality if and only if L 1 and L 2 are dilates.

## Proof

Since 0 < q < n , thus 0 < n q n + q < 1 . From (1.2), (2.2) and Minkowski’s integral inequality , we get that for any Q K o n ,

V ˜ p , q ( K , Q , λ L 1 + ^ q μ L 2 ) n + q n q V ( λ L 1 + ^ q μ L 2 ) = 1 n S n 1 h Q h K p ( α K ( u ) ) ρ K q ( u ) ρ λ L 1 + ^ q μ L 2 n q ( u ) d u n + q n q V ( λ L 1 + ^ q μ L 2 ) = 1 n S n 1 h Q h K p ( n + q ) n q ( α K ( u ) ) ρ K ( u ) q ( n + q ) n q ρ λ L 1 + ^ q μ L 2 n + q ( u ) V ( λ L 1 + ^ q μ L 2 ) n q n + q d u n + q n q = 1 n S n 1 h Q h K p ( n + q ) n q ( α K ( u ) ) ρ K ( u ) q ( n + q ) n q λ ρ L 1 n + q ( u ) V ( L 1 ) + μ ρ L 2 n + q ( u ) V ( L 2 ) n q n + q d u n + q n q λ 1 n S n 1 h Q h K p ( α K ( u ) ) ρ K q ( u ) ρ L 1 n q ( u ) d u n + q n q V ( L 1 ) + μ 1 n S n 1 h Q h K p ( α K ( u ) ) ρ K q ( u ) ρ L 2 n q ( u ) d u n + q n q V ( L 2 ) = λ V ˜ p , q ( K , Q , L 1 ) n + q n q V ( L 1 ) + μ V ˜ p , q ( K , Q , L 2 ) n + q n q V ( L 2 ) .

This yields inequality (3.5).

By the equality condition of Minkowski’s integral inequality, we see that equality holds in (3.5) if and only if L 1 and L 2 are dilates.□

## Proof of Theorem 1.4

Because of 0 < q < n , thus n + q n q > 0 . Hence, by (1.6) and (3.5) we have

ω n p n G ˜ p , q ( K , λ L 1 + ^ q μ L 2 ) n + q n q V ( λ L 1 + ^ q μ L 2 ) = inf [ n V ˜ p , q ( K , Q , λ L 1 + ^ q μ L 2 ) ] n + q n q V ( λ L 1 + ^ q μ L 2 ) V ( Q ) p n n + q n q : Q K o n λ inf [ n V ˜ p , q ( K , Q , L 1 ) ] n + q n q V ( L 1 ) V ( Q ) p n n + q n q : Q K o n + μ inf [ n V ˜ p , q ( K , Q , L 2 ) ] n + q n q V ( L 2 ) V ( Q ) p n n + q n q : Q K o n = λ ω n p n G ˜ p , q ( K , L 1 ) n + q n q V ( L 1 ) + μ ω n p n G ˜ p , q ( K , L 2 ) n + q n q V ( L 2 ) ,

i.e.,

G ˜ p , q ( K , λ L 1 + ^ q μ L 2 ) n + q n q V ( λ L 1 + ^ q μ L 2 ) λ G ˜ p , q ( K , L 1 ) n + q n q V ( L 1 ) + μ G ˜ p , q ( K , L 2 ) n + q n q V ( L 2 ) .

This gives inequality (1.10).

According to the equality condition of inequality (3.5), we see that the equality holds in (1.10) if and only if L 1 and L 2 are dilates.□

## Acknowledgments

The authors would like to express their cordial gratitude to the referee for valuable comments which improved the paper.

1. Funding information: This work was supported by the NNSF of China (Grant Nos 11871275 and 11371194).

2. Conflict of interest: The authors state no conflict of interest.

## References

 E. Lutwak, D. Yang, and G. Y. Zhang, Lp dual curvature measures, Adv. Math. 329 (2018), 85–132. 10.1016/j.aim.2018.02.011Search in Google Scholar

 Y. B. Feng and B. W. He, The (p,q)-mixed geominimal surface areas, Quaest. Math. 42 (2019), 1031–1043. 10.2989/16073606.2018.1504137Search in Google Scholar

 R. J. Gardner, Geometric Tomography, Second ed, Cambridge Univ. Press, Cambridge, 2006. 10.1017/CBO9781107341029Search in Google Scholar

 R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second ed., Cambridge Univ. Press, New York, 2014. 10.1017/CBO9781139003858Search in Google Scholar

 E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. 10.4310/jdg/1214454097Search in Google Scholar

 E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118 (1996), 244–294. 10.1006/aima.1996.0022Search in Google Scholar

 C. M. Petty, Geominimal surface area, Geom. Dedicata. 3 (1974), no. 1, 77–97. 10.1007/BF00181363Search in Google Scholar

 C. Haberl, Lp intersection bodies, Adv. Math. 217 (2008), 2599–2624. 10.1016/j.aim.2007.11.013Search in Google Scholar

 C. Haberl and M. Ludwig, A characterization of Lp intersection bodies, Int. Math. Res. Not. IMRN 2006 (2006), 10548. 10.1155/IMRN/2006/10548Search in Google Scholar

 C. Haberl and F. Schuster, General Lp affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. 10.4310/jdg/1253804349Search in Google Scholar

 D. Hug, E. Lutwak, D. Yang, and G. Y. Zhang, On the Lp Minkowski problem for polytopes, Discrete Comput. Geom. 33 (2005), 699–715. 10.1007/s00454-004-1149-8Search in Google Scholar

