General (p,q)-mixed projection bodies

Yibin Feng; Yanping Zhou

doi:10.1515/math-2020-0055

Article Open Access

General (p,q)-mixed projection bodies

Yibin Feng and Yanping Zhou

Published/Copyright: September 15, 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 this article, the general (p,q)-mixed projection bodies are introduced. Then, some basic properties of the general (p,q)-mixed projection bodies are discussed, and the extreme values of volumes of the general (p,q)-mixed projection bodies and their polar bodies are established.

Keywords: projection bodies; general (p,q)-mixed projection bodies; extreme value

MSC 2010: 52A20; 52A40

1 Introduction

Projection bodies, introduced by Minkowski at the end of nineteenth century, are a central notion in the classical Brunn-Minkowski theory. In recent years, projection bodies and their polars have received considerable attention, see, e.g., [1,2,3,4,5,6,7,8,9].

The classical Brunn-Minkowski theory has a natural extension called the L _p Brunn-Minkowski theory, which was initiated in the early 1960s when Firey introduced his concept of L _p composition of convex bodies (see [10]) and was born in the works of Lutwak [11,12]. Since Lutwak’s two seminal works, this theory has attracted increasing interest in recent years, see, e.g., [13,14,15,16,17,18,19,20]. One of the central concepts in the L _p Brunn-Minkowski theory is the L _p projection bodies introduced by Lutwak et al. [17]. Afterward, Ludwig [21] (see also Haberl and Schuster [22]) extended Lutwak, Yang and Zhang’s L _p projection bodies to an entire class which may be called general L _p projection bodies. See, e.g., [23,24,25,26,27,28,29,30] for the L _p and general L _p projection bodies and their extensions.

The dual Brunn-Minkowski theory, developed by Lutwak [31] in 1975, is a dual concept of the classical Brunn-Minkowski theory. It is a theory of dual mixed volumes of star bodies and has already attracted considerable interest, see, e.g., [32,33,34,35,36,37,38,39,40]. In particular, a recent groundbreaking work of Huang et al. [41] showed that there exists a new family of geometry measures, in the dual Brunn-Minkowski theory, called dual curvature measures which are the long-sought duals of Federer’s curvature measures. Now, the dual curvature measures become the core of the dual Brunn-Minkowski theory.

Very recently, a unification of the L _p Brunn-Minkowski theory and the dual Brunn-Minkowski theory was discovered by Lutwak et al. [42], where they introduced the L _p dual curvature measures which include L _p surface area measures and L _p integral curvatures in the L _p Brunn-Minkowski theory as well as the dual curvature measures in the dual Brunn-Minkowski theory. In the same way that the L _p surface area and dual curvature measures, respectively, play a critical role in the L _p and dual Brunn-Minkowski theories, the L _p dual curvature measures can be seen to be a central concept within the unifying theory. See, e.g., [43,44,45,46] for some recent developments related to the L _p dual curvature measures.

In this article, motivated by the works of Lutwak et al. [42], Ludwig [21] and Haberl and Schuster [22], we introduce a more general definition of projection bodies according to the L _p dual curvature measures, which is called the general (p,q)-mixed projection bodies. Special cases include the classical projection bodies, L _p projection bodies and general L _p projection bodies. Then, we establish the extreme values of volumes for the general (p,q)-mixed projection bodies and their polar bodies. The detailed descriptions for the definition and main results are provided below.

Throughout ℝ n denotes the n-dimensional Euclidean space. A convex body is a compact convex subset of ℝ n with nonempty interior. We denote by K n the set of convex bodies and by K o n the set of convex bodies containing the origin in their interiors. For a convex body K ∈ K n , let ∂ K and V ( K ) be its boundary and n-dimensional volume, respectively. The unit sphere in ℝ n will be denoted by S n − 1 . For x ∈ ℝ n , | x | = x ⋅ x denotes the Euclidean norm of x, and for x ∈ ℝ n \ { 0 } , the unit vector x / | x | ∈ S n − 1 will be abbreviated by 〈 x 〉 .

For all x ∈ ℝ n , the support function of K ∈ K n is defined by

h ( K , x ) = h K ( x ) = max { x ⋅ y : y ∈ K } ,

where x ⋅ y denotes the standard inner product of x and y.

The radial function, ρ K = ρ ( K , ⋅ ) : ℝ n \ { 0 } → ℝ , of a compact and star-shaped set, with respect to the origin, K ⊂ ℝ n , is defined by

ρ ( K , x ) = max { λ : λ x ∈ K } .

If ρ K is positive and continuous, then K is called a star body with respect to the origin. The set of all star bodies about the origin in ℝ n is denoted by S o n . Two star bodies K and L are dilates (of one another) if ρ K ( u ) / ρ L ( u ) is independent of u ∈ S n − 1 . For a star body Q ∈ S o n , ∥ ⋅ ∥ Q : ℝ n → [ 0 , ∞ ) is a continuous and positively homogeneous function of degree 1, which was defined, in [42], by

∥ x ∥ Q = 1 / ρ Q ( x ) x ≠ 0 ; 0 x = 0 .

For a convex body K ∈ K o n , its polar body K ∗ is defined by

K ∗ = { x ∈ ℝ n : x ⋅ y ≤ 1 for all y ∈ K } .

Suppose p , q ∈ ℝ . If K ∈ K o n while Q ∈ S o n , then the L _p dual curvature measure, C ˜ p , q ( K , Q , ⋅ ) , on S n − 1 was defined, in [42], by

(1.1) ∫ S n − 1 g ( v ) d C ˜ p , q ( K , Q , v ) = 1 n ∫ S n − 1 g ( α K ( u ) ) h K ( α K ( u ) ) − p ρ K ( u ) q ρ Q ( u ) n − q d u

for each continuous g : S n − 1 → ℝ , where α K is the radial Gauss map that associates with almost each u ∈ S n − 1 the unique outer unit normal at the point ρ K ( u ) u ∈ ∂ K .

In [42], the L _p surface area measures were shown to be special cases of the L _p dual curvature measures:

(1.2) C ˜ p , q ( K , K , ⋅ ) = 1 n S p ( K , ⋅ ) ;

(1.3) C ˜ p , n ( K , Q , ⋅ ) = 1 n S p ( K , ⋅ ) .

