# Extreme inequalities of general $L_{p}$ $\mu$-projection body and general
$L_{p}$ $\mu$-centroid body

Chao Li  School of Mathematics and Statistics, Ningxia University, Yinchuan,
Ningxia, 750021, China<EMAIL_ADDRESS>and  Gangyi Chen  Gangyi Chen
School of Mathematics and Information Science, Guangzhou University,
Guangzhou, 510006, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce the concept of general $L_{p}$ projection body and
general $L_{p}$ centroid body of general measures with positive homogeneity
density function, and prove the corresponding extreme inequalities. Meanwhile,
we also study their measure comparison problem and monotone inequalities.

###### Key words and phrases:

general $L_{p}$ $\mu$-projection body, general $L_{p}$ $\mu$-centroid body,
positive homogeneity measure, extreme inequality, measure comparison problem

###### 2020 Mathematics Subject Classification:

52A39; 52A40; 52A20

Corresponding author: Gangyi Chen

##  1\. Introduction and main results

Let $\mathscr{K}^{n}$ denote the set of convex bodies (compact, convex subsets
with non-empty interiors) in Euclidean space $\mathbb{R}^{n}$. Let
$\mathscr{K}^{n}_{o}$ denote the set of convex bodies containing the origin in
their interiors in $\mathbb{R}^{n}$. For the set of star bodies (about the
origin) in $\mathbb{R}^{n}$, we write $\mathscr{S}^{n}_{o}$. Besides,
$\mathbb{S}^{n-1}$ denotes the unit sphere and $V_{n}(K)$ denotes the
$n$-dimensional volume of the body $K$. The volume of the Euclidean unit ball
$B^{n}$ is given by $\kappa_{n}=\pi^{n/2}/\Gamma(1+n/2)$.

As is known to all , the projection body has always been one of the main
research contents of Brunn-Minkowski theory, which can be described as
follows: Let $K\in\mathscr{K}^{n}$, the projection body $\Pi K$ of $K$ is an
origin-symmetric convex body whose support function is defined by (see [38])

$\displaystyle h({\Pi K},u)=V_{n-1}(K|u^{\bot}),\qquad u\in\mathbb{S}^{n-1},$

where $V_{n-1}$ is the ($n-1$)-dimensional volume and $K|u^{\bot}$ represents
the image of the orthogonal projection of $K$ onto the subspace orthogonal to
$u$.

Since the concept of projection body was proposed by Minkowski, it has
attracted extensive attention of many scholars studying convex geometry.
Moreover, the Shephard problem [39] of projection body, Petty (affine)
projection inequality [32, 33], Zhang’s projection inequality [44], Schneider
conjecture of projection inequality [36] and a series of far-reaching and
meaningful research topics of projection body are well known to scholars in
convex geometry. In recent years, the concept of projection body has been
extended to $L_{p}$ projection body [25], nonsymmetric $L_{p}$ projection body
[12], Orlicz projection body [26] and complex projection body [1], which
further enriches the research content of projective body.

The research of centroid body, which has the same status as projection body,
has also attracted the attention of scholars. For $K\in\mathscr{S}^{n}_{o}$,
the centroid body $\Gamma K$ of $K$ is a convex body, whose support function
is defined by (see [38])

$\displaystyle h(\Gamma K,u)=\frac{1}{V(K)}\int_{K}|u\cdot x|dx,\qquad
u\in\mathbb{S}^{n-1}.$

In particular, the notion of centroid body is also extended to
$L_{p}$-centroid body [24], general $L_{p}$-centroid body [12, 21], Orlicz
centroid body [27], Orlicz-Lorentz centroid body [29] and complex centroid
body [11]. Moreover, the study of Busemann-Petty centroid inequality [31] and
Shephard type problem [23, 37] of centroid body has had a far-reaching impact,
see [5, 6, 8, 18, 43, 45]. We can also refer to Gardner [9] and Schneider’s
[38] two golden classics, which contain a relatively complete collection of a
large number of research results on projection body and centroid body.

In 2014, Milman and Rotem [28] studied the complementary Brunn-Minkowski
inequality and isoperimetric inequality of homogeneous and non-homogeneous
measures. Afterwards, Livshyts [20] proposed the concept of mixed $\mu$
measure and put forward a new notion which relates projections of convex
bodies to a given measure $\mu$, which is a generalization of the Lebesgue
volume of a projection.

Definition 1.A (Page 5 of [20]). Suppose $K\in\mathscr{K}^{n}$ and $\mu$ is a
measure on $\mathbb{R}^{n}$ with continuous density $\omega$. Consider a unit
vector $u\in\mathbb{S}^{n-1}$. The function on the cylinder
$\mathbb{S}^{n-1}\times[0,1]$ is defined by

$\displaystyle p_{\mu,K}(u,t):=\frac{n}{2}\int_{\mathbb{S}^{n-1}}|u\cdot
v|dS_{\mu}(tK,v).$

Then the $\mu$-projection function on the unit sphere is defined by

$\displaystyle P_{\mu,K}(u):=\int_{0}^{1}p_{\mu,K}(u,t)dt,$

where $dS_{\mu}(K,\cdot)$ denote the surface $\mu$-area measure.

In the particular case of Lebesgue measure $\mathcal{L}$,
$P_{\mathcal{L},K}(u)=V_{n-1}(K|u^{\perp})$.

Afterwords, Wu [41] extended the definition of $\mu$-projection function [20]
to $L_{p}$ cases and studied the Firey-Shephard problem for homogeneous
measure. Recently, Langhurst, Roysdon and Zvavitch [16] proved Zhang’s
inequality with respect to $h(\Pi_{\mu}K,u)=\frac{1}{n}p_{\mu,K}(u,1)$, for
$u\in\mathbb{S}^{n-1}$.

In 2022, Wu and Bu [42] introduced the following concept of the
$L_{p}$-centroid body for general measures and studied the measure-comparison
problem.

Definition 1.B (Page 3 of [42]). Let $p\geq 1$ and $\mu$ be a measure on
$\mathbb{R}^{n}$ with continuous density $\omega$. For
$K\in\mathscr{S}^{n}_{o}$, the $(p,\mu)$-centroid body $\Gamma_{\mu,p}K$ of
$K$ is the body defined by,

$\displaystyle h(\Gamma_{\mu,p}K,x)^{p}=\frac{1}{\mu(K)}\int_{K}|x\cdot
y|^{p}d\mu(y),$

for every $x\in\mathbb{R}^{n}$.

In the case of Lebesgue measure $\mathcal{L},\Gamma_{\mathcal{L},p}K$ is just
the classical $L_{p}$-centroid body $\Gamma_{p}K$.

In recent years, the research on extending some conclusions of the classic
Brunn-Minkowski theory to arbitrary measures has attracted much attention. For
more recent developments with respect to the convex body of arbitrary
measures, also see [2, 13, 14, 15, 19, 30, 34, 35, 46].

In this paper, we apply the method of Haberl and Schuster [12], and first put
forward the following concept of the projection body of general measures to
the asymmetric case.

