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

Juan Zhang; Weidong Wang; Peibiao Zhao

doi:10.1515/math-2022-0024

Open Access Published by De Gruyter Open Access March 19, 2022

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

Juan Zhang , Weidong Wang and Peibiao Zhao

From the journal Open Mathematics

https://doi.org/10.1515/math-2022-0024

Abstract

Motivated by the celebrated work of Lutwak, Yang and Zhang [1] on ( p , q ) -mixed volumes and that of Feng and He [2] 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.

Keywords: (p, q)-mixed volume; (p, q)-mixed geominimal surface area; monotonic inequality; isoperimetric inequality; Brunn-Minkowski-type inequality

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 [3]

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 [4]

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 [5]

(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 [6] 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 [7]. In particular, Lutwak [6] 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. [1] 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 [1], Section 3).

Using the L p dual curvature measures, Lutwak et al. [1] 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. [1] 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 [1], 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 [1] 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 [2] 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 [6]. 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 [2] 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 [2] 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 [1]

(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 [36]

(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

[37] 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

[37] 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

[2] 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 [38], 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.

Funding information: This work was supported by the NNSF of China (Grant Nos 11871275 and 11371194).
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-07-19

Revised: 2022-02-13

Accepted: 2022-02-23

Published Online: 2022-03-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

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

Abstract

1 Introduction

Theorem 1.A

Theorem 1.B

Definition 1.1

Theorem 1.C

Theorem 1.D

2 Main results

Theorem 1.1

Theorem 1.2

Remark 1.1

Theorem 1.3

Remark 1.2

Theorem 1.4

3 Background

4 Proofs of main theorems

Lemma 3.1

Proof of Theorem 1.1

Lemma 3.2

Proof of Theorem 1.2

Lemma 3.3

Proof of Theorem 1.3

Lemma 3.4

Proof

Proof of Theorem 1.4

Acknowledgments

References

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