For τ ∈ [ − 1 , 1 ] , the function φ τ : ℝ → [ 0 , ∞ ) was defined, in [22], by

(1.4) φ τ ( t ) = | t | + τ t .

Now, we begin to define the general (p,q)-mixed projection bodies.

Definition 1.1

Suppose p ≥ 1 , q ∈ ℝ and τ ∈ [ − 1 , 1 ] . If K ∈ K o n while Q ∈ S o n , then for u ∈ S n − 1 , define the general (p,q)-mixed projection body, Π p , q τ ( K , Q ) , to be the convex body whose support function is given by

(1.5) h Π p , q τ ( K , Q ) , u p = n c n , p ( τ ) ∫ S n − 1 φ τ ( u ⋅ v ) p d C ˜ p , q ( K , Q , v ) .

Here,

c n , p ( τ ) = c n , p ( 1 + τ ) p + ( 1 − τ ) p , where c n , p = Γ n + p 2 π ( n − 1 ) / 2 Γ 1 + p 2 .

If we take Q = K or q = n in (1.5), then from (1.2) or (1.3) we have

Π p , q τ ( K , Q ) = Π p τ K ,

where Π p τ K is the general L _p projection body [22] of K, whose support function is given by

(1.6) h Π p τ K , u p = c n , p ( τ ) ∫ S n − 1 φ τ ( u ⋅ v ) p d S p ( K , v ) .

For τ = 0 in (1.6), Π p τ K = Π p K which is the L _p projection body [17]. For τ = 0 and p = 1 in (1.6), the body Π p τ K is the classical projection body, Π K , of K.

Our main results are as follows for the general (p,q)-mixed projection bodies. The first result is to establish the extreme values of V Π p , q τ , ∗ . Note that Π p , q τ , ∗ is used to denote the polar body of Π p , q τ .

Theorem 1.1

Suppose p ≥ 1 and q ∈ ℝ . If K ∈ K o n while L ∈ S o n , then for every τ ∈ [ − 1 , 1 ] ,

(1.7) V Π p , q ∗ ( K , Q ) ≤ V Π p , q τ , ∗ ( K , Q ) ≤ V Π p , q ± , ∗ ( K , Q ) .

Suppose that both K and Q are not origin-symmetric. If τ ≠ 0 , then equality holds in the left inequality if and only if Π p , q τ ( K , Q ) is origin-symmetric; if τ ≠ ± 1 , then equality holds in the right inequality if and only if both Π p , q ± ( K , Q ) are origin-symmetric.

Here, Π p , q + ( K , Q ) is the nonsymmetric (p,q)-mixed projection body, which is the convex body defined by

(1.8) h Π p , q + ( K , Q ) , u p = n c n , p ∫ S n − 1 ( u ⋅ v ) + p d C ˜ p , q ( K , Q , v )

for u ∈ S n − 1 , where ( u ⋅ v ) + = max { u ⋅ v , 0 } . Moreover, Π p , q ( K , Q ) and Π p , q − ( K , Q ) are, respectively, defined by

(1.9) Π p , q ( K , Q ) ≔ Π p , q 0 ( K , Q )

and

(1.10) Π p , q − ( K , Q ) ≔ Π p , q + ( − K , − Q ) .

The special case Q = K or q = n of Theorem 1.1 can be found in [22], which gives rise to the strongest L _p Petty projection inequality. L _p Petty projection inequality is one of the crucial tools used for establishing the sharp affine L _p Sobolev inequalities and the affine Pólya-Szegö principle which are stronger than the usual sharp Sobolev inequalities and the usual Pólya-Szegö principle in the Euclidean space, see, e.g., [24,25,26,27].

The following is to provide the extreme values of V ( Π p , q τ ) .

Theorem 1.2

Suppose p ≥ 1 and q ∈ ℝ . If K ∈ K o n while L ∈ S o n , then for every τ ∈ [ − 1 , 1 ] ,

(1.11) V Π p , q ( K , Q ) ≥ V Π p , q τ ( K , Q ) ≥ V Π p , q ± ( K , Q ) .

When Q = K or q = n , Theorem 1.2 was given by Haberl and Schuster [22].

For quick later reference, we list in Section 2 some basic and well-known facts of convex and star bodies, radial and reverse radial Gauss images and L _p dual curvature measures. The basic properties of the general and nonsymmetric (p,q)-mixed projection bodies are developed in Section 3. Section 4 is devoted to prove Theorems 1.1 and 1.2.

2 Preliminaries

2.1 Basics regarding convex and star bodies

Good general references about the theory of convex bodies are the books of Gardner [47] and Schneider [10].

Let S L ( n ) denote the group of special linear transformation. If ϕ ∈ S L ( n ) , then we write ϕ t for the transpose of ϕ , ϕ − 1 for the inverse of ϕ and ϕ − t for the inverse of the transpose of ϕ . From the definitions of the support function, radial function and polar body, it is easy to see that for ϕ ∈ S L ( n ) , K ∈ K o n and L ∈ S o n ,

(2.1) h ( ϕ K , x ) = h ( K , ϕ t x ) , x ∈ ℝ n ;

(2.2) ρ ( ϕ K , x ) = ρ ( K , ϕ − 1 x ) , x ∈ ℝ n \ { 0 } ;

(2.3) ( ϕ K ) ∗ = ϕ − t K ∗ .

Moreover, it is easy to verify that for K ∈ K o n and c > 0 ,

(2.4) ( c K ) ∗ = 1 c K ∗ .

Suppose p ∈ ℝ . If μ is a Borel measure on S n − 1 and ϕ ∈ S L ( n ) , then the L _p image of μ under ϕ , ϕ p ⊣ μ , is a Borel measure defined, in [42], by

(2.5) ∫ S n − 1 f ( u ) d ϕ p ⊣ μ ( u ) = ∫ S n − 1 | ϕ − 1 u | p f ( 〈 ϕ − 1 u 〉 ) d μ ( u ) ,

for each Borel f : S n − 1 → ℝ .