Let $p\geq 1$ and $\mu$ be a $1/s$ homogeneous Borel measure on
$\mathbb{R}^{n}$ with $1/r$ homogeneous density $\omega$, where $r>0$ and $s$
satisfy the equation $s=\frac{1}{n+\frac{1}{r}}$. For
$K\in\mathscr{K}_{\mathrm{o}}^{n}$ and function
$\psi_{\tau}:\mathbb{R}\rightarrow[0,\infty)$ defined by
$\psi_{\tau}(t)=|t|+\tau t$, $\tau\in[-1,1]$, the general $L_{p}$
$\mu$-projection body $\Pi_{\mu,p}^{\tau}K\in\mathscr{K}_{\mathrm{o}}^{n}$ of
$K$ is defined by (see Section 3)

$\displaystyle
h(\Pi_{\mu,p}^{\tau}K,u)^{p}=c_{n,p}(\tau)\int_{\mathbb{S}^{n-1}}\psi_{\tau}(u\cdot
v)^{p}dS_{\mu,p}(K,v),$

where $u\in\mathbb{S}^{n-1}$ and $dS_{\mu,p}(K,\cdot)$ denotes the
$L_{p}$-surface $\mu$-area measure of $K$. In the case of Lebesgue measure
$\mathcal{L}$, $\Pi_{\mathcal{L},p}^{\tau}K$ is just the general
$L_{p}$-projection body $\Pi_{p}^{\tau}K$ defined by Haberl and Schuster [12].

Suppose $\mu$ is a $1/s$ homogeneous Borel measure on $\mathbb{R}^{n}$ with
$1/r$ homogeneous density $\omega$, where $r>0$ and $s$ satisfy the equation
$s=\frac{1}{n+\frac{1}{r}}$. Let $p\geq 1$ and $K\in\mathscr{S}^{n}_{o}$, for
$u\in\mathbb{S}^{n-1}$, the general $L_{p}$ $\mu$-centroid body
$\Gamma_{\mu,p}^{\tau}K\in\mathscr{K}^{n}_{\mathrm{o}}$ of $K$ is defined by
(see Section 3)

$\displaystyle h(\Gamma_{\mu,p}^{\tau}K,u)^{p}=$
$\displaystyle\frac{2}{\alpha_{n,p}(\tau)\mu(K)}\int_{K}\psi_{\tau}(u\cdot
x)^{p}d\mu(x).$

In particular, if $\mu$ is the Lebesgue measure $\mathcal{L}$, then
$\Gamma_{\mathcal{L},p}^{\tau}K$ is the general $L_{p}$-centroid body
$\Gamma_{p}^{\tau}K$ defined by Feng, Wang and Lu [7]. In addition,
$[(\alpha_{n,p}(\tau)V(K))/(2c_{n,p}(\tau))]^{\frac{1}{p}}\Gamma_{\mathcal{L},p}^{\tau}K$
is the general $L_{p}$-moment body $M^{\tau}_{p}K$ (see Page 10 of [12]).

The following are the main results of this paper. Firstly, we study the
extreme inequality of the polar body of general $L_{p}$ $\mu$-projection body.

######  Theorem 1.1.

Let $\mu$ be a $1/s$ homogeneous Borel measure on $\mathbb{R}^{n}$ with $1/r$
homogeneous density function $\omega$, and $\omega$ is even, where $r>0$ and
$s$ satisfy the equation $s=\frac{1}{n+\frac{1}{r}}$, if
$K\in\mathscr{K}^{n}_{o}$, $p\geq 1$, and $\tau\in[-1,1]$, then

$\displaystyle\mu(\Pi_{\mu,p}^{\ast}K)\leq\mu(\Pi_{\mu,p}^{\tau,\ast}K)\leq\mu(\Pi_{\mu,p}^{\pm,\ast}K),$
(1)

when $\tau\neq 0$, equality holds in the left inequality if and only if
$\Pi_{\mu,p}^{\tau}K$ is origin-symmetric, and when $\tau\neq\pm 1$, equality
holds in the right inequality if and only if $\Pi_{\mu,p}^{\pm}K$ is origin-
symmetric.

######  Remark 1.2.

If $K$ is origin-symmetric, then (1) is an identical equation.

######  Remark 1.3.

If $\mu$ is the Lebesgue measure, then Theorem 1.1 is proved by Haberl and
Schuster (See Theorem 6.3 of [12]).

Furthermore, the extreme inequality of the polar body of the general $L_{p}$
$\mu$-centroid body is also proved.

######  Theorem 1.4.

Let $\mu$ be a $1/s$ homogeneous Borel measure on $\mathbb{R}^{n}$ with $1/r$
homogeneous density function $\omega$, and $\omega$ is even, where $r>0$ and
$s$ satisfy the equation $s=\frac{1}{n+\frac{1}{r}}$, if
$K\in\mathscr{S}^{n}_{o}$, $p\geq 1$, and $\tau\in[-1,1]$, then

$\displaystyle\mu(\Gamma_{\mu,p}^{\ast}K)\leq\mu(\Gamma_{\mu,p}^{\tau,\ast}K)\leq\mu(\Gamma_{\mu,p}^{\pm,\ast}K),$
(2)

when $\tau\neq 0$, equality holds in the left inequality if and only if
$\Gamma_{\mu,p}^{\tau}K$ is origin-symmetric, and when $\tau\neq\pm 1$,
equality holds in the right inequality if and only if $\Gamma_{\mu,p}^{\pm}K$
is origin-symmetric.

######  Remark 1.5.

If $K$ is origin-symmetric, then (2) is an identical equation.

This paper is organized as follows: In Section 2, we list some basic knowledge
of the convex geometry. In Section 3, we introduce the general $L_{p}$
$\mu$-projection body and the general $L_{p}$ $\mu$-centroid body, and study
some of their basic properties. We also prove the measure comparison
inequality of general $L_{p}$ $\mu$-projection body and general $L_{p}$
$\mu$-centroid body and their polar bodies. In Section 4, We prove Theorem 1.1
and Theorem 1.4. Finally, in Section 5, we prove the Brunn-Minkowski type
inequality of general $L_{p}$ $\mu$-projection body, and also obtain the
monotone inequality of general $L_{p}$ $\mu$-centroid body.

##  2\. Preliminaries

Below we will list some basic facts of convex geometry.

###  2.1. $L_{p}$-Minkowski combination and $L_{p}$-radial combination

A convex body $K\in\mathscr{K}^{n}$, is uniquely determined by its support
function, $h(K,\cdot)=h_{K}:\mathbb{R}^{n}\rightarrow\mathbb{R}$, which is
defined by

$\displaystyle h(K,x)=\max\\{x\cdot y:y\in K\\},\ \ \ \ x\in\mathbb{R}^{n},$
(3)

where $x\cdot y$ denotes the standard inner product of $x$ and $y$ in
$\mathbb{R}^{n}$. It is also clear from the definition that $h(K,u)\leq
h(L,u)$ for $u\in\mathbb{S}^{n-1}$, if and only if $K\subseteq L$.

