## Abstract

Abardia and Bernig proposed the notions of complex projection body and complex mixed projection body. In this paper, we introduce the concepts of the general complex

## 1 Introduction

The classical Brunn-Minkowski theory appeared at the turn of the nineteenth into the twentieth century. One of the core concepts that Minkowski introduced within the Brunn-Minkowski theory is the projection body. There are important inequalities involving the volume of the projection body and its polar, such as Petty projection inequality [1] and Zhang projection inequality [2].

In the early 1960s, Firey [3] introduced the Firey-Minkowski

A mixed projection body was introduced in the classic volume of Bonnesen-Fenchel [9]. In [10] and [11], Lutwak considered the volume of the mixed projection body and established the classical mixed volume inequalities, such as Aleksandrov-Fenchel inequalities and Brunn-Minkowski inequalities.

Let us mention that the projection bodies described above are all real. The theory of the real projection body continues to be a very active field now. For additional information and some results on real projection body see, e.g., [8,12, 13,14,15, 16,17]. However, some classical concepts of convex geometry in real vector space were extended to complex cases, such as complex difference body [18], complex intersection body [19], complex centroid body [20,21], and complex projection body [22,23, 24,25].

In this paper, we mainly study the projection body in complex vector space. Let

In 2011, the complex

for every

Very recently, the concept of asymmetric complex

Based on the fact that

for all

for all

Motivated by the works of Abardia and Bernig [22], Haberl [20], and Liu and Wang [24], we introduce more general definitions of complex

## Definition 1.1

If

for every

and

We use

and the complex

It is clear that if

## Definition 1.2

If

for every

Moreover, if

Before stating our main results, let us introduce the

where

Our main results can be stated as the following Theorems 1.1–1.4 and among them, Theorems 1.1–1.2 are the Brunn-Minkowski-type inequalities for the general complex

## Theorem 1.1

*If*
*and*
*is an asymmetric*
*zonoid, then*

*with equality if and only if*
*and*
*are real dilates*.

## Theorem 1.2

*If*
*and*
*is an asymmetric*
*zonoid, then*

*with equality if and only if*
*and*
*are real dilates*.

## Theorem 1.3

*If*
*and*
*is an asymmetric*
*zonoid, then*

*If*
*is an ellipsoid centered at the origin or an Hermitian ellipsoid, the equality holds*.

## Theorem 1.4

*If*
*and*
*is an asymmetric*
*zonoid. Let*
*If*
*then*

*If*
*is an ellipsoid centered at the origin or an Hermitian ellipsoid, the equality holds*.

## Remark 1.1

Note that the cases of

If

## 2 Preliminaries

### 2.1 Support function, radial function, and polar of convex body

For a complex number

where

We collect complex reformulations of well-known results from convex geometry. These complex version can be directly deduced from their real counterparts by an appropriate application of

A convex body

For every Borel set

where

If

If

### 2.2 The
L
p
mixed quermassintegrals

For

where

If

The extension of the Brunn-Minkowski inequality (see [4]) is as follows: If

with equality if and only if

If

for all

In view of the

with equality for

with equality if and only if

### 2.3 The dual
L
p
mixed quermassintegrals

For

where

If

For

The dual

with equality if and only if

### 2.4 The general complex
L
p
projection body and complex
L
p
mixed projection body

Since the integral representations of the general complex

First of all, by combining (1.3) and (1.4), we obtain the integral representation of the general complex

for all

In order to give the integral representation of

### Definition 2.1

For

for each Borel

Let us mention that

If

Next, with respect to (1.7) and (2.10), we have the following integral representation of the general complex

for all

## 3 Proofs of main results

In this section, we give the proofs of Theorems 1.1–1.4. First, the proof of Theorem 1.1 needs the following Lemma 3.1.

## Lemma 3.1

*Let*
*and*
*be an asymmetric*
*zonoid. If*
*and*
*then for all*
*we have*

## Proof

From (2.3), (1.4), and the conjugate linear of Hermitian inner product, we know that for all

where

Now, we are in a position to prove Theorem 1.1.

## Proof of Theorem 1.1

Since

According to Lemma 3.1, for all

By (2.4), we have

with equality if and only if

Taking

with equality if and only

The following lemma provides a connection of

## Lemma 3.2

*Let*
*and*
*be an asymmetric*
*zonoid. If*
*and*
*then for all*
*we have*

*where*
*is the general complex*
*moment body* [24]. *If*
*we write*
*as*

## Proof

By (2.7), (2.9), and the conjugate linear of Hermitian inner product, we have

which ends the proof of Lemma 3.2.□

## Proof of Theorem 1.2

From (2.8), (3.2), and Lemma 3.2, we have for all

with equality if and only if

Taking

with equality if and only if

From now on, we pay attention to prove Theorems 1.3 and 1.4.

## Proof of Theorem 1.3

From (1.3), (2.11), and the Hölder’s integral inequality [32], one has

According to the equality condition of Hölder’s integral inequality, the equality holds if and only if

For each

Taking

with equality if and only if

By the extension of the Brunn-Minkowski inequality (2.2), we obtain

with equality if and only if

Combining (3.9) and (3.10), we obtain

If

After that, let

Let us turn toward the equality condition. If

Similarly, we also obtain