Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms

Xia Zhao; Weidong Wang; Youjiang Lin

doi:10.1515/math-2020-0058

Article Open Access

Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms

Xia Zhao , Weidong Wang and Youjiang Lin

Published/Copyright: October 14, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

Keywords: radial Blaschke-Minkowski homomorphism; star duality; Brunn-Minkowski inequality; monotonic inequality; Shephard-type problem; dual quermassintegral

MSC 2010: 52A20; 52A39; 52A40

1 Introduction and main results

For a compact star-shaped (about the origin) K in n-dimensional Euclidean space ℝ n , the radial function of K, ρ K = ρ ( K , ⋅ ) : ℝ n \ { 0 } → [ 0 , + ∞ ) , is defined by [1]

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

If ρ K is positive and continuous, then K will be called a star body (with respect to the origin). Let S o n and S o s n denote the set of star bodies in ℝ n with respect to the origin and origin-symmetric. 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 , where S n − 1 denotes the unit sphere in ℝ n .

The intersection body was introduced by Lutwak [2]. For each K ∈ S o n , n ≥ 2 , the intersection body, IK, of K is an origin-symmetric star body whose radial function is defined by

ρ ( I K , u ) = V n − 1 ( K ∩ u ⊥ ) ,

for all u ∈ S n − 1 . Here, u ⊥ is the ( n − 1 ) -dimensional hyperplane orthogonal to u and V n − 1 denotes the ( n − 1 ) -dimensional volume.

In 2006, based on the properties of intersection bodies, Schuster [3] introduced the radial Blaschke-Minkowski homomorphism, which is more general than intersection body operator as follows.

Definition 1.A

A map Ψ : S o n → S o n is called a radial Blaschke-Minkowski homomorphism, if it satisfies the following conditions:

Ψ is continuous with respect to radial metric;
For all K , L ∈ S o n , Ψ ( K + ^ L ) = Ψ K + ˜ Ψ L , where K + ^ L denotes the radial Blaschke sum of K and L, and Ψ K + ˜ Ψ L denotes the radial Minkowski sum of Ψ K and Ψ L ;
For all K ∈ S o n and every φ ∈ S O ( n ) , Ψ ( φ K ) = φ Ψ ( K ) , where S O ( n ) denotes the group of rotations in n-dimensions.

Furthermore, Schuster [3] introduced the mixed radial Blaschke-Minkowski homomorphism. Let Ψ : S o n → S o n be a radial Blaschke-Minkowski homomorphism, then there is a continuous operator

Ψ : S o n × ⋯ × S o n ︸ n − 1 → S o n ,

symmetric in its arguments such that for K 1 , … , K r ∈ S o n and λ 1 , … , λ r ≥ 0 ,

Ψ λ 1 K 1 + ˜ ⋯ + ˜ λ r K r = ∑ i 1 , … , i n − 1 λ i 1 … λ i n − 1 Ψ ( K i 1 , … , K i n − 1 ) ,

where “ + ˜ ” denotes the radial Minkowski addition. For measure [3] μ ∈ ℳ + ( S n − 1 , e ˆ ) , there is

(1.1) ρ ( Ψ ( K 1 , … , K n − 1 ) , ⋅ ) = ρ ( K 1 , ⋅ ) … ρ ( K n − 1 , ⋅ ) ∗ μ .

Clearly, the aforementioned continuous operator generalizes the notion of radial Blaschke-Minkowski homomorphism, we call Ψ : S o n × ⋯ × S o n ︸ n − 1 → S o n the mixed radial Blaschke-Minkowski homomorphism and write Ψ ( K 1 , K 2 , … , K n − 1 ) . If K 1 = ⋯ = K n − i − 1 = K , K n − i = ⋯ = K n − 1 = L , we write Ψ i ( K , L ) for Ψ K , … , K ︸ n − i − 1 , L , … , L ︸ i and call Ψ i ( K , L ) the i-mixed radial Blaschke-Minkowski homomorphism of K and L. If L = B , we write Ψ i K for Ψ i ( K , B ) . In particular, denote Ψ 0 K by Ψ K .

By (1.1) and allow i is any real, Wang, Chen and Zhang [4] defined the ith radial Blaschke-Minkowski homomorphisms as follows.

Definition 1.B

For K ∈ S o n , real i satisfies 0 ≤ i < n − 1 , the ith radial Blaschke-Minkowski homomorphism, Ψ i K , of K is given by

ρ ( Ψ i K , ⋅ ) = ρ ( K , ⋅ ) n − i − 1 ∗ μ ,

where μ ∈ ℳ + ( S n − 1 , e ˆ ) .

Regarding the studies of the (mixed) radial Blaschke-Minkowski homomorphisms, many results have been found in the previous studies, see, e.g., [5,6,7,9,10,11,12,13,14,15,16,17,18,20].

In 1999, Moszyńska [19] introduced the star duality of star bodies. For K ∈ S o n , the star duality, K ∘ , of K is given by

(1.2) ρ ( K ∘ , u ) = 1 ρ ( K , u ) ,

for every u ∈ S n − 1 . With the emergence of this notion, a number of characterizations and inequalities were established about star bodies, see [20,21,22,23,24,25,26].

We use Ψ j ∘ K to denote the star duality of the jth radial Blaschke-Minkowski homomorphism Ψ j K . The purpose of this article is to establish Brunn-Minkowski inequalities, monotonic inequality and consider Shephard-type problems for star duality of radial Blaschke-Minkowski homomorphisms. First, associated with the L p radial combination, we give the following Brunn-Minkowski inequality.