For $K,L\in\mathscr{K}_{o}^{n}$, $p\geq 1$ and $\alpha,\beta\geq 0$ (not both
zero), the $L_{p}$-Minkowski combination, $\alpha\cdot K+_{p}\beta\cdot L$, of
$K$ and $L$ is defined by (see [38])

$\displaystyle h(\alpha\cdot K+_{p}\beta\cdot L,\cdot)^{p}=\alpha
h(K,\cdot)^{p}+\beta h(L,\cdot)^{p},$ (4)

where $``+_{p}"$ denotes the $L_{p}$-sum and $\alpha\cdot K=\alpha^{1/p}K$ is
the $L_{p}$-Minkowski scalar multiplication. If $p=1$, (4) is just the
classical Minkowski linear combination.

If $K$ is a compact star-shaped set (about the origin) in $\mathbb{R}^{n}$,
then its radial function,
$\rho_{K}=\rho(K,\cdot):\mathbb{R}^{n}\setminus\\{0\\}\rightarrow[0,\infty)$,
is defined by

$\displaystyle\rho(K,x)=\max\\{c\geq 0:cx\in K\\},\ \
x\in\mathbb{R}^{n}\setminus\\{0\\}.$ (5)

If $\rho_{K}$ is positive and continuous, $K$ will be called a star body (with
respect to the origin). Obviously, if $\alpha>0$, then $\rho_{\alpha
K}=\alpha\rho_{K}$.

By (2.3), we easily see that if $K\subseteq L$, then

$\displaystyle\rho(K,\cdot)\leq\rho(L,\cdot),$ (6)

with equality if and only if $K=L$.

If $K\in\mathscr{K}^{n}_{o}$, the polar body $K^{\ast}$, of $K$ is defined by

$\displaystyle K^{\ast}=\\{x\in\mathbb{R}^{n}:x\cdot y\leq 1,y\in K\\}.$ (7)

From (3), (5) and (7), it follows that if $K\in\mathscr{K}^{n}_{o}$, then

$\displaystyle\rho(K^{\ast},\cdot)=\frac{1}{h(K,\cdot)},\quad\quad\quad
h(K^{*},\cdot)=\frac{1}{\rho(K,\cdot)}.$ (8)

If $K,L\in\mathscr{S}^{n}_{o}$, $p\neq 0$ and $\alpha,\beta\geq 0$ (not both
zero), the $L_{p}$-radial combination, $\alpha\circ
K\widetilde{+}_{p}\beta\circ L\in\mathscr{S}^{n}_{o}$, of $K$ and $L$ is
defined by (see [38])

$\displaystyle\rho(\alpha\circ K\widetilde{+}_{p}\beta\circ
L,\cdot)^{p}=\alpha\rho(K,\cdot)^{p}+\beta\rho(L,\cdot)^{p},$ (9)

where $\widetilde{+}_{p}$ is called the $L_{p}$-radial addition and
$\alpha\circ K=\alpha^{1/p}K$ is the $L_{p}$-radial scalar multiplication. If
$p=1$, then $\alpha\circ K\widetilde{+}\beta\circ L$ is the classical radial
combination. If $p\geq 0$, then $\alpha\circ K\widetilde{+}_{-p}\beta\circ L$
is called $L_{p}$-harmonic radial combination.

From (4), (8) and (9), for $K,L\in\mathscr{K}_{o}^{n}$, $p\geq 1$ and
$\alpha,\beta\geq 0$ (not both zero), then

$\displaystyle(\alpha\cdot K+_{p}\beta\cdot L)^{*}=\alpha\circ
K^{*}\widetilde{+}_{-p}\beta\circ L^{*}.$ (10)

###  2.2. $L_{p}$ mixed $\mu$-measure

A measurable function $\omega:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ is
called $1/r$-homogeneous, $r\in(-\infty,\infty)\backslash\\{0\\}$, if for
$x\in\mathbb{R}^{n}$ and $\lambda>0$, then (see Definition 2.6 of [28])

$\omega(\lambda x)=\lambda^{1/r}\omega(x),$

with the interpretation when $r=\infty$ that $\omega(\lambda x)=\omega(x)$ for
all $\lambda>0$, i.e. that $\omega$ is constant along rays from the origin.

A Borel measure $\mu$ on $\mathbb{R}^{n}$ is called $1/s$-homogeneous,
$s\in(-\infty,\infty)\backslash\\{0\\}$, if for a Borel subset
$\mathcal{A}\subset\mathbb{R}^{n}$, and $\lambda>0$, then (see Definition 2.7
of [28])

$\mu(\lambda\mathcal{A})=\lambda^{1/s}\mu(\mathcal{A}),$

with the interpretation when $s=\infty$ that
$\mu(\lambda\mathcal{A})=\mu(\mathcal{A})$ for all $\lambda>0$.

For convenience, we call the measure $\mu$ satisfying the homogeneity
condition as:

(I) Let $\mu$ be a $1/s$ homogeneous Borel measure on $\mathbb{R}^{n}$ with
$1/r$ homogeneous density $\omega$, where $r>0$ and $s$ satisfy the equation
$s=\frac{1}{n+\frac{1}{r}}$.

A function $\omega:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}$ is called
$r$-concave if for every $x,y\in\mathbb{R}^{n}$ such that
$\omega(x)\omega(y)>0$, and for every $\alpha\in[0,1]$, one has

$\displaystyle\omega(\alpha
x+(1-\alpha)y)\geq(\alpha\omega(x)^{r}+(1-\alpha)\omega(y)^{r})^{1/r}.$

We remark that $0$-concave functions are also called log-concave functions.

The following generalized Brunn-Minkowski inequality is well known (see [3,
4]). Let $r\in[-\frac{1}{n},+\infty]$, and let $\mu$ be a measure on
$\mathbb{R}^{n}$ with $r$-concave density $\omega$. Let

$\displaystyle\frac{1}{s}=\frac{1}{r}+n.$

Then the measure $\mu$ is $s$-concave on $\mathbb{R}^{n}$. That is, for every
pair of Borel sets $\mathcal{A}$ and $\mathcal{B}$ and for every
$\alpha\in[0,1]$ one has

$\displaystyle\mu(\alpha\mathcal{A}+(1-\alpha)\mathcal{B})\geq(\alpha\mu(\mathcal{A})^{s}+(1-\alpha)\mu(\mathcal{B})^{s})^{1/s}.$

For convenience, we call the measure $\mu$ satisfying the concavity condition
as:

(II) Let $\mu$ be a $s$-concave Borel measure on $\mathbb{R}^{n}$ with
$r$-concave density $\omega$, where $r>0$ and $s$ satisfy the equation
$s=\frac{1}{n+\frac{1}{r}}$.