The support and radial functions of a convex body K ∈ K o n and its polar body are related by

(2.6) ρ K = 1 / h K ∗ and h K = 1 / ρ K ∗ .

If K i ∈ K n , we say that K i → K 0 ∈ K n provided

| h K i − h K 0 | ∞ ≔ max u ∈ S n − 1 | h K i ( u ) − h K 0 ( u ) | → 0 .

For K , L ∈ K n and α , β ≥ 0 (both not zero), the Minkowski combination α K + β L is defined by

α K + β L = { α x + β y : x ∈ K , y ∈ L }

whose support function is

(2.7) h α K + β L = α h K + β h L .

In the early 1960s, Firey [48] introduced the L _p Minkowski combination, which is also known as the Minkowski-Firey combination. For p ≥ 1 , K , L ∈ K o n and α , β ≥ 0 (both not zero), the L _p Minkowski combination, α ⋅ K + p β ⋅ L , was defined by

(2.8) h ( α ⋅ K + p β ⋅ L , ⋅ ) p = α h ( K , ⋅ ) p + β h ( L , ⋅ ) p ,

where α ⋅ K = α 1 p K .

In [48], Firey also established the L _p Brunn-Minkowski inequality: if p ≥ 1 and K , L ∈ K o n , then

(2.9) V ( K + p L ) p n ≥ V ( K ) p n + V ( L ) p n ,

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

The L _p harmonic radial combination of two star bodies was introduced by Lutwak [12]. For p ≥ 1 , K , L ∈ S o n and α , β ≥ 0 (both not zero), the L _p harmonic radial combination, α ∘ K + ˜ p β ∘ L , is a star body whose radial function is given by

(2.10) ρ ( α ∘ K + ˜ p β ∘ L , ⋅ ) − p = α ρ ( K , ⋅ ) − p + β ρ ( L , ⋅ ) − p .

Note that α ∘ K = α − 1 p K .

Lutwak’s L _p dual Brunn-Minkowski inequality, see [12], is as follows: if p ≥ 1 and K , L ∈ S o n , then

(2.11) V ( K + ˜ p L ) − p n ≥ V ( K ) − p n + V ( L ) − p n ,

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

2.2 Radial and reverse radial Gauss images

In the following, we state some necessary facts with regard to the radial and reverse radial Gauss images, cf. [41,42].

For K ∈ K o n and v ∈ S n − 1 , the set

(2.12) H K ( v ) = { x ∈ ℝ n : x ⋅ v = h K ( v ) }

is called the supporting hyperplane of K with the outer unit normal v.

The spherical image of σ ⊂ ∂ K , K ∈ K o n , is defined by

ν K ( σ ) = { v ∈ S n − 1 : x ∈ H K ( v ) for some x ∈ σ } ⊂ S n − 1 .

Let σ K ⊂ ∂ K be the set consisting of all x ∈ ∂ K for which the set ν K ( { x } ) , often abbreviated as ν K ( x ) , contains more than a single element. Then, ℋ n − 1 ( σ K ) = 0 based on Schneider [10, p. 84], where ℋ n − 1 is the ( n − 1 ) -dimensional Hausdorff measure on ∂ K . For x ∈ ∂ K \ σ K , ν K ( x ) has the unique element denoted by ν K ( x ) called the spherical image map of K. For convenience, we abbreviate ∂ K \ σ K as ∂ ′ K . If the integration is with respect to ℋ n − 1 , then it follows from ℋ n − 1 ( σ K ) = 0 that it will be immaterial over subsets of ∂ ′ K or ∂ K .

For K ∈ K o n and ω ⊂ S n − 1 , the radial Gauss image of ω , α K ( ω ) , is defined by

α K ( ω ) = { v ∈ S n − 1 : ρ K ( u ) u ∈ H K ( v ) for some u ∈ ω } .

Thus for u ∈ S n − 1 ,

(2.13) α K ( u ) = { v ∈ S n − 1 : ρ K ( u ) u ∈ H K ( v ) } .

Let ω K = { u ∈ S n − 1 : ρ K ( u ) u ∈ σ K } ⊂ S n − 1 . Clearly, if u ∈ S n − 1 \ ω K , then α K ( u ) contains only a single element denoted by α K ( u ) , i.e.,

α K : S n − 1 \ ω K → S n − 1 ,

which is named as the radial Gauss map of K. Combining (2.12) and (2.13), we obtain two essential facts needed: for λ > 0 ,

(2.14) α λ K = α K ;

(2.15) α − K ( u ) = − α K ( − u ) , for u ∈ S n − 1 .

Let K ∈ K o n and η ⊂ S n − 1 . The reverse radial Gauss image of η , α K ∗ ( η ) , is defined by

(2.16) α K ∗ ( η ) = { u ∈ S n − 1 : ρ K ( u ) u ∈ H K ( v ) for some v ∈ η } .

Together (2.12) with (2.16), it is easy to see that for λ > 0 ,

(2.17) α λ K ∗ = α K ∗ .

2.3 L _p dual curvature measures

The following are some key properties, shown in [42] and needed in next section, of the L _p dual curvature measures.

In [42], Lutwak et al. showed that for p , q ∈ ℝ , K ∈ K o n and Q ∈ S o n , definition (1.1) can also be written as:

(2.18) ∫ S n − 1 g ( v ) d C ˜ p , q ( K , Q , v ) = 1 n ∫ ∂ ′ K g ( ν K ( x ) ) ( x ⋅ ν K ( x ) ) 1 − p ∥ x ∥ Q q − n d ℋ n − 1 ( x ) .

The L _p dual curvature measures have the following integral representation (see [42]): if p , q ∈ ℝ , K ∈ K o n and Q ∈ S o n , then

(2.19) C ˜ p , q ( K , Q , η ) = 1 n ∫ a K ∗ ( η ) h K ( α K ( u ) ) − p ρ K q ( u ) ρ Q n − q ( u ) d u

for each Borel set η ⊆ S n − 1 . Glancing (2.14) and (2.17), it immediately follows from (2.19) that for λ > 0 ,

(2.20) C ˜ p , q ( λ K , Q , η ) = λ q − p C ˜ p , q ( K , Q , η ) .

