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

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

Juan Zhang , Weidong Wang and Peibiao Zhao

Published/Copyright: March 19, 2022

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$Open Mathematics$

From the journal Open Mathematics Volume 20 Issue 1

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.

Articles in the same Issue

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

Keywords for this article

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

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