If $\mu$ satisfies the homogeneity condition (I), then

$\displaystyle\mu(K)=\int_{K}d\mu(x),$

from the polar coordinate transformation, we get

$\displaystyle\mu(K)=s\int_{\mathbb{S}^{n-1}}\rho(K,u)^{\frac{1}{s}}d\mu(u).$
(11)

Let $\omega=1$ in (11), then we get

$\displaystyle V_{n}(K)=\frac{1}{n}\int_{\mathbb{S}^{n-1}}\rho(K,u)^{n}du.$

In 2017, Wu [40] extended the mixed $\mu$-measure to $L_{p}$ case and proposed
the following $L_{p}$ mixed $\mu$-measure: Let $K,L\in\mathscr{K}^{n}_{o}$,
$p\geq 1$ and $\mu$ be a measure on $\mathbb{R}^{n}$, the $L_{p}$ mixed
$\mu$-measure $\mu_{p}(K,L)$, of $K$ and $L$, is defined by

$\displaystyle\mu_{p}(K,L)=\liminf_{\varepsilon\rightarrow
0}\frac{\mu(K+_{p}\varepsilon\cdot L)-\mu(K)}{\varepsilon}.$

Moreover, Wu [40] also got the following integral expression:

$\displaystyle\mu_{p}(K,L)=\frac{1}{p}\int_{\mathbb{S}^{n-1}}h(L,u)^{p}dS_{\mu,p}(K,u).$
(12)

Here, $dS_{\mu,p}(K,\cdot)$ denotes the $L_{p}$-surface $\mu$-area measure of
$K$.

Especially, if $K$ is a $C^{2}$-smooth and strictly convex body with the
origin in its interior, then for all $u\in\mathbb{S}^{n-1}$, the $L_{p}$
surface $\mu$-area measure $S_{\mu,p}(K,u)$ has representation

$\displaystyle dS_{\mu,p}(K,u)=h_{K}^{1-p}(u)f_{K}(u)\omega\left(\nabla
h_{K}(u)\right)du.$

where $f_{K}$ denotes the curvature function of $K$.

In particular, if $p=1$, then $dS_{\mu,1}(K,\cdot)=dS_{\mu}(K,\cdot)$ is the
surface $\mu$-area measure defined by Livshyts [20], and its expression is

$\displaystyle dS_{\mu}(K,u)=f_{K}(u)\omega\left(\nabla h_{K}(u)\right)du.$

In addition, Wu [40] obtained that the measure $S_{\mu,p}(K,\cdot)$ is
absolutely continuous with respect to $S_{\mu}(K,\cdot)$, and has Radon-
Nikodym derivative,

$\displaystyle\frac{dS_{\mu,p}(K,\cdot)}{dS_{\mu}(K,\cdot)}=h_{K}^{1-p}.$

Note that $\mu_{p}(K,L)$ is not homogeneous in $K$, since $\mu$ is not
homogeneous. However, $\mu_{p}(K,L)$ is $p$-homogeneous in $L$ i.e.,
$\mu_{p}(K,\lambda L)=\lambda^{p}\mu_{p}(K,L)$, for $\lambda>0$.

Meanwhile, from $L_{p}$ mixed $\mu$-measure, Wu [40] defined the following
concept of the $L_{p}$ mixed volume with respect to $\mu$: Let $p\geq 1$,
$K,L\in\mathscr{K}^{n}_{o}$ and $\mu$ be a measure on $\mathbb{R}^{n}$, then
the $L_{p}$ mixed volume with respect to the measure $\mu$, $V_{\mu,p}(K,L)$,
of $K$ and $L$, is defined by

$\displaystyle V_{\mu,p}(K,L)=\int_{0}^{1}\mu_{p}(t\cdot K,L)dt.$ (13)

If $\mu$ satisfies the homogeneity condition (I), from (12) and (13), Wu [40]
obtained

$\displaystyle
V_{\mu,p}(K,L)=sp\mu_{p}(K,L)=s\int_{\mathbb{S}^{n-1}}h(L,u)^{p}dS_{\mu,p}(K,u).$
(14)

Obviously, if $\mu$ satisfies the homogeneity condition (I), then

$\displaystyle V_{\mu,p}(K,K)=\mu(K),\qquad\mu_{p}(K,K)=\frac{1}{sp}\mu(K).$

If $\mu$ satisfies the homogeneity condition (I), Wu [40] also deduced
$\mu_{p}(\lambda K,L)=\lambda^{\frac{1-sp}{s}}\mu_{\mathrm{p}}(K,L)$, for
$\lambda>0$, thus

$\displaystyle dS_{\mu,p}(\lambda
K,\cdot)=\lambda^{\frac{1-sp}{s}}dS_{\mu,p}(K,\cdot).$ (15)

Clearly, in the case of Lebesgue measure $\mathcal{L}$, we have

$\displaystyle V_{\mathcal{L},p}(K,L)=V_{p}(K,L).$

Note that, for $p=1,(\ref{lphmt})$ is just

$\displaystyle\mu_{1}(K,L)=\int_{\mathbb{S}^{n-1}}h_{L}(u)dS_{\mu,1}(K,u),$

which is given by Livshyts [20].

In addition, Wu [40] also proved the following Minkowski inequality for
$L_{p}$-mixed volume with respect to the measure $\mu$: Let $\mu$ satisfy the
homogeneity condition (I) and concavity condition (II), if
$K,L\in\mathscr{K}^{n}_{o}$ and $p\geq 1$, then

$\displaystyle\mu_{p}(K,L)\geq\frac{1}{s}\mu(K)^{1-s}\mu(L)^{s}+\frac{1-p}{sp}\mu(K),$
(16)

and

$\displaystyle V_{\mu,p}(K,L)\geq p\mu(K)^{1-s}\mu(L)^{s}+(1-p)\mu(K).$ (17)

######  Lemma 2.1.

Let $\mu$ satisfy the homogeneity condition (I) and concavity condition (II).
If $K,L\in\mathscr{K}^{n}_{o}$, $\alpha,\beta\geq 0$ and $p\geq 1$, then

$\displaystyle\frac{1+(\alpha+\beta)(p-1)}{p}\mu(\alpha\cdot K+_{p}\beta\cdot
L)^{s}\geq\alpha\mu(K)^{s}+\beta\mu(L)^{s}.$

###### Proof.

For $p\geq 1$, by (12), (4) and (16), we know that for any
$M\in\mathscr{K}_{o}^{n}$

$\displaystyle\mu_{p}(M,\alpha\cdot K+_{p}\beta\cdot L)=$
$\displaystyle\alpha\mu_{p}(M,K)+\beta\mu_{p}(M,L)$ $\displaystyle\geq$
$\displaystyle\frac{1}{s}\mu(M)^{1-s}[\alpha\mu(K)^{s}+\beta\mu(L)^{s}]+\frac{(\alpha+\beta)(1-p)}{sp}\mu(M),$

Let $M=\alpha\cdot K+_{p}\beta\cdot L$ in the above formula, we can deduce

$\displaystyle\frac{1+(\alpha+\beta)(p-1)}{p}\mu(\alpha\cdot K+_{p}\beta\cdot
L)^{s}\geq\alpha\mu(K)^{s}+\beta\mu(L)^{s}.$

This completes the proof. ∎

In particular, if $\alpha=\beta=1$, Lemma 2.1 is exactly Wu’s result (see Page
171 of [40]).

###  2.3. $L_{p}$ dual mixed $\mu$ measure

######  Definition 2.2 ([17]).

Let $K,L\in\mathscr{S}^{n}_{o}$, $p\neq 0$ and $r>0$. If $\mu$ satisfies the
homogeneity condition (I), then the $L_{p}$ dual mixed $\mu$ measure can be
defined by

$\displaystyle\widetilde{V}_{\mu,p}(K,L)=sp\lim_{\varepsilon\rightarrow
0^{+}}\frac{\mu\left(K\widetilde{+}_{p}\varepsilon\circ
L\right)-\mu(K)}{\varepsilon}.$

Next, we will give the integral expression of the $L_{p}$ dual mixed $\mu$
measure.