It was shown, in [42], that if p ≠ 0 and q ≠ 0 , then for K ∈ K o n , Q ∈ S o n and all ϕ ∈ S L ( n ) ,

(2.21) C ˜ p , q ( ϕ K , ϕ Q , ⋅ ) = ϕ p ⊣ t C ˜ p , q ( K , Q , ⋅ ) .

The L _p dual curvature measures are weakly convergent (see [42]): if p , q ∈ ℝ , Q ∈ S o n and K i ∈ K o n with K i → K 0 ∈ K o n , then for each continuous function g : S n − 1 → ℝ ,

(2.22) lim i → ∞ ∫ S n − 1 g ( v ) d C ˜ p , q ( K i , Q , v ) = ∫ S n − 1 g ( v ) d C ˜ p , q ( K 0 , Q , v ) .

Moreover, they are a valuation (see [42]), i.e., if K , L ∈ K o n are such that K ∪ L ∈ K o n , then

(2.23) C ˜ p , q ( K , Q , ⋅ ) + C ˜ p , q ( L , Q , ⋅ ) = C ˜ p , q ( K ∩ L , Q , ⋅ ) + C ˜ p , q ( K ∪ L , Q , ⋅ ) .

3 General and nonsymmetric (p,q)-mixed projection bodies

In this section, we first show the relation between general and nonsymmetric (p,q)-mixed projection bodies. Then, some of their basic properties are established.

Let g ( v ) = φ τ ( u ⋅ v ) p , with p ≥ 1 , in (2.18). Then, it will be easy to see that definition (1.5) can be rewritten as:

(3.1) h Π p , q τ ( K , Q ) , u p = c n , p ( τ ) ∫ ∂ K φ τ ( u ⋅ ν K ( x ) ) p ( x ⋅ ν K ( x ) ) 1 − p ∥ x ∥ Q q − n d ℋ n − 1 ( x ) .

From the definition of Π p , q − ( K , Q ) , (1.8), (1.1) and (2.15), it follows that for p ≥ 1 , K ∈ K o n and Q ∈ S o n ,

(3.2) h Π p , q − ( K , Q ) , u p = n c n , p ∫ S n − 1 ( − u ⋅ v ) + p d C ˜ p , q ( K , Q , v ) , u ∈ S n − 1 .

Since for any u , v ∈ S n − 1 ,

φ τ ( u ⋅ v ) p = ( 1 + τ ) p ( u ⋅ v ) + p + ( 1 − τ ) p ( − u ⋅ v ) + p ,

by (1.5), (1.8), (3.2) and (2.8) we have

(3.3) Π p , q τ ( K , Q ) = f 1 ( τ ) ⋅ Π p , q + ( K , Q ) + p f 2 ( τ ) ⋅ Π p , q − ( K , Q ) ,

where

f 1 ( τ ) = ( 1 + τ ) p ( 1 + τ ) p + ( 1 − τ ) p and f 2 ( τ ) = ( 1 − τ ) p ( 1 + τ ) p + ( 1 − τ ) p .

Note that

f 1 ( τ ) + f 2 ( τ ) = 1 ,

f 1 ( − τ ) = f 2 ( τ ) and f 2 ( − τ ) = f 1 ( τ ) .

From (3.3), we see that if τ = ± 1 , then

(3.4) Π p , q + 1 ( K , Q ) = Π p , q + ( K , Q ) and Π p , q − 1 ( K , Q ) = Π p , q − ( K , Q ) ;

if τ = 0 , then

(3.5) Π p , q ( K , Q ) = 1 2 ⋅ Π p , q + ( K , Q ) + p 1 2 ⋅ Π p , q − ( K , Q ) .

We first show the property of Π p , q ± .

Proposition 3.1

Suppose p ≥ 1 , q ∈ ℝ . If K ∈ K o n while Q ∈ S o n , then

(3.6) − Π p , q + ( K , Q ) = Π p , q − ( K , Q ) = Π p , q + ( − K , − Q ) .

Proof

We just need to prove the left equality. From (3.2) and (1.8), it follows that for any u ∈ S n − 1 ,

h Π p , q − ( K , Q ) , u p = n c n , p ∫ S n − 1 ( − u ⋅ v ) + p d C ˜ p , q ( K , Q , v ) = h Π p , q + ( K , Q ) , − u p = h − Π p , q + ( K , Q ) , u p ,

i.e.,

Π p , q − ( K , Q ) = − Π p , q + ( K , Q ) .

This finishes the proof.□

The following is to show the properties of Π p , q τ .

Proposition 3.2

Suppose p ≥ 1 , q ∈ ℝ , K ∈ K o n while Q ∈ S o n .

If τ ∈ [ − 1 , 1 ] , then
(3.7) Π p , q τ ( − K , − Q ) = Π p , q − τ ( K , Q ) = − Π p , q τ ( K , Q ) ;
If τ ≠ 0 , then

(3.8) Π p , q τ ( K , Q ) = Π p , q − τ ( K , Q ) ⇔ Π p , q + ( K , Q ) = Π p , q − ( K , Q ) .

Proof

By (3.3) and (3.6), we have
(3.9) Π p , q − τ ( K , Q ) = f 2 ( τ ) ⋅ Π p , q + ( K , Q ) + p f 1 ( τ ) ⋅ Π p , q − ( K , Q ) = f 2 ( τ ) ⋅ Π p , q − ( − K , − Q ) + p f 1 ( τ ) ⋅ Π p , q + ( − K , − Q ) = Π p , q τ ( − K , − Q ) .
Moreover, from (3.3), (2.8), (3.6) and (2.8) again, we get that for any u ∈ S n − 1 ,
h Π p , q − τ ( K , Q ) , u p = f 2 ( τ ) h Π p , q + ( K , Q ) , u p + f 1 ( τ ) h Π p , q − ( K , Q ) , u p = f 2 ( τ ) h Π p , q − ( K , Q ) , − u p + f 1 ( τ ) h Π p , q + ( K , Q ) , − u p = h f 1 ( τ ) ⋅ Π p , q + ( K , Q ) + p f 2 ( τ ) ⋅ Π p , q − ( K , Q ) , − u p = h − Π p , q τ ( K , Q ) , u p ,
i.e.,
(3.10) Π p , q − τ ( K , Q ) = − Π p , q τ ( K , Q ) .
Together (3.9) with (3.10) gives
Π p , q τ ( − K , − Q ) = Π p , q − τ ( K , Q ) = − Π p , q τ ( K , Q ) .
Let τ ≠ 0 and Π p , q + ( K , Q ) = Π p , q − ( K , Q ) . We know that for any u ∈ S n − 1 ,

(3.11) h Π p , q τ ( K , Q ) , u p = f 1 ( τ ) h Π p , q + ( K , Q ) , u p + f 2 ( τ ) h Π p , q − ( K , Q ) , u p

and

(3.12) h Π p , q − τ ( K , Q ) , u p = f 2 ( τ ) h Π p , q + ( K , Q ) , u p + f 1 ( τ ) h Π p , q − ( K , Q ) , u p .