 T. Li and W. D. Wang, Lp-mixed affine surface area, Math. Inequal. Appl. 20 (2017), no. 4, 949–962. 10.7153/mia-2017-20-59Search in Google Scholar

 E. Lutwak, D. Yang, and G. Y. Zhang, Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. 10.4310/jdg/1090347527Search in Google Scholar

 E. Lutwak, D. Yang, and G. Y. Zhang, On the Lp-Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), 4359–4370. 10.1090/S0002-9947-03-03403-2Search in Google Scholar

 E. Lutwak and G. Y. Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), 1–16. 10.4310/jdg/1214460036Search in Google Scholar

 C. Schütt and E. Werner, Surface bodies and p-affine surface area, Adv. Math. 187 (2004), 98–145. 10.1016/j.aim.2003.07.018Search in Google Scholar

 X. Y. Wan and W. D. Wang, Lp-dual mixed affine surface areas, Ukrainian Math. J. 68 (2016), 601–609. 10.1007/s11253-016-1250-1Search in Google Scholar

 W. Wang and B. W. He, Lp-dual affine surface area, J. Math. Anal. Appl. 348 (2008), no. 2, 746–751. 10.1016/j.jmaa.2008.08.006Search in Google Scholar

 W. D. Wang and G. S. Leng, Lp-dual mixed quermassintegrals, Indian J. Pure Appl. Math. 36 (2005), no. 4, 177–188. Search in Google Scholar

 W. D. Wang and G. S. Leng, Lp-mixed affine surface area, J. Math. Anal. Appl. 335 (2007), no. 1, 341–354. 10.1016/j.jmaa.2007.01.046Search in Google Scholar

 E. Werner, On Lp affine surface areas, Indiana Univ. Math. J. 56 (2007), 2305–2323. 10.1512/iumj.2007.56.3099Search in Google Scholar

 E. Werner and D. P. Ye, New Lp affine isoperimetric inequalities, Adv. Math. 218 (2008), 762–780. 10.1016/j.aim.2008.02.002Search in Google Scholar

 T. Zhang, W. D. Wang, and L. Si, The mixed Lp-dual affine surface area for multiple star bodies, J. Nonlinear Sci. Appl. 9 (2016), 2813–2822. 10.22436/jnsa.009.05.76Search in Google Scholar

 H. P. Chen and W. D. Wang, Lp-dual mixed geominimal surface area, J. Wuhan Univ. Natur. Sci. Ed. 22 (2017), no. 4, 307–312. 10.1007/s11859-017-1251-4Search in Google Scholar

 Y. B. Feng and W. D. Wang, Lp dual mixed geominimal surface area, Glasg. Math. J. 56 (2014), no. 4, 229–239. 10.1017/S0017089513000244Search in Google Scholar

 Y. N. Li and W. D. Wang, The Lp-dual mixed geominimal surface area for multiple star bodies, J. Inequal. Appl. 2014 (2014), 1–10. 10.1186/1029-242X-2014-456Search in Google Scholar

 Z. H. Shen, Y. N. Li, and W. D. Wang, Lp-dual geominimal surface areas for the general Lp-intersection bodies, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3519–3529. 10.22436/jnsa.010.07.14Search in Google Scholar

 X. Y. Wan and W. D. Wang, Lp-dual geominimal surface area, J Wuhan Univ (Natural Science Edition) 59 (2013), no. 6, 515–518 (in Chinese).Search in Google Scholar

 W. D. Wang and C. Qi, Lp-dual geominimal surface area, J. Inequal. Appl. 2011 (2011), no. 6. Search in Google Scholar

 L. Yan, W. D. Wang, and L. Si, Lp-dual mixed geominimal surface areas, J. Nonlinear Sci. Appl. 9 (2016), no. 3, 1143–1152. 10.22436/jnsa.009.03.41Search in Google Scholar

 D. P. Ye, Lp geominimal surface areas and their inequalities, Int. Math. Res. Not. IMRN 2015 (2015), 2465–2498. Search in Google Scholar

 D. P. Ye, B. C. Zhu, and J. Z. Zhou, The mixed Lp geominimal surface areas for multiple convex bodies, Indiana Univ. Math. J. 64 (2015), no. 5, 1513–1552. 10.1512/iumj.2015.64.5623Search in Google Scholar

 B. C. Zhu, N. Li, and J. Z. Zhou, Isoperimetric inequalities for Lp geominimal surface area, Glasg. Math. J. 53 (2011), 717–726. 10.1017/S0017089511000292Search in Google Scholar

 B. C. Zhu, J. Z. Zhou, and W. X. Xu, Affine isoperimetric inequalities for Lp geominimal surface area, in: Y. J. Suh et al. (eds.), Real and complex submanifolds, Springer Proceedings in Mathematics & Statistics, 106, Springer, Tokyo, 2014, pp. 167–176. 10.1007/978-4-431-55215-4_15Search in Google Scholar

 B. C. Zhu, J. Z. Zhou, and W. X. Xu, Lp mixed geominimal surface area, J. Math. Anal. Appl. 422 (2015), 1247–1263. 10.1016/j.jmaa.2014.09.035Search in Google Scholar

 Y. B. Feng and W. D. Wang, Shephard type problems for Lp-centroid bodies, Math. Inequal. Appl. 17 (2014), no. 3, 865–877. 10.7153/mia-17-63Search in Google Scholar

 B. Chen and W. D. Wang, Some inequalities for (p,q)-mixed volume, J. Inequal. Appl. 2018 (2018), 247. 10.1186/s13660-018-1836-2Search in Google Scholar

 E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261. 10.1016/0001-8708(88)90077-1Search in Google Scholar 