######  Lemma 2.3 ([17]).

Let $K,L\in\mathscr{S}^{n}_{o}$ and $p\neq 0$. If $\mu$ satisfies the
homogeneity condition (I), then

$\displaystyle\widetilde{V}_{\mu,p}(K,L)=s\int_{\mathbb{S}^{n-1}}\rho(K,u)^{\frac{1-sp}{s}}\rho(L,u)^{p}d\mu(u).$
(18)

Note that Lemma 2.3 is a special case of Gardner, Hug, Weil, Xing and Ye (See
(16) of [10]). By Lemma 2.3 and applying Hölder’s integral inequality and
Minkowski’s integral inequality, we will easily prove the following Minkowski
inequality and Brunn-Minkowski inequality for $L_{p}$ dual mixed $\mu$
measure.

######  Lemma 2.4 ([17]).

Let $\mu$ satisfy the homogeneity condition (I) and
$K,L\in\mathscr{S}^{n}_{o}$, if $0<p<n+\frac{1}{r}$, then

$\displaystyle\widetilde{V}_{\mu,p}(K,L)\leq\mu(K)^{1-sp}\mu(L)^{sp}.$ (19)

If $p<0$ or $p>n+\frac{1}{r}$, then inequality (19) is reversed. Equality
holds if and only if $K$ and $L$ are dilatates.

######  Lemma 2.5 ([17]).

Let $\mu$ satisfy the homogeneity condition (I) and
$K,L\in\mathscr{S}^{n}_{o}$, if $0<p<n+\frac{1}{r}$ and $\alpha,\beta>0$, then

$\displaystyle\mu(\alpha\circ K\widetilde{+}_{p}\beta\circ
L)^{sp}\leq\alpha\mu(K)^{sp}+\beta\mu(L)^{sp}.$ (20)

If $p<0$ or $p>n+\frac{1}{r}$, then inequality (20) is reversed. Equality
holds if and only if $K$ and $L$ are dilatates.

##  3\. General $L_{p}$ $\mu$-projection bodies and General $L_{p}$
$\mu$-centroid bodies

In this part, we will introduce the concepts of nonsymmetric $L_{p}$
$\mu$-projection body and nonsymmetric $L_{p}$ $\mu$-centroid body, and then
study the corresponding measure comparison problem according to their
properties.

The nonsymmetric $L_{p}$ cosine transform of $\nu$ is defined by Haberl and
Schuster [12] as follows: For each finite Borel measure $\nu$ on
$\mathbb{S}^{n-1}$, a continuous function $\mathcal{C}_{p}^{+}\nu$ on
$\mathbb{S}^{n-1}$, the nonsymmetric $L_{p}$ cosine transform of $\nu$, is
defined by

$\displaystyle(\mathcal{C}_{p}^{+}\nu)(u)=c_{n,p}\int_{\mathbb{S}^{n-1}}(u\cdot
v)_{+}^{p}d\nu(v),\qquad u\in\mathbb{S}^{n-1},$

where $(u\cdot v)_{+}=\max\\{u\cdot v,0\\}$, and

$\displaystyle
c_{n,p}=\frac{\Gamma\left(\mbox{$\frac{n+p}{2}$}\right)}{\pi^{\frac{n-1}{2}}\Gamma\left(\frac{1+p}{2}\right)}.$

According to the nonsymmetric $L_{p}$ cosine transform, we give the following
concept of nonsymmetric $L_{p}$ $\mu$-projection body by: Let $\mu$ satisfy
the homogeneity condition (I), $K\in\mathscr{K}^{n}_{\mathrm{o}}$ and $p\geq
1$, then the nonsymmetric $L_{p}$ $\mu$-projection body $\Pi_{\mu,p}^{+}K$ of
$K$, is the convex body defined by

$\displaystyle
h(\Pi_{\mu,p}^{+}K,\cdot)^{p}=\mathcal{C}_{p}^{+}S_{\mu,p}(K,\cdot),$

i.e., for $u\in\mathbb{S}^{n-1}$,

$\displaystyle h(\Pi_{\mu,p}^{+}K,u)^{p}=c_{n,p}\int_{\mathbb{S}^{n-1}}(u\cdot
v)_{+}^{p}dS_{\mu,p}(K,v).$

Let $(u\cdot v)_{-}=((-u)\cdot v)_{+}=\max\\{(-u)\cdot v,0\\}$. For
$K\in\mathscr{K}^{n}_{\mathrm{o}}$ and $p\geq 1$, we define $\Pi_{\mu,p}^{-}K$
by

$\displaystyle h(\Pi_{\mu,p}^{-}K,u)^{p}=c_{n,p}\int_{\mathbb{S}^{n-1}}(u\cdot
v)_{-}^{p}dS_{\mu,p}(K,v),$

for $u\in\mathbb{S}^{n-1}$. Thus, we have

$\displaystyle h(-\Pi_{\mu,p}^{+}K,u)^{p}$
$\displaystyle=h(\Pi_{\mu,p}^{+}K,-u)^{p}$
$\displaystyle=c_{n,p}\int_{\mathbb{S}^{n-1}}(-u\cdot
v)_{+}^{p}dS_{\mu,p}(K,v)$
$\displaystyle=c_{n,p}\int_{\mathbb{S}^{n-1}}(u\cdot
v)_{-}^{p}dS_{\mu,p}(K,v)$ $\displaystyle=h(\Pi_{\mu,p}^{-}K,u)^{p},$

that is,

$\displaystyle\Pi_{\mu,p}^{-}K=-\Pi_{\mu,p}^{+}K.$ (21)

Let $\mu$ satisfy the homogeneity condition (I), $K\in\mathscr{S}^{n}_{o}$ and
$p\geq 1$, then the nonsymmetric $L_{p}$ $\mu$-centroid body
$\Gamma_{\mu,p}^{+}K$ of $K$ is a convex body defined by

$\displaystyle
h(\Gamma_{\mu,p}^{+}K,u)^{p}=\frac{2}{c_{n,p}\mu(K)}\int_{K}(u\cdot
x)_{+}^{p}d\mu(x),\qquad u\in\mathbb{S}^{n-1}.$ (22)

Using the polar coordinates, it is easy to verify that for all
$u\in\mathbb{S}^{n-1}$,

$\displaystyle
h(\Gamma_{\mu,p}^{+}K,u)^{p}=\frac{2}{(p+\frac{1}{s})c_{n,p}\mu(K)}\int_{\mathbb{S}^{n-1}}(u\cdot
v)_{+}^{p}\rho(K,v)^{p+\frac{1}{s}}d\mu(v).$ (23)

Similarly, we also define $\Gamma_{\mu,p}^{-}K$ by

$\displaystyle
h(\Gamma_{\mu,p}^{-}K,u)^{p}=\frac{2}{c_{n,p}\mu(K)}\int_{K}(u\cdot
x)_{-}^{p}d\mu(x),\qquad u\in\mathbb{S}^{n-1}.$ (24)