Together (3.11) with (3.12), it follows that

Π p , q τ ( K , Q ) = Π p , q − τ ( K , Q ) .

Moreover, let Π p , q τ ( K , Q ) = Π p , q − τ ( K , Q ) . Combining (3.11) with (3.12), we have

(3.13) [ f 1 ( τ ) − f 2 ( τ ) ] h Π p , q + ( K , Q ) , u p = [ f 1 ( τ ) − f 2 ( τ ) ] h Π p , q − ( K , Q ) , u p .

Since f 1 ( τ ) − f 2 ( τ ) ≠ 0 when τ ≠ 0 , it follows from (3.13) that

Π p , q + ( K , Q ) = Π p , q − ( K , Q ) .

□Next, we will show that the operator Π p , q τ ( ⋅ , ⋅ ) is S L ( n ) contravariant and homogeneous of q / p − 1 with respect to the first part.

Proposition 3.3

Suppose p ≥ 1 , q ∈ ℝ , τ ∈ [ − 1 , 1 ] , K ∈ K o n while Q ∈ S o n .

For all ϕ ∈ S L ( n ) ,
(3.14) Π p , q τ ( ϕ K , ϕ Q ) = ϕ − t Π p , q τ ( K , Q ) ;
For λ > 0 ,

(3.15) Π p , q τ ( λ K , Q ) = λ q / p − 1 Π p , q τ ( K , Q ) .

Proof

From (1.5), (2.21), (2.5), (1.5) again and (2.1), we have that for any u ∈ S n − 1 and all ϕ ∈ S L ( n ) ,
h Π p , q τ ( ϕ K , ϕ Q ) , u p = n c n , p ( τ ) ∫ S n − 1 φ τ ( u ⋅ v ) p d C ˜ p , q ( ϕ K , ϕ Q , v ) = n c n , p ( τ ) ∫ S n − 1 φ τ ( u ⋅ v ) p d ϕ p ⊣ t C ˜ p , q ( K , Q , v ) = n c n , p ( τ ) ∫ S n − 1 | ϕ − t v | p φ τ ( u ⋅ 〈 ϕ − t v 〉 ) p d C ˜ p , q ( K , Q , v ) = n c n , p ( τ ) ∫ S n − 1 φ τ ( u ⋅ ϕ − t v ) p d C ˜ p , q ( K , Q , v ) = n c n , p ( τ ) ∫ S n − 1 φ τ ( ϕ − 1 u ⋅ v ) p d C ˜ p , q ( K , Q , v ) = h Π p , q τ ( K , Q ) , ϕ − 1 u p = h ϕ − t Π p , q τ ( K , Q ) , u p ,
i.e.,
Π p , q τ ( ϕ K , ϕ Q ) = ϕ − t Π p , q τ ( K , Q ) .
By (1.5) and (2.20), it follows that for any u ∈ S n − 1 and λ > 0 ,

h Π p , q τ ( λ K , Q ) , u p = n c n , p ( τ ) ∫ S n − 1 φ τ ( u ⋅ v ) p d C ˜ p , q ( λ K , Q , v ) = λ q − p h Π p , q τ ( K , Q ) , u p = h λ q / p − 1 Π p , q τ ( K , Q ) , u p .

That is,

Π p , q τ ( λ K , Q ) = λ q / p − 1 Π p , q τ ( K , Q ) .

For a fixed Q ∈ S o n , we will prove that Π p , q τ ( ⋅ , Q ) : K o n → K o n is continuous.□

Proposition 3.4

Suppose p ≥ 1 , q ∈ ℝ and Q ∈ S o n . If K i ∈ K o n with K i → K 0 ∈ K o n , then for every τ ∈ [ − 1 , 1 ] ,

(3.16) Π p , q τ ( K i , Q ) → Π p , q τ ( K 0 , Q ) .

Proof

Suppose u 0 ∈ S n − 1 . From (1.5) and (2.22), we get for a fixed Q ∈ S o n ,

h Π p , q τ ( K i , Q ) , u 0 → h Π p , q τ ( K 0 , Q ) , u 0 , as i → ∞ .

The support functions h Π p , q τ ( K i , Q ) → h Π p , q τ ( K 0 , Q ) pointwise on S n − 1 imply that they converge uniformly (see [10, Theorem 1.8.15]). This finishes the proof.□

An operator Z : K o n → K o n is called an L _p Minkowski valuation if

Z K 1 + p Z K 2 = Z ( K 1 ∪ K 2 ) + p Z ( K 1 ∩ K 2 ) ,

whenever K 1 , K 2 , K 1 ∪ K 2 ∈ K o n .

The theory of real valued valuations lies at the very core of geometry, see, e.g., [49,50]. In the 1970s, Schneider [51] first obtained the results on a special class of Minkowski valuations. Recently, since the seminal work of Ludwig [3,21], the investigations of these Minkowski valuations have become the focus of increased attention, see, e.g., [52,53,54,55,56,57,58,59].

An immediate result of (2.23) is that for a fixed Q ∈ S o n , Π p , q τ ( ⋅ , Q ) : K o n → K o n is an L _p Minkowski valuation.