Theorem 1.1

For j ∈ ℝ , let Ψ j be the jth radial Blaschke-Minkowski homomorphism. If reals i and j satisfy 0 ≤ i , j < n − 1 , and K , L ∈ S o n , then for p > 0 ,

(1.3) W ˜ i Ψ j ∘ ( K + ˜ p L ) − p ( n − j − 1 ) ( n − i ) ≥ W ˜ i ( Ψ j ∘ K ) − p ( n − j − 1 ) ( n − i ) + W ˜ i ( Ψ j ∘ L ) − p ( n − j − 1 ) ( n − i ) ;

for j − n + 1 < p < 0 ,

(1.4) W ˜ i ( Ψ j ∘ ( K + ˜ p L ) ) − p ( n − j − 1 ) ( n − i ) ≤ W ˜ i ( Ψ j ∘ K ) − p ( n − j − 1 ) ( n − i ) + W ˜ i ( Ψ j ∘ L ) − p ( n − j − 1 ) ( n − i ) .

In each case, with equality if and only if K and L are dilates. Here, W ˜ i ( M ) denotes the dual quermassintegrals of M and K + ˜ p L denotes the L p radial Minkowski sum of K and L.

In particular, let i = 0 and p = 1 in Theorem 1.1 and note that W ˜ 0 ( M ) = V ( M ) (see (2.7)), we have as follows.

Corollary 1.1

For j ∈ ℝ , let Ψ j be the jth radial Blaschke-Minkowski homomorphism. If K , L ∈ S o n and 0 ≤ j < n − 1 , then

(1.5) V Ψ j ∘ ( K + ˜ L ) − 1 n ( n − j − 1 ) ≥ V ( Ψ j ∘ K ) − 1 n ( n − j − 1 ) + V ( Ψ j ∘ L ) − 1 n ( n − j − 1 ) ,

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

Remark 1.1

Recall that Schuster [3] gave the notion of Blaschke-Minkowski homomorphism and proved the following Brunn-Minkowski inequality for polar duality of Blaschke-Minkowski homomorphisms: let Φ j be the jth Blaschke-Minkowski homomorphism, if K and L are convex bodies, 0 ≤ j ≤ n − 3 , then

(1.6) V Φ j ∗ ( K + L ) − 1 n ( n − j − 1 ) ≥ V Φ j ∗ K − 1 n ( n − j − 1 ) + V Φ j ⁎ L − 1 n ( n − j − 1 ) ,

with equality if and only if K and L are homothetic. Here, K + L denotes the Minkowski sum of K and L, and Φ j ∗ Q denotes the polar duality of Φ j Q .

Comparing (1.5) with (1.6), we can feel that inequality (1.5) is an analogue of inequality (1.6).

Note that the L p harmonic radial sum is just the L − p radial Minkowski sum of star bodies (see (2.1) and (2.2)). From this, if p ≤ − 1 , we replace p by − p in Theorem 1.1, then inequality (1.4) yields the following Brunn-Minkowski inequality for the L p harmonic radial sum.

Corollary 1.2

For j ∈ ℝ , let Ψ j be the jth radial Blaschke-Minkowski homomorphism. If K , L ∈ S o n , reals i and j satisfy 0 ≤ i , j < n − 1 , then for 1 ≤ p < n − j − 1 ,

W ˜ i Ψ j ∘ ( K + ˜ − p L ) p ( n − j − 1 ) ( n − i ) ≤ W ˜ i Ψ j ∘ K p ( n − j − 1 ) ( n − i ) + W ˜ i Ψ j ∘ L p ( n − j − 1 ) ( n − i ) ,

with equality if and only if K and L are dilates. Here, K + ˜ − p L denotes the L p harmonic radial addition of K and L.

Next, according to the L p harmonic Blaschke sum, another Brunn-Minkowski inequality for the star duality of radial Blaschke-Minkowski homomorphisms is established as follows.

Theorem 1.2

For j ∈ ℝ , let Ψ j be the jth radial Blaschke-Minkowski homomorphism. If K , L ∈ S o n , and reals i and j satisfy 0 ≤ i , j < n − 1 , then for p > 0 ,

(1.7) W ˜ i Ψ j ∘ ( K ∓ p L ) − n + p ( n − j − 1 ) ( n − i ) V ( K ∓ p L ) ≥ W ˜ i Ψ j ∘ K − n + p ( n − j − 1 ) ( n − i ) V ( K ) + W ˜ i Ψ j ∘ L − n + p ( n − j − 1 ) ( n − i ) V ( L ) ,

equality holds if and only if K and L are dilates. Here, K ∓ p L denotes the L p Blaschke sum of K and L.

Furthermore, we consider the Shephard-type problems for the star duality of radial Blaschke-Minkowski homomorphisms and get a positive form and a negative answer, respectively.

Theorem 1.3

Let Ψ i be the ith radial Blaschke-Minkowski homomorphism, L ∈ S o n and 0 ≤ i < n − 1 . If K ∈ Ψ i S o n and Ψ i ∘ K ⊆ Ψ i ∘ L , then

(1.8) W ˜ i ( K ∘ ) ≤ W ˜ i ( L ∘ ) ,

equality holds if and only if K = L .

Theorem 1.4

Let Ψ i be an even ith radial Blaschke-Minkowski homomorphism, K ∈ S o n and 0 ≤ i < n − 1 . If K is not origin-symmetric, then there exists L ∈ S o n , such that

Ψ i ∘ K ⊂ Ψ i ∘ L .

But

W ˜ i ( K ∘ ) > W ˜ i ( L ∘ ) .

In this article, the proofs of Theorems 1.1–1.4 are given in Section 3. In addition, in Section 3 we also give a monotonic inequality of star duality of the radial Blaschke-Minkowski homomorphisms.

2 Preliminaries

2.1 Polar duality

If E is an arbitrary nonempty subset of ℝ n , the polar duality [1,27], E ∗ , of E is defined by

E ∗ = { x ∈ ℝ n : x ⋅ y ≤ 1 , y ∈ E } .

