General (p,q)-mixed projection bodies

Yibin Feng; Yanping Zhou

doi:10.1515/math-2020-0055

Open Access Published by De Gruyter Open Access September 15, 2020

General (p,q)-mixed projection bodies

Yibin Feng and Yanping Zhou

From the journal Open Mathematics

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

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.

General (p,q)-mixed projection bodies

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

Journal and Issue

Articles in the same Issue

General (p,q)-mixed projection bodies

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

Journal and Issue

Articles in the same Issue

2.3 L _p dual curvature measures