Proposition 3.5

Suppose, q ∈ ℝ and Q ∈ S o n . Then, for every τ ∈ [ − 1 , 1 ] ,

Π p , q τ ( ⋅ , Q ) : K o n → K o n

is an L _p Minkowski valuation, i.e., if K , L ∈ K o n are such that K ∪ L ∈ K o n , then

Π p , q τ ( K , Q ) + p Π p , q τ ( L , Q ) = Π p , q τ ( K ∪ L , Q ) + p Π p , q τ ( K ∩ L , Q ) .

4 Proofs of Theorems 1.1–1.2

This section is dedicated to give the proofs of Theorems 1.1–1.2.

Proof of Theorem 1.1

Suppose that both K and Q are not origin-symmetric (otherwise the result is trivial). We first show the right inequality. Let − 1 < τ < 1 . From (3.3), (2.8), (2.6) and (2.10), it follows that

(4.1) Π p , q τ , ∗ ( K , Q ) = f 1 ( τ ) ∘ Π p , q + , ∗ ( K , Q ) + ˜ p f 2 ( τ ) ∘ Π p , q − , ∗ ( K , Q ) .

Using the L _p dual Brunn-Minkowski inequality (2.11), (3.6) and (2.4), and note that α ∘ K = α − 1 p K , we have

V Π p , q τ , ∗ ( K , Q ) ≤ V Π p , q ± , ∗ ( K , Q ) ,

with equality if and only if Π p , q + , ∗ ( K , Q ) and Π p , q − , ∗ ( K , Q ) are dilates, which is equivalent to Π p , q + ( K , Q ) = Π p , q − ( K , Q ) . Since Π p , q − ( K , Q ) = − Π p , q + ( K , Q ) and Π p , q + ( K , Q ) = − Π p , q − ( K , Q ) , Π p , q + ( K , Q ) = Π p , q − ( K , Q ) implies that both Π p , q + ( K , Q ) and Π p , q − ( K , Q ) are origin-symmetric and vice versa.

Next, we prove the left inequality. Let τ ≠ 0 . From (4.1) and (2.10), it follows that for any u ∈ S n − 1 ,

(4.2) ρ Π p , q τ , ∗ ( K , Q ) , u − p = f 1 ( τ ) ρ Π p , q + , ∗ ( K , Q ) , u − p + f 2 ( τ ) ρ Π p , q − , ∗ ( K , Q ) , u − p

and

(4.3) ρ Π p , q − τ , ∗ ( K , Q ) , u − p = f 2 ( τ ) ρ Π p , q + , ∗ ( K , Q ) , u − p + f 1 ( τ ) ρ Π p , q − , ∗ ( K , Q ) , u − p .

Combining (4.2) and (4.3), we get

1 2 ρ Π p , q τ , ∗ ( K , Q ) , u − p + 1 2 ρ Π p , q − τ , ∗ ( K , Q ) , u − p = 1 2 ρ Π p , q + , ∗ ( K , Q ) , u − p + 1 2 ρ Π p , q − , ∗ ( K , Q ) , u − p .

This together (2.10), (2.8) with (3.5) yields

(4.4) Π p , q ∗ ( K , Q ) = 1 2 ∘ Π p , q τ , ∗ ( K , Q ) + ˜ p 1 2 ∘ Π p , q − τ , ∗ ( K , Q ) .

By inequalities (2.11), (3.7) and (2.4), this has

V Π p , q ∗ ( K , Q ) ≤ V Π p , q τ , ∗ ( K , Q ) ,

with equality if and only if Π p , q τ , ∗ ( K , Q ) and Π p , q − τ , ∗ ( K , Q ) are dilates, which is equivalent to Π p , q τ ( K , Q ) = Π p , q − τ ( K , Q ) . From (3.7), Π p , q τ ( K , Q ) = Π p , q − τ ( K , Q ) yields that Π p , q τ ( K , Q ) is origin-symmetric and vice versa.□

We will use the following consequence (see [22]).

Lemma 5.1

If K ∈ K o n and p is not an odd integer, then

Π p + K = Π p − K ⇔ K i s o r i g i n ‐ s y m m e t r i c .

Here, Π p ± K denote the nonsymmetric L _p projection bodies (see [22]).

Let Q = K or q = n in Theorem 1.1. Then, we immediately have the following result given by Haberl and Schuster [22], where the equality conditions follow from Lemma 5.1 and the case Q = K or q = n of (3.8).

Corollary 5.1

[22] For p ≥ 1 , τ ∈ [ − 1 , 1 ] and K ∈ K o n ,

V Π p ∗ K ≤ V Π p τ , ∗ K ≤ V Π p ± , ∗ K .

If K is not origin-symmetric and p is not an odd integer, equality holds in the left inequality if and only if τ = 0 and equality holds in the right inequality if and only if τ = ± 1 .

Proof of Theorem 1.2

Assume that both K and Q are not origin-symmetric. First, we deduce the right inequality. Let − 1 < τ < 1 . Then it follows from (3.3), the L _p Brunn-Minkowski inequality (2.9) and the fact that α ⋅ K = α 1 p K that

V Π p , q τ ( K , Q ) ≥ V Π p , q ± ( K , Q ) ,

with equality if and only if Π p , q + ( K , Q ) and Π p , q − ( K , Q ) are dilates, which is equivalent to Π p , q + ( K , Q ) = Π p , q − ( K , Q ) , i.e., Π p , q ± ( K , Q ) are origin-symmetric.

In order to see the left inequality, we suppose τ ≠ 0 . From (4.4), (2.6) and (2.8), we have

Π p , q ( K , Q ) = 1 2 ⋅ Π p , q τ ( K , Q ) + p 1 2 ⋅ Π p , q − τ ( K , Q ) .

Using inequality (2.9) and equality (3.7), this gets

V Π p , q ( K , Q ) ≥ V Π p , q τ ( K , Q ) ,

with equality if and only if Π p , q τ ( K , Q ) and Π p , q − τ ( K , Q ) are dilates, which is equivalent to Π p , q τ ( K , Q ) = Π p , q − τ ( K , Q ) . That is, Π p , q τ ( K , Q ) is origin-symmetric.

As a special case of Theorem 1.2, the following is a direct result. Note that the proof of the equality conditions is similar to that of Corollary 5.1.