2.2 Some L p combinations

For K , L ∈ S o n , λ , μ ≥ 0 (both not zero) and real p ≠ 0 , the L p radial Minkowski combination [27,28], λ ∘ K + ˜ p μ ∘ L ∈ S o n , of K and L is given by

(2.1) ρ ( λ ∘ K + ˜ p μ ∘ L , ⋅ ) p = λ ρ ( K , ⋅ ) p + μ ρ ( L , ⋅ ) p ,

where “ + ˜ p ” denotes the L p radial Minkowski addition, λ ∘ K denotes the L p radial scalar multiplication and λ ∘ K = λ 1 p K . Obviously, if p = 1 , we can get classical radial Minkowski combination.

If K , L ∈ S o n , λ , μ ≥ 0 (both not zero) and real p ≥ 1 , then the L p harmonic radial combination [29], λ ∘ K + ˜ − p μ ∘ L ∈ S o n , of K and L is defined by

(2.2) ρ ( λ ∘ K + ˜ − p μ ∘ L , ⋅ ) − p = λ ρ ( K , ⋅ ) − p + μ ρ ( L , ⋅ ) − p ,

where “ + ˜ − p ” denotes the L p harmonic radial addition. Obviously, L − p radial combination is the L p harmonic radial combination.

Let K , L ∈ S o n , λ , μ ≥ 0 (both not zero) and real p > 0 , the L p harmonic Blaschke combination [30,31], λ ∗ K ∓ p μ ∗ L ∈ S o n , of K and L is given by

(2.3) ρ ( λ ∗ K ∓ p μ ∗ L , ⋅ ) n + p V ( λ ∗ K ∓ p μ ∗ L ) = λ ρ ( K , ⋅ ) n + p V ( K ) + μ ρ ( L , ⋅ ) n + p V ( L ) ,

where “ ∓ p ” denotes the L p harmonic Blaschke addition, and λ ∗ K denotes the L p harmonic Blaschke scalar multiplication and λ ∗ K = λ 1 p K . When p = 1 , (2.3) is the classical harmonic Blaschke combination.

2.3 Dual mixed quermassintegrals

Lutwak [32] gave the notion of dual mixed volume. For K 1 , K 2 , … , K n ∈ S o n , the dual mixed volume, V ˜ ( K 1 , K 2 , … , K n ) , of K 1 , K 2 , … , K n is given by

(2.4) V ˜ ( K 1 , K 2 , … , K n ) = 1 n ∫ S n − 1 ρ ( K 1 , u ) … ρ ( K n , u ) d u .

In (2.4), if K 1 = ⋯ = K n − i − 1 = K , K n − i = ⋯ = K n − 1 = B , K n = L , then we write W ˜ i ( K , L ) = V ˜ K , … , K ︸ n − i − 1 , B , … , B ︸ i , L ( i = 0 , 1 , … , n − 2 ) . For K , L ∈ S o n , if allow i is any real, then the dual mixed quermassintegral, W ˜ i ( K , L ) , of K and L is defined by

(2.5) W ˜ i ( K , L ) = 1 n ∫ S n − 1 ρ ( K , u ) n − i − 1 ρ ( L , u ) d u .

Taking L = K , then (2.5) yields the dual quermassintegral, W ˜ i ( K ) , of K by

(2.6) W ˜ i ( K ) = W ˜ i ( K , K ) = 1 n ∫ S n − 1 ρ ( K , u ) n − i d u .

Obviously, if i = 0 , then

(2.7) W ˜ 0 ( K ) = 1 n ∫ S n − 1 ρ ( K , u ) n d u = V ( K ) .

The Minkowski inequality [32] for the dual mixed quermassintegrals can be showed as follows.

Theorem 2.A

For K , L ∈ S o n , i is any real. If i < n − 1 , then

(2.8) W ˜ i ( K , L ) ≤ W ˜ i ( K ) n − i − 1 n − i W ˜ i ( L ) 1 n − i ,

with equality if and only if K is a dilate of L. If i > n − 1 , inequality (2.8) is reverse. If i = n − 1 , (2.8) becomes an equality.

2.4 General ith radial Blaschke bodies

For K , L ∈ S o n , 0 ≤ i < n − 1 and λ , μ ≥ 0 (both not zero), the ith radial Blaschke combination [4], λ ⋅ K + ^ i μ ⋅ L ∈ S o n , of K and L is defined by

(2.9) ρ ( λ ⋅ K + ^ i μ ⋅ L , ⋅ ) n − i − 1 = λ ρ ( K , ⋅ ) n − i − 1 + μ ρ ( L , ⋅ ) n − i − 1 .

Taking i = 0 in (2.9), then λ ⋅ K + ^ 0 μ ⋅ L is the radial Blaschke combination [1] λ ⋅ K + ^ μ ⋅ L .

For τ ∈ [ − 1 , 1 ] , let

(2.10) λ = f 1 ( τ ) = ( 1 + τ ) 2 2 ( 1 + τ 2 ) , μ = f 2 ( τ ) = ( 1 − τ ) 2 2 ( 1 + τ 2 ) ,

and L = − K in (2.9), writing

(2.11) ∇ ^ i τ K = f 1 ( τ ) ⋅ K + ^ i f 2 ( τ ) ⋅ ( − K ) ,

and called ∇ ^ i τ K the general ith radial Blaschke body of K.

For the general ith radial Blaschke body [4], the following fact was also given.

Theorem 2.B

For K ∈ S o n , i < n − 1 . If K ∉ S o s n , then for τ ∈ [ − 1 , 1 ] ,

∇ ^ i τ K ∈ S o s n ⇔ τ = 0 .

3 Proofs of theorems

In this section, we will complete proofs of Theorems 1.1–1.4. In order to prove Theorem 1.1, we require the following lemmas.

Lemma 3.1

For M , N ∈ S o n , reals i, j satisfy 0 ≤ i , j < n − 1 , then

(3.1) W ˜ i ( M ∘ , Ψ j ∘ N ) = W ˜ j ( N ∘ , Ψ i ∘ M ) .