Obviously, for $u\in\mathbb{S}^{n-1}$, from (22) and (24),we easily obtain

$\displaystyle h(-\Gamma_{\mu,p}^{+}K,u)^{p}$
$\displaystyle=h(\Gamma_{\mu,p}^{+}K,-u)^{p}=\frac{2}{c_{n,p}\mu(K)}\int_{K}((-u)\cdot
x)_{+}^{p}d\mu(x)$ $\displaystyle=\frac{2}{c_{n,p}\mu(K)}\int_{K}(u\cdot
x)_{-}^{p}d\mu(x)$ $\displaystyle=h(\Gamma_{\mu,p}^{-}K,u)^{p},$

i.e.,

$\displaystyle\Gamma_{\mu,p}^{-}K=-\Gamma_{\mu,p}^{+}K.$ (25)

In 2006, Ludwig [22] introduced a function
$\psi_{\tau}:\mathbb{R}\rightarrow[0,\infty)$ by

$\displaystyle\psi_{\tau}(t)=|t|+\tau t,\qquad\tau\in[-1,1].$

Let $\mu$ satisfy the homogeneity condition (I),
$K\in\mathscr{K}_{\mathrm{o}}^{n}$, $p\geq 1$ and $\tau\in[-1,1]$, the general
$L_{p}$ $\mu$-projection body
$\Pi_{\mu,p}^{\tau}K\in\mathscr{K}_{\mathrm{o}}^{n}$ of $K$ is defined by

$\displaystyle
h(\Pi_{\mu,p}^{\tau}K,u)^{p}=c_{n,p}(\tau)\int_{\mathbb{S}^{n-1}}\psi_{\tau}(u\cdot
v)^{p}dS_{\mu,p}(K,v),\qquad u\in\mathbb{S}^{n-1},$ (26)

where

$\displaystyle c_{n,p}(\tau)=\frac{c_{n,p}}{(1+\tau)^{p}+(1-\tau)^{p}}.$

Usually, we use $\Pi_{\mu,p}^{\tau,\ast}K$ to represent the polar body of
$\Pi_{\mu,p}^{\tau}K$ instead of $(\Pi_{\mu,p}^{\tau}K)^{\ast}$.

Since $\mu$ satisfies the homogeneity condition (I), for $\lambda>0$, from
(15) and (26), it follows that

$\displaystyle\Pi_{\mu,p}^{\tau}(\lambda
K)=\lambda^{\frac{1-sp}{sp}}\Pi_{\mu,p}^{\tau}K.$

For all $u,v\in\mathbb{S}^{n-1}$, it is easy to verify that

$\displaystyle\psi_{\tau}(u\cdot v)^{p}=(1+\tau)^{p}(u\cdot
v)_{+}^{p}+(1-\tau)^{p}(u\cdot v)_{-}^{p}.$ (27)

From (27) and the definition of $\Pi_{\mu,p}^{\pm}K$, we obtain

$\displaystyle\Pi_{\mu,p}^{\tau}K=g_{1}(\tau)\cdot\Pi_{\mu,p}^{+}K+_{p}g_{2}(\tau)\cdot\Pi_{\mu,p}^{-}K.$
(28)

where

$\displaystyle
g_{1}(\tau)=\frac{(1+\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}},\qquad
g_{2}(\tau)=\frac{(1-\tau)^{p}}{(1+\tau)^{p}+(1-\tau)^{p}}.$ (29)

Obviously,

$\displaystyle g_{1}(\tau)+g_{2}(\tau)=1,\qquad
g_{1}(-\tau)=g_{2}(\tau),\qquad g_{1}(\tau)=g_{2}(-\tau).$ (30)

Note that, let $\tau=\pm 1$, then

$\displaystyle\Pi_{\mu,p}^{+1}K=\Pi_{\mu,p}^{+}K,\qquad\Pi_{\mu,p}^{-1}K=\Pi_{\mu,p}^{-}K.$

If $\tau=0$, we define that $\Pi_{\mu,p}^{0}K=\Pi_{\mu,p}K$, by (28) and (29),
then

$\displaystyle\Pi_{\mu,p}K=\frac{1}{2}\cdot\Pi_{\mu,p}^{+}K+_{p}\frac{1}{2}\cdot\Pi_{\mu,p}^{-}K.$
(31)

Let $\mu$ satisfy the homogeneity condition (I), $K\in\mathscr{S}^{n}_{o}$,
$p\geq 1$ and $\tau\in[-1,1]$, the general $L_{p}$ $\mu$-centroid body
$\Gamma_{\mu,p}^{\tau}K\in\mathscr{K}^{n}_{\mathrm{o}}$ of $K$ is defined by

$\displaystyle h(\Gamma_{\mu,p}^{\tau}K,u)^{p}=$
$\displaystyle\frac{2}{\alpha_{n,p}(\tau)\mu(K)}\int_{K}\psi_{\tau}(u\cdot
x)^{p}d\mu(x)$ (32) $\displaystyle=$
$\displaystyle\frac{2}{\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(K)}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(u\cdot
v)^{p}\rho(K,v)^{p+\frac{1}{s}}d\mu(v),$

for $u\in\mathbb{S}^{n-1}$, and

$\displaystyle\alpha_{n,p}(\tau)=c_{n,p}[(1+\tau)^{p}+(1-\tau)^{p}].$

Then for every $\tau\in[-1,1]$, by (32), (27) and (29), we can obtain

$\displaystyle\Gamma_{\mu,p}^{\tau}K=g_{1}(\tau)\cdot\Gamma_{\mu,p}^{+}K+_{p}g_{2}(\tau)\cdot\Gamma_{\mu,p}^{-}K.$
(33)

Usually, we use $\Gamma_{\mu,p}^{\tau,\ast}K$ to represent the polar body of
$\Gamma_{\mu,p}^{\tau}K$ other than $(\Gamma_{\mu,p}^{\tau}K)^{\ast}$.

Since $\mu$ satisfies the homogeneity condition (I), for $\lambda>0$, from
(32), it follows that

$\displaystyle\Gamma_{\mu,p}^{\tau}(\lambda K)=\lambda\Gamma_{\mu,p}^{\tau}K.$

Note that, let $\tau=\pm 1$, then

$\displaystyle\Gamma_{\mu,p}^{+1}K=\Gamma_{\mu,p}^{+}K,\qquad\Gamma_{\mu,p}^{-1}K=\Gamma_{\mu,p}^{-}K.$

If $\tau=0$, we define that $\Gamma_{\mu,p}^{0}K=\Gamma_{\mu,p}K$, then

$\displaystyle\Gamma_{\mu,p}K=\frac{1}{2}\cdot\Gamma_{\mu,p}^{+}K+_{p}\frac{1}{2}\cdot\Gamma_{\mu,p}^{-}K.$
(34)

Next, the measure comparison theorems of the general $L_{p}$ $\mu$-projection
body and the general $L_{p}$ $\mu$-centroid body are obtained. Before that, we
give the following properties.