Corollary 5.2

[22] For p ≥ 1 , τ ∈ [ − 1 , 1 ] and K ∈ K o n ,

V Π p K ≥ V Π p τ K ≥ V Π p ± K .

If K is not origin-symmetric and p is not an odd integer, equality holds in the left inequality if and only if τ = 0 and equality holds in the right inequality if and only if τ = ± 1 .

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11901346) and Innovation Foundation of Institutions of Higher Learning of Gansu (Grant No. 2020A-108). The authors want to express earnest thankfulness for the referees who provided extremely precious and helpful comments and suggestions.

References

[1] J. Abardia and A. Bernig, Projection bodies in complex vector spaces, Adv. Math. 227 (2011), 830–846.10.1016/j.aim.2011.02.013Search in Google Scholar

[2] E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. Lond. Math. Soc. 78 (1999), 77–115.10.1112/S0024611599001653Search in Google Scholar

[3] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158–168.10.1016/S0001-8708(02)00021-XSearch in Google Scholar

[4] E. Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (1985), no. 1, 91–105.10.1090/S0002-9947-1985-0766208-7Search in Google Scholar

[5] E. Lutwak, Selected affine isoperimetric inequalities, in: P. M. Gruber and J. M. Wills (eds.), Handbook of Convex Geometry, vol. A, North-Holland, Amsterdam, 1993, pp. 151–176.10.1016/B978-0-444-89596-7.50010-9Search in Google Scholar

[6] E. Lutwak, Inequalities for mixed projection bodies, Trans. Amer. Math. Soc. 339 (1993), no. 2, 901–916.10.1090/S0002-9947-1993-1124171-8Search in Google Scholar

[7] C. M. Petty, Isoperimetric problems, Proc. Conf. on Convexity and Combinatorial Geometry (Univ. Oklahoma, 1971), University of Oklahoma, 1972, pp. 26–41.Search in Google Scholar

[8] F. E. Schuster, Volume inequalities and additive maps of convex bodies, Mathematica 53 (2006), 211–234.10.1112/S0025579300000103Search in Google Scholar

[9] G. Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991), 213–222.10.1007/BF00182294Search in Google Scholar

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

[11] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131–150.10.4310/jdg/1214454097Search in Google Scholar

[12] E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294.10.1006/aima.1996.0022Search in Google Scholar

[13] W. Chen, Lp Minkowski problem with not necessarily positive data, Adv. Math. 201 (2006), 77–89.10.1016/j.aim.2004.11.007Search in Google Scholar

[14] K. Chou and X. Wang, The Lp Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006), 33–83.10.1016/j.aim.2005.07.004Search in Google Scholar

[15] B. Fleury, O. Guédon, and G. Paouris, A stability result for mean width of Lp centroid bodies, Adv. Math. 214 (2007), 865–877.10.1016/j.aim.2007.03.008Search in Google Scholar

[16] Y. Huang, E. Lutwak, D. Yang, and G. Zhang, The Lp Alexandrov problem for the Lp integral curvature, J. Differential Geom. 110 (2018), no. 1, 1–29, 10.4310/jdg/1536285625.Search in Google Scholar

[17] E. Lutwak, D. Yang, and G. Zhang, Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111–132.10.4310/jdg/1090347527Search in Google Scholar

[18] E. Lutwak, D. Yang, and G. Zhang, On the Lp Minkowski problem, Trans. Amer. Math. Soc. 356 (2004), 4359–4370.10.1090/S0002-9947-03-03403-2Search in Google Scholar

[19] E. Lutwak, D. Yang, and G. Zhang, Lp John ellipsoids, Proc. Lond. Math. Soc. 90 (2005), 497–520.10.1112/S0024611504014996Search in Google Scholar

[20] E. Werner and D. Ye, New Lp affine isoperimetric inequalities, Adv. Math. 218 (2008), 762–780.10.1016/j.aim.2008.02.002Search in Google Scholar

[21] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191–4213.10.1090/S0002-9947-04-03666-9Search in Google Scholar

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

[23] S. Campi and P. Gronchi, The Lp Busemann-Petty centroid inequality, Adv. Math. 167 (2002), 128–141.10.1006/aima.2001.2036Search in Google Scholar

[24] A. Cianchi, E. Lutwak, D. Yang, and G. Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), 419–436.10.1007/s00526-009-0235-4Search in Google Scholar

[25] C. Haberl and F. Schuster, Asymmetric affine Lp Sobolev inequalities, J. Funct. Anal. 257 (2009), 641–658.10.1016/j.jfa.2009.04.009Search in Google Scholar

[26] E. Lutwak, D. Yang, and G. Zhang, Sharp affine Lp Sobolev inequalities, J. Differential Geom. 62 (2002), 17–38.10.4310/jdg/1090425527Search in Google Scholar

[27] E. Lutwak, D. Yang, and G. Zhang, Optimal Sobolev norms and the Lp Minkowski problem, Int. Math. Res. Not. IMRN 2006 (2006), 62987, 10.1155/IMRN/2006/62987.Search in Google Scholar

[28] E. Lutwak, D. Yang, and G. Zhang, Orlicz projection bodies, Adv. Math. 223 (2010), 220–242.10.1016/j.aim.2009.08.002Search in Google Scholar

[29] C. A. Rogers and G. C. Shephard, Some extremal problems for convex bodies, Mathematika 5 (1958), 93–102.10.1112/S0025579300001418Search in Google Scholar

[30] G. C. Shephard, Shadow systems of convex sets, Israel J. Math. 2 (1964), 229–236.10.1007/BF02759738Search in Google Scholar

[31] E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531–538.10.2140/pjm.1975.58.531Search in Google Scholar

[32] A. Bernig, The isoperimetrix in the dual Brunn-Minkowski theory, Adv. Math. 254 (2014), 1–14, 10.1016/j.aim.2013.12.020.Search in Google Scholar

[33] P. Dulio, R. J. Gardner, and C. Peri, Characterizing the dual mixed volume via additive functionals, Indiana Univ. Math. J. 65 (2016), no. 1, 69–91.10.1512/iumj.2016.65.5765Search in Google Scholar