Proof

From (2.5) and combined (1.2) with Definition 1.B, we have

W ˜ i ( M ∘ , Ψ j ∘ N ) = 1 n ∫ S n − 1 ρ ( M ∘ , u ) n − i − 1 ρ ( Ψ j ∘ N , u ) d u = 1 n ∫ S n − 1 1 ρ ( M , u ) n − i − 1 ρ ( Ψ j N , u ) d u = 1 n ∫ S n − 1 1 ρ ( M , u ) n − i − 1 ρ ( N , u ) n − j − 1 ∗ μ d u = 1 n ∫ S n − 1 1 ρ ( Ψ i M , u ) ρ ( N , u ) n − j − 1 d u = 1 n ∫ S n − 1 ρ ( N ∘ , u ) n − j − 1 ρ ( Ψ i ∘ M , u ) d u = W ˜ j ( N ∘ , Ψ i ∘ M ) .

This is equality (3.1).□

Lemma 3.2

Let K , L ∈ S o n , real j satisfies 0 ≤ j < n − 1 . If p > 0 , then for any Q ∈ S o n ,

(3.2) W ˜ j ( ( K + ˜ p L ) ∘ , Q ) − p n − j − 1 ≥ W ˜ j ( K ∘ , Q ) − p n − j − 1 + W ˜ j ( L ∘ , Q ) − p n − j − 1 ;

if j − n + 1 < p < 0 , then for any Q ∈ S o n ,

(3.3) W ˜ j ( ( K + ˜ p L ) ∘ , Q ) − p n − j − 1 ≤ W ˜ j ( K ∘ , Q ) − p n − j − 1 + W ˜ j ( L ∘ , Q ) − p n − j − 1 .

In each case, equality holds if and only if K and L are dilates.

Proof

If p > 0 , note that 0 ≤ j < n − 1 , then − n − j − 1 p < 0 . By (2.5), (1.2), (2.1) and associated with the Minkowski integral inequality, we get

W ˜ j ( ( K + ˜ p L ) ∘ , Q ) − p n − j − 1 = 1 n ∫ S n − 1 ρ ( ( K + ˜ p L ) ∘ , u ) n − j − 1 ρ ( Q , u ) d u − p n − j − 1 = 1 n ∫ S n − 1 ρ ( K + ˜ p L , u ) − ( n − j − 1 ) ρ ( Q , u ) d u − p n − j − 1 = 1 n ∫ S n − 1 ( ρ ( K , u ) p + ρ ( L , u ) p ) − n − j − 1 p ρ ( Q , u ) d u − p n − j − 1 ≥ 1 n ∫ S n − 1 ρ ( K , u ) − ( n − j − 1 ) ρ ( Q , u ) d u − p n − j − 1 + 1 n ∫ S n − 1 ρ ( L , u ) − ( n − j − 1 ) ρ ( Q , u ) d u − p n − j − 1 = W ˜ j ( K ∘ , Q ) − p n − j − 1 + W ˜ j ( L ∘ , Q ) − p n − j − 1 .

This is just inequality (3.2).

For j − n + 1 < p < 0 , then − n − j − 1 p > 1 , similar to the aforementioned method, using the Minkowski integral inequality, we can obtain inequality (3.3).

According to equality condition of the Minkowski integral inequality, we know that equalities hold in (3.2) and (3.3) if and only if K and L are dilates.□

Proof of Theorem 1.1

For K , L ∈ S o n , 0 ≤ j < n − 1 and p > 0 , by (3.1) and (3.2), we obtain for any Q ∈ S o n ,

W ˜ i ( Q ∘ , Ψ j ∘ ( K + ˜ p L ) ) − p n − j − 1 = W ˜ j ( ( K + ˜ p L ) ∘ , Ψ i ∘ Q ) − p n − j − 1 ≥ W ˜ j ( K ∘ , Ψ i ∘ Q ) − p n − j − 1 + W ˜ j ( L ∘ , Ψ i ∘ Q ) − p n − j − 1 = W ˜ i ( Q ∘ , Ψ j ∘ K ) − p n − j − 1 + W ˜ i ( Q ∘ , Ψ j ∘ L ) − p n − j − 1 .

Since 0 ≤ i < n − 1 , from (2.8) and note that − p n − j − 1 < 0 , we have

W ˜ i ( Q ∘ , Ψ j ∘ ( K + ˜ p L ) ) − p n − j − 1 ≥ W ˜ i ( Q ∘ ) − p ( n − i − 1 ) ( n − j − 1 ) ( n − i ) W ˜ i ( Ψ j ∘ K ) − p ( n − j − 1 ) ( n − i ) + W ˜ i ( Ψ j ∘ L ) − p ( n − j − 1 ) ( n − i ) .

Taking Q = Ψ j ( K + ˜ p L ) , by (2.6) we get the desired inequality (1.3).

Similarly, if j − n + 1 < p < 0 , then − p n − j − 1 > 1 , because of 0 ≤ i < n − 1 , thus we can obtain inequality (1.4) by (2.8) and inequality (3.3).