######  Proposition 3.1.

Let $\mu$ satisfy the homogeneity condition (I), if
$K,L\in\mathscr{K}^{n}_{\mathrm{o}}$, $p\geq 1$ and $\tau\in[-1,1]$, then

$\displaystyle
V_{\mu,p}(K,\Pi_{\mu,p}^{\tau}L)=V_{\mu,p}(L,\Pi_{\mu,p}^{\tau}K).$

###### Proof.

From (14) together with the (26) and Fubini’s theorem, we obtain

$\displaystyle V_{\mu,p}(K,\Pi_{\mu,p}^{\tau}L)$
$\displaystyle=s\int_{\mathbb{S}^{n-1}}h(\Pi_{\mu,p}^{\tau}L,u)^{p}dS_{\mu,p}(K,u)$
$\displaystyle=sc_{n,p}(\tau)\int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(u\cdot
v)^{p}dS_{\mu,p}(L,v)dS_{\mu,p}(K,u),$
$\displaystyle=sc_{n,p}(\tau)\int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(v\cdot
u)^{p}dS_{\mu,p}(K,u)dS_{\mu,p}(L,v),$
$\displaystyle=s\int_{\mathbb{S}^{n-1}}h(\Pi_{\mu,p}^{\tau}K,v)^{p}dS_{\mu,p}(L,v)$
$\displaystyle=V_{\mu,p}(L,\Pi_{\mu,p}^{\tau}K).$

∎

######  Proposition 3.2.

Let $\mu$ satisfy the homogeneity condition (I), if
$K\in\mathscr{K}^{n}_{\mathrm{o}}$, $L\in\mathscr{S}^{n}_{o}$, $p\geq 1$ and
$\tau\in[-1,1]$, then

$\displaystyle
V_{\mu,p}(K,\Gamma_{\mu,p}^{\tau}L)=\frac{2}{c_{n,p}(\tau)\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(L)}\widetilde{V}_{\mu,-p}(L,\Pi_{\mu,p}^{\tau,*}K).$

###### Proof.

Since $K\in\mathscr{K}^{n}_{\mathrm{o}}$ and $L\in\mathscr{S}^{n}_{o}$, by
(14) and (32), we obtain

$\displaystyle V_{\mu,p}(K,\Gamma_{\mu,p}^{\tau}L)$
$\displaystyle=s\int_{\mathbb{S}^{n-1}}h(\Gamma_{\mu,p}^{\tau}L,u)^{p}dS_{\mu,p}(K,u)$
$\displaystyle=\frac{2s}{\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(L)}\int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(u\cdot
v)^{p}\rho(L,v)^{p+\frac{1}{s}}d\mu(v)dS_{\mu,p}(K,u),$

from Fubini’s theorem, (26), (8) and (18), it follow that

$\displaystyle V_{\mu,p}(K,\Gamma_{\mu,p}^{\tau}L)$
$\displaystyle=\frac{2s}{\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(L)}\int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(v\cdot
u)^{p}dS_{\mu,p}(K,u)\rho(L,v)^{p+\frac{1}{s}}d\mu(v)$
$\displaystyle=\frac{2s}{c_{n,p}(\tau)\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(L)}\int_{\mathbb{S}^{n-1}}\rho(L,v)^{p+\frac{1}{s}}\rho(\Pi_{\mu,p}^{\tau,*}K,v)^{-p}d\mu(v)$
$\displaystyle=\frac{2}{c_{n,p}(\tau)\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(L)}\widetilde{V}_{\mu,-p}(L,\Pi_{\mu,p}^{\tau,*}K).$

This leads to the desired result. ∎

Next, let $\mathcal{P}^{\tau}_{\mu,p}$ and $\mathcal{P}^{\tau,*}_{\mu,p}$
denotes the set of general $L_{p}$ $\mu$-projection bodies and their polar
bodies, respectively. Let $\mathcal{C}^{\tau}_{\mu,p}$ and
$\mathcal{C}^{\tau,*}_{\mu,p}$ denotes the set of general $L_{p}$
$\mu$-centroid bodies and their polar bodies, respectively.

######  Theorem 3.3.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II), let $K\in\mathscr{K}^{n}_{\mathrm{o}}$, $p\geq 1$ and
$\tau\in[-1,1]$.

(a) If $\Pi_{\mu,p}^{\tau}K\subseteq\Pi_{\mu,p}^{\tau}L$ and
$L\in\mathcal{P}^{\tau}_{\mu,p}$, then

$\displaystyle 1\geq
p\left(\frac{\mu(K)}{\mu(L)}\right)^{1-s}+(1-p)\frac{\mu(K)}{\mu(L)}.$

(b) If $\Pi_{\mu,p}^{\tau,\ast}K\supseteq\Pi_{\mu,p}^{\tau,\ast}L$ and
$L\in\mathcal{C}^{\tau}_{\mu,p}$, then

$\displaystyle 1\geq
p\left(\frac{\mu(K)}{\mu(L)}\right)^{1-s}+(1-p)\frac{\mu(K)}{\mu(L)}.$

###### Proof.

For (a), since $\Pi_{\mu,p}^{\tau}K\subseteq\Pi_{\mu,p}^{\tau}L$ and
$L\in\mathcal{P}^{\tau}_{\mu,p}$, then there exist $M\in\mathscr{K}^{n}_{o}$,
such that $L=\Pi_{\mu,p}^{\tau}M$, by (14), we deduce

$\displaystyle V_{\mu,p}(M,\Pi_{\mu,p}^{\tau}K)\leq
V_{\mu,p}(M,\Pi_{\mu,p}^{\tau}L).$

From these and Proposition 3.1, it follows that

$\displaystyle
V_{\mu,p}(K,\Pi_{\mu,p}^{\tau}M)=V_{\mu,p}(M,\Pi_{\mu,p}^{\tau}K)\leq
V_{\mu,p}(M,\Pi_{\mu,p}^{\tau}L)=V_{\mu,p}(L,\Pi_{\mu,p}^{\tau}M).$

Let $\Pi_{\mu,p}^{\tau}M=L$ in the above inequality, from (17), it follows
that

$\displaystyle\mu(L)\geq V_{\mu,p}(K,L)\geq
p\mu(K)^{1-s}\mu(L)^{s}+(1-p)\mu(K),$

i.e.,

$\displaystyle 1\geq
p\left(\frac{\mu(K)}{\mu(L)}\right)^{1-s}+(1-p)\frac{\mu(K)}{\mu(L)}.$

Therefore, (a) is established.