[34] A. Fish, F. Nazarov, D. Ryabogin, and A. Zvavitch, The unit ball is an attractor of the intersection body operator, Adv. Math. 226 (2011), 2629–2642.10.1016/j.aim.2010.07.018Search in Google Scholar

[35] R. J. Gardner, The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities, Adv. Math. 216 (2007), 358–386.10.1016/j.aim.2007.05.018Search in Google Scholar

[36] R. J. Gardner and A. Zvavitch, Gaussian Brunn-Minkowski inequalities, Trans. Amer. Math. Soc. 362 (2010), 5333–5353.10.1090/S0002-9947-2010-04891-3Search in Google Scholar

[37] R. J. Gardner, D. Hug, and W. Weil, Operations between sets in geometry, J. Eur. Math. Soc. (JEMS) 15 (2013), 2297–2352.10.4171/JEMS/422Search in Google Scholar

[38] C. H. Jiménez and I. Villanueva, Characterization of dual mixed volumes via polymeasures, J. Math. Anal. Appl. 426 (2015), no. 2, 688–699.10.1016/j.jmaa.2015.01.068Search in Google Scholar

[39] A. Koldobsky, G. Paouris, and M. Zymonopoulou, Isomorphic properties of intersection bodies, J. Funct. Anal. 261 (2011), 2697–2716.10.1016/j.jfa.2011.07.011Search in Google Scholar

[40] E. Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006), 566–598.10.1016/j.aim.2005.09.008Search in Google Scholar

[41] Y. Huang, E. Lutwak, D. Yang, and G. Zhang, Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems, Acta Math. 216 (2016), no. 2, 325–388.10.1007/s11511-016-0140-6Search in Google Scholar

[42] E. Lutwak, D. Yang, and G. Zhang, Lp dual curvature measures, Adv. Math. 329 (2018), 85–132.10.1016/j.aim.2018.02.011Search in Google Scholar

[43] K. J. Böröczky and F. Fodor, The Lp dual Minkowski problem for p > 1 and q > 0, J. Differential Equations 266 (2019), 7980–8033.10.1016/j.jde.2018.12.020Search in Google Scholar

[44] C. Chen, Y. Huang, and Y. Zhao, Smooth solutions to the Lp dual Minkowski problem, Math. Ann. 373 (2019), no. 3, 953–976.10.1007/s00208-018-1727-3Search in Google Scholar

[45] R. J. Gardner, D. Hug, W. Weil, S. Xing, and D. Ye, General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differential Equations 58 (2019), 12, 10.1007/s00526-018-1449-0.Search in Google Scholar

[46] Y. Huang and Y. Zhao, On the Lp dual Minkowski problem, Adv. Math. 332 (2018), 57–84.10.1016/j.aim.2018.05.002Search in Google Scholar

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

[48] W. J. Firey, p-Means of convex bodies, Math. Scand. 10 (1962), 17–24.10.7146/math.scand.a-10510Search in Google Scholar

[49] D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge Univ. Press, Cambridge, 1997.Search in Google Scholar

[50] P. McMullen, Valuations and dissections, in: P. M. Gruber and J. M. Wills (eds.), Handbook of Convex Geometry, Vol. B, North-Holland, Amsterdam, 1993, pp. 933–990.10.1016/B978-0-444-89597-4.50010-XSearch in Google Scholar

[51] R. Schneider, Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc. 194 (1974), 53–78.10.1090/S0002-9947-1974-0353147-1Search in Google Scholar

[52] M. Kiderlen, Blaschke- and Minkowski-endomorphisms of convex bodies, Trans. Amer. Math. Soc. 358 (2006), 5539–5564.10.1090/S0002-9947-06-03914-6Search in Google Scholar

[53] L. Parapatits, SL(n)-covariant Lp Minkowski valuations, J. Lond. Math. Soc. 89 (2014), 397–414.10.1112/jlms/jdt068Search in Google Scholar

[54] L. Parapatits, SL(n)-contravariant Lp Minkowski valuations, Trans. Amer. Math. Soc. 366 (2014), 1195–1211.10.1090/S0002-9947-2013-05750-9Search in Google Scholar

[55] L. Parapatits and F. E. Schuster, The Steiner formula for Minkowski valuations, Adv. Math. 230 (2012), 978–994.10.1016/j.aim.2012.03.024Search in Google Scholar PubMed PubMed Central

[56] R. Schneider and F. E. Schuster, Rotation equivariant Minkowski valuations, Int. Math. Res. Not. IMRN 2006 (2006), no. 9, 72894, 10.1155/IMRN/2006/72894.Search in Google Scholar

[57] F. E. Schuster, Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), no. 1, 1–30, 10.1215/00127094-2010-033.Search in Google Scholar

[58] F. E. Schuster and T. Wannerer, GL(n) contravariant Minkwoski valuations, Trans. Amer. Math. Soc. 364 (2012), 815–826.10.1090/S0002-9947-2011-05364-XSearch in Google Scholar

[59] F. E. Schuster and T. Wannerer, Even Minkowski valuations, Amer. J. Math. 137 (2015), 1651–1683.10.1353/ajm.2015.0041Search in Google Scholar

Received: 2019-12-09

Revised: 2020-06-25

Accepted: 2020-07-01

Published Online: 2020-09-15

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

Articles in the same Issue

https://doi.org/10.1515/math-2020-0055

Keywords for this article

projection bodies; general (p,q)-mixed projection bodies; extreme value

Creative Commons

BY 4.0

General (p,q)-mixed projection bodies

Article

Abstract

1 Introduction

Definition 1.1

Theorem 1.1

Theorem 1.2

2 Preliminaries

2.1 Basics regarding convex and star bodies

2.2 Radial and reverse radial Gauss images

2.3 L p dual curvature measures

3 General and nonsymmetric (p,q)-mixed projection bodies

Proposition 3.1

Proof

Proposition 3.2

Proof

Proposition 3.3

Proof

Proposition 3.4

Proof

Proposition 3.5

4 Proofs of Theorems 1.1–1.2

Proof of Theorem 1.1

Lemma 5.1

Corollary 5.1

Proof of Theorem 1.2

Corollary 5.2

Acknowledgments

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

2.3 L _p dual curvature measures