According to equality conditions of Lemma 3.2 and (2.8), we see that equalities hold in (1.3) and (1.4) if and only if K and L are dilates.□

Lemma 3.3

If K , L ∈ S o n , p > 0 and real j satisfies 0 ≤ j < n − 1 , then for any Q ∈ S o n ,

(3.4) W ˜ j ( ( K ∓ p L ) ∘ , Q ) − n + p n − j − 1 V ( K ∓ p L ) ≥ W ˜ j ( K ∘ , Q ) − n + p n − j − 1 V ( K ) + W ˜ j ( L ∘ , Q ) − n + p n − j − 1 V ( L ) ,

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

Proof

Since p > 0 and 0 < j < n − 1 , thus − n − j − 1 n + p < 0 . By (2.5), (1.2), (2.3) and combined with the Minkowski integral inequality, we obtain

W ˜ j ( ( K ∓ p L ) ∘ , Q ) − n + p n − j − 1 V ( K ∓ p L ) = 1 n ∫ S n − 1 ρ ( ( K ∓ p L ) ∘ , u ) n − j − 1 ρ ( Q , u ) d u − n + p n − j − 1 V ( K ∓ p L ) = 1 n ∫ S n − 1 ρ ( K ∓ p L , u ) − ( n − j − 1 ) ρ ( Q , u ) d u − n + p n − j − 1 V ( K ∓ p L ) = 1 n ∫ S n − 1 ( ρ ( K ∓ p L , u ) n + p ) − n − j − 1 n + p ρ ( Q , u ) d u V ( K ∓ p L ) − n − j − 1 n + p − n + p n − j − 1 = 1 n ∫ S n − 1 ρ ( K , u ) n + p V ( K ) + ρ ( L , u ) n + p V ( L ) − n − j − 1 n + p ρ ( Q , u ) d u − n + p n − j − 1 ≥ 1 V ( K ) 1 n ∫ S n − 1 ρ ( K , u ) − ( n − j − 1 ) ρ ( Q , u ) d u − n + p n − j − 1 + 1 V ( L ) 1 n ∫ S n − 1 ρ ( L , u ) − ( n − j − 1 ) ρ ( Q , u ) d u − n + p n − j − 1 = W ˜ j ( K ∘ , Q ) − n + p n − j − 1 V ( K ) + W ˜ j ( L ∘ , Q ) − n + p n − j − 1 V ( L ) .

This yields inequality (3.4).

According to the equality condition of the Minkowski integral inequality, we know that equality holds in (3.4) if and only if K and L are dilates.□

Proof of Theorem 1.2

For K , L ∈ S o n , 0 ≤ j < n − 1 and p > 0 , by (3.1) and (3.4), we obtain for any Q ∈ S o n ,

W ˜ i ( Q ∘ , Ψ j ∘ ( K ∓ p L ) ) − n + p n − j − 1 V ( K ∓ p L ) = W ˜ j ( ( K ∓ p L ) ∘ , Ψ i ∘ Q ) − n + p n − j − 1 V ( K ∓ p L ) ≥ W ˜ j ( K ∘ , Ψ i ∘ Q ) − n + p n − j − 1 V ( K ) + W ˜ j ( L ∘ , Ψ i ∘ Q ) − n + p n − j − 1 V ( L ) = W ˜ i ( Q ∘ , Ψ j ∘ K ) − n + p n − j − 1 V ( K ) + W ˜ i ( Q ∘ , Ψ j ∘ L ) − n + p n − j − 1 V ( L ) .

By (2.8), for 0 ≤ i < n − 1 and − n + p n − j − 1 < 0 , we have

W ˜ i ( Q ∘ , Ψ j ∘ ( K ∓ p L ) ) − n + p n − j − 1 V ( K ∓ p L ) ≥ W ˜ i ( Q ∘ ) − ( n + p ) ( n − i − 1 ) ( n − i ) ( n − j − 1 ) W ˜ i ( Ψ j ∘ K ) − n + p ( n − j − 1 ) ( n − i ) V ( K ) + W ˜ i ( Ψ j ∘ L ) − n + p ( n − j − 1 ) ( n − i ) V ( L ) .

Let Q = Ψ j ( K ∓ p L ) in the aforementioned inequality, by (2.6), this yields inequality (1.7). Equality holds in (1.7) if and only if K and L are dilates.□

Proof of Theorem 1.3

Since Ψ i ∘ K ⊆ Ψ i ∘ L ( 0 ≤ i < n − 1 ) , then ρ ( Ψ i ∘ K , ⋅ ) ≤ ρ ( Ψ i ∘ L , ⋅ ) , thus using (2.5) we know for any M ∈ S o n and 0 ≤ j < n − 1 ,

W ˜ j ( M ∘ , Ψ i ∘ K ) ≤ W ˜ j ( M ∘ , Ψ i ∘ L ) .

This together with (3.1) yields

W ˜ i ( K ∘ , Ψ j ∘ M ) ≤ W ˜ i ( L ∘ , Ψ j ∘ M ) .

Because of K ∈ Ψ i S o n , taking Ψ j M = K , then by (2.6) and (2.8) we obtain

W ˜ i ( K ∘ ) ≤ W ˜ i ( L ∘ , K ∘ ) ≤ W ˜ i ( L ∘ ) n − i − 1 n − i W ˜ i ( K ∘ ) 1 n − i ,

i.e.,

W ˜ i ( K ∘ ) ≤ W ˜ i ( L ∘ ) .

This is just inequality (1.8).

According to the equality condition of inequality (2.8), we see that W ˜ i ( K ∘ ) = W ˜ i ( L ∘ ) if and only if K ∘ and L ∘ are dilates. From this, let K ∘ = c L ∘ ( c > 0 ) and together (1.2) with W ˜ i ( K ∘ ) = W ˜ i ( L ∘ ) , we obtain c = 1 . Therefore, equality holds in (1.8) if and only if K = L .□

Lemma 3.4

If K ∈ S o n , 0 ≤ i < n − 1 and τ ∈ [ − 1 , 1 ] , then

(3.5) W ˜ i ( ∇ ^ i τ , ∘ K ) ≤ W ˜ i ( K ∘ ) ,

with equality for τ ∈ ( − 1 , 1 ) if and only if K is origin-symmetric. For τ = ± 1 , the aforementioned inequality becomes an equality. Here, ∇ ^ i τ , ∘ K denotes the star duality of ∇ ^ i τ K .

