General (p,q)-mixed projection bodies

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.

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 n denotes the n-dimensional Euclidean space. A convex body is a compact convex subset of n with nonempty interior. We denote by n the set of convex bodies and by o n the set of convex bodies containing the origin in their interiors. For a convex body ∈ K n , let ∂K and ( ) V K be its boundary and n-dimensional volume, respectively. The unit sphere in n will be denoted by − S n 1 . For ∈ x n , | | = ⋅ x x x denotes the Euclidean norm of x, and for ∈ { } x \ 0 n , the unit vector /| | ∈ − x x S n 1 will be abbreviated by 〈 〉 x . For all ∈ x n , the support function of ∈ K n is defined by where ⋅ x y denotes the standard inner product of x and y. The radial function, = ( ⋅) { } → ρ ρ K, : \ 0 K n , of a compact and star-shaped set, with respect to the origin, ⊂ K n , is defined by is a continuous and positively homogeneous function of degree 1, which was defined, in [42], by For a convex body ∈ K o n , its polar body * K is defined by , on − S n 1 was defined, in [42], by for each continuous → − g S : n 1 , 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 u u K K . In [42], the L p surface area measures were shown to be special cases of the L p dual curvature measures: was defined, in [22], by Now, we begin to define the general (p,q)-mixed projection bodies.
n , then for ∈ − u S n 1 , define the general (p,q)-mixed projection body, , to be the convex body whose support function is given by If we take = Q K or = q n in (1.5), then from (1.2) or (1.3) we have is the general L p projection body [22] of K, whose support function is given by p which is the L p projection body [17]. For = τ 0 and = p 1 in (1.6), the body K Π p τ is the classical projection body, K Π , 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 ( , is used to denote the polar body of Π p q τ , .

7)
Suppose that both K and Q are not origin-symmetric. If ≠ τ 0, then equality holds in the left inequality if and only if is origin-symmetric; if ≠ ± τ 1, then equality holds in the right inequality if and only if both are origin-symmetric. Here, is the nonsymmetric (p,q)-mixed projection body, which is the convex body defined by are, respectively, defined by

10)
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 ( )

11)
Suppose that both K and Q are not origin-symmetric. If ≠ τ 0, then equality holds in the left inequality if and only if is origin-symmetric; if ≠ ± τ 1, then equality holds in the right inequality if and only if both are origin-symmetric. 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.

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 ( ) SL n denote the group of special linear transformation. If ∈ ( ) ϕ SL 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 ∈ ( ) Moreover, it is easy to verify that for ∈ K o n and > c 0, If μ is a Borel measure on − S n 1 and ∈ ( ) ϕ SL n , then the L p image of μ under ϕ, ⊣ ϕ μ p , is a Borel measure defined, in [42], by The support and radial functions of a convex body ∈ K o n and its polar body are related by , 0 (both not zero), the Minkowski combination + αK βL is defined by αx βy x K y L : , whose support function is In the early 1960s, Firey [48] introduced the L p Minkowski combination, which is also known as the Minkowski-Firey combination. For ≥ p 1, In [48], Firey also established the L p Brunn-Minkowski inequality: if ≥ p 1 and ∈ K L , o n , then 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].
, is a star body whose radial function is given by Lutwak's L p dual Brunn-Minkowski inequality, see [12], is as follows: if ≥ p 1 and ∈ K L , o n , then with equality if and only if K and L are dilates.

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 o n and ∈ − v S n 1 , the set is called the supporting hyperplane of K with the outer unit normal v.
The spherical image of ⊂ ∂ σ K , ∈ K o n , is defined by for some .
for some .
Thus for ∈ − u S n 1 , The L p dual curvature measures have the following integral representation (see [42] for each Borel set ⊆ − η S n 1 . Glancing (2.14) and (2.17), it immediately follows from (2.19) that for > λ 0,

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It was shown, in [42], that if ≠ p 0 and ≠ q 0, then for ∈ Moreover, they are a valuation (see [42]), i.e., if ∈ K L , o n are such that ∪ ∈ K L o n , then

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.
Then, it will be easy to see that definition (1.5) can be rewritten as:

(3.2)
Since for any ∈ − u v S , n 1 , Note that

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

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Proof. We just need to prove the left equality. From (3.2) and (1.8), it follows that for any ∈ − u S n 1 , i.e., This finishes the proof. □ The following is to show the properties of Π p q τ , .
An immediate result of (2.23) is that for a fixed ∈ 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 ,