Inequalities on Complex L p Centroid Bodies

: Based on the notion of the complex p L centroid body, we establish Brunn-Minkowski type inequalities and monotonicity inequalities for complex p L centroid bodies in this article. Moreover, we obtain the affirmative form of Shephard type problem for the complex p L centroid bodies and its negative form.


Introduction
Let n  denote the set of convex bodies (compact, convex sets with non-empty interiors) in Euclidean space n R .For the set of convex bodies containing the origin in their interiors, the set of origin-symmetric convex bodies, we write o n  and os n  , respectively.Let V(K) denote the volume of K and 1 n S  the unit sphere.Centroid bodies are a classical notion from geometry which have attracted increasing attention in recent years [1-11] .In 1997, Lutwak and Zhang [12] introduced the concept of p L centroid body as follows: For each compact star-shaped about the origin n K  R and 1 p ∨ , the p L centroid body, p K  , of K is the origin-symmetric convex body whose support function is defined by for any . Refs. [13][14][15][16][17][18][19] had conducted a series of studies on the p L centroid body, and many scholars were attracted.The p L centroid body has got many results from these articles.Particularly, Refs.[20,21] gave the Brunn-Minkowski inequality and monotonicity inequality for the p L centroid body.Grinberg and Zhang [22] gave the Shephard problems for the p L centroid body.However, complex convex geometry has been studied in many works [23-28] .In this paper, we mainly study the complex centroid body.First, we introduce some notations in complex vector space n C .Let ( ) the set of compact convex subsets of complex vector space n C .Let  denote the set of complex star bodies, the set of complex star bodies containing the origin in their interiors, and the set of origin symmetric complex star bodies, respectively.2 1 n S  stands for the complex unit sphere.
Harberl [29] firstly proposed the complex centroid body of K and established the Busemann-Petty centroid inequality.In 2021, Wu [30] introduced the concept of the where the integration is with respect to the push forward of the Lebesgue measure under the canonical isomorphism  and as for  , it is the canonical isomorphism between n C and 2n R , i.e., In this article, associated with the definition of complex p L centroid body, we continuously study the complex p L centroid body.Let with equality if and only if K and L are real dilation.
Theorem 2 If os 1, , ( ) with equality if and only if L and K are real dilation.
Then we obtain monotonicity inequalities for complex p L centroid bodies.
with equality if and only if K=L.Theorem 4 For o 1, , ( ) Throughout this paper, we assume that dim 0 C ∨ .

Preliminaries
In this section, we collect complex reformulations of well-known results from convex geometry.These complex versions can be directly deduced from their real counterparts by an appropriate application of  .For standard reference, the readers may consult the books of Gardner [31] and Schneider [32] .

Complex Support Functions and Radial Functions
For a complex number n c C , we write c for its conjugate and c for its norm.If where "•" means the standard Hermitian inner production and continuous, K will be called a star body.Moreover, if o ( ) An application of polar coordinates to the volume of a complex star body o ( ) By (9) we have ( , ) ( , ) where 2 1 n  stands for (2n  1)-dimensional Hausdorff measure on 2n R .
In addition, the complex surface area measures are translation invariant and and each Borel set , with equality if and only if K and L are real dilation.The real p L Minkowski inequality and its proof are shown in Ref .[32] .
For o 1, , ( ) is defined by (see Ref. [33]) The polar coordinate formula for volume yields Particularly, ( , ) ( ) The integral representation (12), together with the Hӧlder inequality [34] immediately gives that 2 2 ( , ) ( ) ( ) with equality if and only if K and L are real dilation.For the real p L harmonic radial combination and real p L dual Minkowski inequality, we refer to Ref. [35].

The Complex L p Harmonic Blaschke Combination
The notion of real p L harmonic Blaschke combination was given by Lu and Leng [36] .Then, we extend real p L harmonic Blaschke combination to the complex case.

Proofs of Theorems
In this section, we will prove Theorem 1-Theorem 6.

Proof of
From ( 10) and for any o ( ) Therefore, by (11), we get Together (16) with the equality condition of ( 17), we know that the equality holds if and only if K and L are real dilation.
Proof of Theorem 2 From ( 8) and ( 16), one has Then by (12) and the inverse Minkowski's integral inequality [34] , we obtain Taking * , ( ) 19) and by (13), one yields the inequality (6).According to the equality conditions of Minkowski's integral inequalities, we see that equality holds in (19) if and only if K and L are real dilation.
Next, we turn to prove Theorem 3 and Theorem 4. Lemma 1 provides a connection of Proof From (3), ( 8), (10) and definition of p L projection body [29] , we have

Proof
From (3), ( 8) and ( 12), it easily gets which yields (21). and ( , ) ( , ) , , with equality if and only if K = L.By Lemma 1, we obtain 23) and by (11), one has with equality in the second inequality of ( 24) if and only if From Lemma 1, we see that inequalities ( 22) and ( 23) are equivalent.Thus, equality holds in (25) if and only if K = L.
Proof of Theorem 4 Since ( , ) with equality if and only if K = L. Combining ( 21) and ( 26), we obtain with equality in the second inequality of (30) with equality if and only if K = L. Now, we are dedicated to proving Theorem 5 and Theorem 6.
Proof of Theorem 5 For 1 p≥ and M  o ( ) for all Combining (30) and (31), we get Proof of Theorem 6 By (3), ( 15) and ( 16), we have Meanwhile, according to (12), ( 13) and ( 14), it yields ing to (3), we see that complex convex bodies, the set of complex convex bodies containing the origin in their interiors, and the set of origin symmetric complex convex bodies, re- are the real part and imaginary part, respectively.It is obvious to get that if

,
ellipsoid or an Hermitian ellipsoid, then the equality holds.Now we are in a position to prove Theorem 3

,
are real dilation.Thus, it follows from (24) that we have and only if K = L.
which ends the proof of Theorem 6.