Proof

For 0 ≤ i < n − 1 , then − n − i n − i − 1 < 0 , thus from (1.2), (2.6), (2.9), (2.11) and associated with the Minkowski integral inequality, we have

(3.6) ( W ˜ i ( ∇ ^ i τ , ∘ K ) ) − n − i − 1 n − i = 1 n ∫ S n − 1 ρ ( ∇ ^ i τ , ∘ K , u ) n − i d u − n − i − 1 n − i = 1 n ∫ S n − 1 ρ ( ∇ ^ i τ K , u ) − ( n − i ) d u − n − i − 1 n − i = 1 n ∫ S n − 1 ( ρ ( f 1 ( τ ) ⋅ K + ^ i f 2 ( τ ) ⋅ ( − K ) , u ) n − i − 1 ) − n − i n − i − 1 d u − n − i − 1 n − i = 1 n ∫ S n − 1 ( f 1 ( τ ) ρ ( K , u ) n − i − 1 + f 2 ( τ ) ρ ( − K , u ) n − i − 1 ) − n − i n − i − 1 d u − n − i − 1 n − i ≥ f 1 ( τ ) 1 n ∫ S n − 1 ρ ( K , u ) − ( n − i ) d u − n − i − 1 n − i + f 2 ( τ ) 1 n ∫ S n − 1 ρ ( − K , u ) − ( n − i ) d u − n − i − 1 n − i = f 1 ( τ ) W ˜ i ( K ∘ ) − n − i − 1 n − i + f 2 ( τ ) W ˜ i ( ( − K ) ∘ ) − n − i − 1 n − i .

Since f 1 ( τ ) > 0 and f 2 ( τ ) > 0 with τ ∈ ( − 1 , 1 ) , from the equality condition of the Minkowski integral inequality, we know that equality holds in (3.6) for τ ∈ ( − 1 , 1 ) if and only if K and − K are dilates, that is, K is origin-symmetric.

If τ = ± 1 , then by ∇ ^ i ± 1 K = ± K , we see (3.6) becomes an equality.

But by (2.6) and (1.2), we have

W ˜ i ( ( − K ) ∘ ) = 1 n ∫ S n − 1 ρ ( − K , u ) − ( n − i ) d u = 1 n ∫ S n − 1 ρ ( K , − u ) − ( n − i ) d ( − u ) = 1 n ∫ S n − 1 ρ ( K ∘ , − u ) n − i d ( − u ) = W ˜ i ( K ∘ ) .

Note that (2.10) gives f 1 ( τ ) + f 2 ( τ ) = 1 . These combined with (3.6) and − n − i − 1 n − i < 0 , we easily obtain

W ˜ i ( ∇ ^ i τ , ∘ K ) ≤ W ˜ i ( K ∘ ) .

Thus, inequality (3.5) is given.

By the equality condition of (3.6), we know that with equality in (3.5) for τ ∈ ( − 1 , 1 ) if and only if K is origin-symmetric. For τ = ± 1 , (3.5) becomes an equality.□

Lemma 3.5

Let Ψ i be an ith radial Blaschke-Minkowski homomorphism, K ∈ S o n and 0 ≤ i < n − 1 , then for c > 0 ,

Ψ i ∘ ( c K ) = c − ( n − i − 1 ) Ψ i ∘ K .

Proof

From (1.2), Definition 1.B and property of radial function ( ρ ( c K , ⋅ ) = c ρ ( K , ⋅ ) , c > 0 ) , we have for c > 0 and any u ∈ S n − 1 ,

ρ ( Ψ i ∘ ( c K ) , u ) = ρ ( Ψ i ( c K ) , u ) − 1 = 1 ρ ( c K , u ) n − i − 1 ∗ μ = 1 c n − i − 1 ρ ( K , u ) n − i − 1 ∗ μ = c − ( n − i − 1 ) ρ ( Ψ i ∘ K , u ) = ρ ( c − ( n − i − 1 ) Ψ i ∘ K , u ) ,

i.e.,

Ψ i ∘ ( c K ) = c − ( n − i − 1 ) Ψ i ∘ K .

Hence, Lemma 3.5 is proven.□

Lemma 3.6

Let Ψ i be an even ith radial Blaschke-Minkowski homomorphism. If K ∈ S o n , 0 ≤ i < n − 1 and τ ∈ [ − 1 , 1 ] , then

Ψ i ∘ ( ∇ ^ i τ K ) = Ψ i ∘ K .

Proof

Since Ψ i is an even ith radial Blaschke-Minkowski homomorphism, thus for any K ∈ S o n , Ψ i ( − K ) = Ψ i K .

From this, associated with Definition 1.B, (1.2), (2.10) and (2.11), it follows that for any u ∈ S n − 1 ,

ρ ( Ψ i ∘ ( ∇ ^ i τ K ) , u ) = 1 ρ ( Ψ i ( ∇ ^ i τ K ) , u ) = 1 ρ ( ∇ ^ i τ K , u ) n − i − 1 ∗ μ = 1 ( f 1 ( τ ) ρ ( K , u ) n − i − 1 + f 2 ( τ ) ρ ( − K , u ) n − i − 1 ) ∗ μ = 1 f 1 ( τ ) ρ ( K , u ) n − i − 1 ∗ μ + f 2 ( τ ) ρ ( − K , u ) n − i − 1 ∗ μ = 1 f 1 ( τ ) ρ ( Ψ i K , u ) + f 2 ( τ ) ρ ( Ψ i ( − K ) , u ) = 1 ρ ( Ψ i K , u ) = ρ ( Ψ i ∘ K , u ) ,

i.e.,

Ψ i ∘ ( ∇ ^ i τ K ) = Ψ i ∘ K .

This gives proof of Lemma 3.6.□

Proof of Theorem 1.4

Since K is not origin-symmetric, thus from Lemma 3.4 we know that

W ˜ i ( ∇ ^ i τ , ∘ K ) < W ˜ i ( K ∘ ) .

Note that for 0 < c < 1 , c ∇ ^ i τ K ⊆ ∇ ^ i τ K implies ∇ ^ i τ , ∘ K ⊂ c − 1 ∇ ^ i τ , ∘ K . Therefore, choose 0 < ε < 1 such that

W ˜ i ( ( 1 − ε ) − 1 ∇ ^ i τ , ∘ K ) < W ˜ i ( K ∘ ) .

From this, let L = ( 1 − ε ) ∇ ^ i τ K , then L ∈ S o n (Theorem 2.B gives that for τ = 0 , L ∈ S o s n ; for τ ∈ ( − 1 , 1 ) and τ ≠ 0 , L ∈ S o n \ S o s n ) and

W ˜ i ( L ∘ ) < W ˜ i ( K ∘ ) .

But Lemma 3.5 gives that Ψ i ∘ ( ( 1 − ε ) K ) = ( 1 − ε ) − ( n − i − 1 ) Ψ i ∘ K , because of 0 ≤ i < n − 1 , then ( 1 − ε ) − ( n − i − 1 ) > 1 , these together with Lemma 3.6, we obtain that

Ψ i ∘ L = Ψ i ∘ ( ( 1 − ε ) ∇ ^ i τ K ) = ( 1 − ε ) − ( n − i − 1 ) Ψ i ∘ ( ∇ ^ i τ K ) = ( 1 − ε ) − ( n − i − 1 ) Ψ i ∘ K ⊃ Ψ i ∘ K .

Therefore, we complete the proof of Theorem 1.4.□

Finally, we give a monotonic inequality of star duality of radial Blaschke-Minkowski homomorphisms.

Theorem 3.1

For j ∈ ℝ , let Ψ j be the jth radial Blaschke-Minkowski homomorphism, K , L ∈ S o n and reals i and j satisfy 0 ≤ i , j < n − 1 . If for any Q ∈ Ψ i S o n ,

W ˜ j ( K ∘ , Q ∘ ) ≤ W ˜ j ( L ∘ , Q ∘ ) ,

then

(3.7) W ˜ i ( Ψ j ∘ K ) ≤ W ˜ i ( Ψ j ∘ L ) .

With equality if and only if K = L . Here, Ψ i S o n denotes the range of Ψ i .

Proof

Since W ˜ j ( K ∘ , Q ∘ ) ≤ W ˜ j ( L ∘ , Q ∘ ) and Q ∈ Ψ i S o n , thus let Q = Ψ i M , we have

(3.8) W ˜ j ( K ∘ , Ψ i ∘ M ) ≤ W ˜ j ( L ∘ , Ψ i ∘ M ) ,

equality holds if and only if K = L .

(3.8) and (3.1) imply that

W ˜ i ( M ∘ , Ψ j ∘ K ) ≤ W ˜ i ( M ∘ , Ψ j ∘ L ) .

Taking M = Ψ j K in the aforementioned inequality, for 0 ≤ i < n − 1 and from (2.6) and (2.8) we have

(3.9) W ˜ i ( Ψ j ∘ K ) ≤ W ˜ i ( Ψ j ∘ K , Ψ j ∘ L ) ≤ W ˜ i ( Ψ j ∘ K ) n − i − 1 n − i W ˜ i ( Ψ j ∘ L ) 1 n − i ,

i.e.,

W ˜ i ( Ψ j ∘ K ) ≤ W ˜ i ( Ψ j ∘ L ) .

This is the desired inequality (3.7).

According to the equality condition of (2.8), with equality in (3.9) if and only if Ψ j ∘ K and Ψ j ∘ L are dilates, thus combined with equality conditions of (3.8) and (3.9), we know that equality holds in (3.7) if and only if K = L .□

Acknowledgment

This research was supported in part by the Natural Science Foundation of China (No. 11371224), the Innovation Foundation of Graduate Student of China Three Gorges University (No. 2019SSPY144), NSFC (No. 11971080) and the funds of the Basic and Advanced Research Project of CQ CSTC (cstc2018jcyjAX0790).

References

[1] R. J. Gardner, Geometric Tomography, 2nd ed., Cambridge Univ. Press, Cambridge, UK, 2006.10.1017/CBO9781107341029Search in Google Scholar

[2] E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), no. 2, 232–261, 10.1016/0001-8708(88)90077-1.Search in Google Scholar

[3] F. E. Schuster, Volume inequalities and additive maps of convex bodies, Mathematika 53 (2006), no. 2, 211–234, 10.1112/S0025579300000103.Search in Google Scholar

[4] W. D. Wang, H. P. Chen, and Y. Y. Zhang, Busemann-Petty problem for the i-th radial Blaschke-Minkowski homomorphisms, Filomat 32 (2018), no. 19, 6819–6827.10.2298/FIL1819819WSearch in Google Scholar

[5] Y. B. Feng, W. D. Wang, and J. Yuan, Differences of quermass- and dual quermassintegrals of Blaschke-Minkowski and radial Blaschke-Minkowski homomorphisms, Bull. Belg. Math. Soc. 21 (2014), no. 4, 577–592, 10.36045/bbms/1414091004.Search in Google Scholar

[6] Y. B. Feng, S. H. Wu, and W. D. Wang, Mixed chord-integrals of index i and radial Blaschke-Minkowski homomorphisms, Rocky Mountain J. Math. 47 (2017), no. 8, 2627–2640, 10.1216/RMJ-2017-47-8-2627.Search in Google Scholar

[7] L. W. Ji and Z. B. Zeng, Some inequalities for radial Blaschke-Minkowski homomorphisms, Czechoslovak Math. J. 67 (2017), no. 3, 779–793, 10.21136/CMJ.2017.0180-16.Search in Google Scholar

[8] Z. F. Li and W. D. Wang, Star dual of radial Blaschke-Minkowski homomorphism, Wuhan Univ. J. Nat. Sci. 23 (2018), no. 4, 295–300, 10.1007/s11859-018-1325-y.Search in Google Scholar

[9] L. J. Liu, Mixed radial Blaschke-Minkowski homomorphisms and comparison of volumes, Math. Inequal. Appl. 67 (2013), no. 2, 401–412, 10.7153/mia-16-30.Search in Google Scholar

[10] F. E. Schuster, Valuations and Busemann-Petty type problems, Adv. Math. 219 (2008), 344–368, 10.1016/j.aim.2008.05.001.Search in Google Scholar

[11] B. Wei, W. D. Wang, and F. H. Lu, Inequalities for radial Blaschke-Minkowski homomorphisms, Ann. Polon. Math. 113 (2015), no. 3, 243–253, 10.4064/ap113-3-2.Search in Google Scholar

[12] C. J. Zhao, On radial Blaschke-Minkowski homomorphisms, Geom. Dedicata 167 (2013), no. 1, 1–10, 10.1007/s10711-012-9798-x.Search in Google Scholar

[13] C. J. Zhao, On radial and polar Blaschke-Minkowski homomorphisms, Proc. Amer. Math. Soc. 141 (2013), no. 2, 1–10, 10.1090/S0002-9939-2012-11318-8.Search in Google Scholar

[14] C. J. Zhao, On Blaschke-Minkowski homomorphisms and radial Blaschke-Minkowski homomorphisms, J. Geom. Anal. 26 (2016), no. 2, 1523–1538, 10.1007/s12220-015-9598-2.Search in Google Scholar

[15] C. J. Zhao, Quotient for radial Blaschke-Minkowski homomorphisms, Bull. Math. Soc. Sci. Math. 108 (2017), no. 2, 147–157.Search in Google Scholar

[16] C. J. Zhao, Orlicz-Brunn-Minkowski inequality for radial Blaschke-Minkowski homomorphisms, Quaest. Math. 41 (2018), no. 7, 1–14, 10.2989/16073606.2017.1417336.Search in Google Scholar

[17] C. J. Zhao and W. S. Cheung, Radial Blaschke-Minkowski homomorphisms and volume differences, Geom. Dedicata 154 (2011), no. 1, 81–91, 10.1007/s10711-010-9568-6.Search in Google Scholar

[18] C. J. Zhao and W. S. Cheung, Dresher type inequalities for radial Blaschke-Minkowski homomorphisms, Moscow Math. J. 16 (2016), no. 2, 371–379.10.17323/1609-4514-2016-16-2-371-379Search in Google Scholar

[19] M. Moszyńska, Quotient star bodies, intersection bodies and star duality, J. Math. Anal. Appl. 232 (1999), no. 1, 45–60, 10.1006/jmaa.1998.6238.Search in Google Scholar

[20] Z. F. Li and W. D. Wang, Star dual of radial Blaschke-Minkowski homomorphism, Wuhan Univ. J. Nat. Sci. 23 (2018), no. 4, 295–300, 10.1007/s11859-018-1325-y.Search in Google Scholar

[21] L. J. Liu, W. Wang, and B. W. He, Brunn-Minkowski inequalities for star duals of intersection bodies and two additions, J. Shanghai Univ. 14 (2010), no. 3, 201–205, 10.1007/s11741-010-0630-1.Search in Google Scholar

[22] F. H. Lu, W. H. Mao, and G. S. Leng, On star duality of mixed intersection bodies, J. Inequal. Appl. 2007 (2007), 039345, 10.1155/2007/39345.Search in Google Scholar

[23] M. Moszyńska, Selected Topics in Convex Geometry, Birkhauser, Boston, 2006, 10.1007/0-8176-4451-2.Search in Google Scholar

[24] J. Yuan, H. W. Zhu, and G. S. Leng, Inequalities for star duals of intersection bodies, J. Korean Math. Soc. 44 (2007), no. 2, 297–306, 10.4134/JKMS.2007.44.2.297.Search in Google Scholar

[25] C. J. Zhao and W. S. Cheung, Lp-Brunn-Minkowski inequality, Indag. Math. 20 (2009), no. 2, 179–190.10.1016/S0019-3577(09)00018-4Search in Google Scholar

[26] X. Y. Zhu and Y. J. Shen, The dual Brunn-Minkowski inequalities for star dual of mixed intersection bodies, Acta Math. Sci. Ser. A Chin. Ed. 30 (2010), no. 3, 739–746, 10.1016/S0252-9602(10)60075-6.Search in Google Scholar

[27] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, 2nd edn., Cambridge Univ. Press, Cambridge, 2014.10.1017/CBO9781139003858Search in Google Scholar

[28] R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), no. 3, 355–405, 10.1090/S0273-0979-02-00941-2.Search in Google Scholar

[29] E. Lutwak, The Brunn-Minkowski-Firey theory II: affine and geominimal surface area, Adv. Math. 118 (1996), 244–294, 10.1006/aima.1996.0022.Search in Google Scholar

[30] 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-17-32.Search in Google Scholar

[31] F. H. Lu, On Lp-Brunn-Minkowski type inequalities of convex bodies, Bol. Soc. Mat. Mexicana 13 (2007), no. 1, 167–176.Search in Google Scholar

[32] E. Lutwak, Dual mixed volume, Pacific J. Math. 58 (1975), no. 2, 531–538.10.2140/pjm.1975.58.531Search in Google Scholar

Received: 2019-04-08

Revised: 2020-06-24

Accepted: 2020-07-01

Published Online: 2020-10-14

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

Articles in the same Issue

https://doi.org/10.1515/math-2020-0058

Keywords for this article

radial Blaschke-Minkowski homomorphism; star duality; Brunn-Minkowski inequality; monotonic inequality; Shephard-type problem; dual quermassintegral

Creative Commons

BY 4.0