For (b), since $\Pi_{\mu,p}^{\tau,\ast}K\supseteq\Pi_{\mu,p}^{\tau,\ast}L$ and
$L\in\mathcal{C}^{\tau}_{\mu,p}$, then there exist $N\in\mathscr{S}^{n}_{o}$,
such that $L=\Gamma_{\mu,p}^{\tau}N$, from (18), we obtain

$\displaystyle\widetilde{V}_{\mu,-p}(N,\Pi_{\mu,p}^{\tau,\ast}K)\leq\widetilde{V}_{\mu,-p}(N,\Pi_{\mu,p}^{\tau,\ast}L),$

from this and Proposition 3.2, we deduce

$\displaystyle V_{\mu,p}(K,\Gamma_{\mu,p}^{\tau}N)\leq
V_{\mu,p}(L,\Gamma_{\mu,p}^{\tau}N).$

Let $\Gamma_{\mu,p}^{\tau}N=L$ in the above inequality, from (17), it follows
that

$\displaystyle\mu(L)\geq V_{\mu,p}(K,L)\geq
p\mu(K)^{1-s}\mu(L)^{s}+(1-p)\mu(K),$

i.e.,

$\displaystyle 1\geq
p\left(\frac{\mu(K)}{\mu(L)}\right)^{1-s}+(1-p)\frac{\mu(K)}{\mu(L)}.$

This completes the proof of the Theorem. ∎

Especially, let $p=1$ and $\tau=0$ in Theorem 3.3, since $1-s>0$, we can get
the following interesting results.

######  Corollary 3.4.

Let $\mu$ satisfy the homogeneity condition (I) and concavity condition (II)
and $K\in\mathscr{K}^{n}_{\mathrm{o}}$.

(a) If $\Pi_{\mu}K\subseteq\Pi_{\mu}L$ and $L\in\mathcal{P}_{\mu}$, then

$\displaystyle\mu(K)\leq\mu(L).$

(b) If $\Pi_{\mu}^{\ast}K\supseteq\Pi_{\mu}^{\ast}L$ and
$L\in\mathcal{C}_{\mu}$, then

$\displaystyle\mu(K)\leq\mu(L).$

Here, $\mathcal{P}_{\mu}$ is the set of $\mu$-projection bodies and
$\mathcal{C}_{\mu}$ denotes the set of $\mu$-centroid bodies.

######  Proposition 3.5.

Let $\mu$ satisfy the homogeneity condition (I),
$K,L\in\mathscr{S}^{n}_{\mathrm{o}}$ and $p\geq 1$, then

$\displaystyle\frac{\widetilde{V}_{\mu,-p}(K,\Gamma_{\mu,p}^{\tau,\ast}L)}{\mu(K)}=\frac{\widetilde{V}_{\mu,-p}(L,\Gamma_{\mu,p}^{\tau,\ast}K)}{\mu(L)}.$

###### Proof.

Combining (18), (32), (8) and Fubini’s theorem, using (8) (32) and (18) again,
we get

$\displaystyle\widetilde{V}_{\mu,-p}(L,\Gamma_{\mu,p}^{\tau,\ast}K)$
$\displaystyle=$ $\displaystyle
s\int_{\mathbb{S}^{n-1}}\rho(L,u)^{\frac{1+sp}{s}}\rho(\Gamma_{\mu,p}^{\tau,\ast}K,u)^{-p}d\mu(u)$
$\displaystyle=$
$\displaystyle\frac{2s}{\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(K)}\int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(u\cdot
v)^{p}\rho(K,v)^{\frac{1+sp}{s}}\rho(L,u)^{\frac{1+sp}{s}}d\mu(v)d\mu(u)$
$\displaystyle=$
$\displaystyle\frac{2s}{\alpha_{n,p}(\tau)(p+\frac{1}{s})\mu(K)}\int_{\mathbb{S}^{n-1}}\int_{\mathbb{S}^{n-1}}\psi_{\tau}(v\cdot
u)^{p}\rho(L,u)^{\frac{1+sp}{s}}\rho(K,v)^{\frac{1+sp}{s}}d\mu(u)d\mu(v)$
$\displaystyle=$
$\displaystyle\frac{\mu(L)}{\mu(K)}s\int_{\mathbb{S}^{n-1}}\rho(K,v)^{\frac{1+sp}{s}}\rho(\Gamma_{\mu,p}^{\tau,\ast}L,v)^{-p}d\mu(v)$
$\displaystyle=$
$\displaystyle\frac{\mu(L)}{\mu(K)}\widetilde{V}_{\mu,-p}(K,\Gamma_{\mu,p}^{\tau,\ast}L).$

This gives the equality. ∎

######  Theorem 3.6.

Let $\mu$ satisfy the homogeneity condition (I),
$K\in\mathscr{S}^{n}_{\mathrm{o}}$, $p\geq 1$ and $\tau\in[-1,1]$.

(a) If $\Gamma_{\mu,p}^{\tau}K\subseteq\Gamma_{\mu,p}^{\tau}L$ and
$L\in\mathcal{P}^{\tau,*}_{\mu,p}$, then

$\displaystyle\mu(K)\leq\mu(L).$

(a) If $\Gamma_{\mu,p}^{\tau,\ast}K\supseteq\Gamma_{\mu,p}^{\tau,\ast}L$ and
$L\in\mathcal{C}^{\tau,*}_{\mu,p}$, then

$\displaystyle\mu(K)\leq\mu(L).$

Equality holds in (a) and (b) if and only if $K=L$.

###### Proof.

For (a), since $\Gamma_{\mu,p}^{\tau}K\subseteq\Gamma_{\mu,p}^{\tau}L$ and
$L\in\mathcal{P}^{\tau,*}_{\mu,p}$, then there exist
$Q\in\mathscr{K}^{n}_{o}$, such that $L=\Pi_{\mu,p}^{\tau,*}Q$, by (14), we
get

$\displaystyle V_{\mu,p}(Q,\Gamma_{\mu,p}^{\tau}K)\leq
V_{\mu,p}(Q,\Gamma_{\mu,p}^{\tau}L)$ (35)

with equality if and only if $\Gamma_{\mu,p}^{\tau}K=\Gamma_{\mu,p}^{\tau}L$.
From these and Proposition 3.5, it follows that

$\displaystyle\frac{1}{\mu(K)}\widetilde{V}_{\mu,-p}(K,\Pi_{\mu,p}^{\tau,*}Q)\leq\frac{1}{\mu(L)}\widetilde{V}_{\mu,-p}(L,\Pi_{\mu,p}^{\tau,*}Q).$

Let $\Pi_{\mu,p}^{\tau,*}Q=L$ in the above inequality, from Lemma 2.4, it
follows that

$\displaystyle
1\geq\frac{1}{\mu(K)}\widetilde{V}_{\mu,-p}(K,L)\geq\frac{1}{\mu(K)}\mu(K)^{1+sp}\mu(L)^{-sp}.$

i.e.,

$\displaystyle\mu(K)\leq\mu(L).$

The equality conditions of Lemma 2.4 and (35) imply that in (a) the equality
holds if and only if $K=L$. This shows that (a) holds.

For (b), if $\Gamma_{\mu,p}^{\tau,\ast}K\supseteq\Gamma_{\mu,p}^{\tau,\ast}L$
and $L\in\mathcal{C}^{\tau,*}_{\mu,p}$, then there exist
$M\in\mathscr{S}^{n}_{o}$, such that $L=\Gamma_{\mu,p}^{\tau,*}M$, by (18) and
(6), we deduce

$\displaystyle\widetilde{V}_{\mu,-p}(M,\Gamma_{\mu,p}^{\tau,\ast}K)\leq\widetilde{V}_{\mu,-p}(M,\Gamma_{\mu,p}^{\tau,\ast}L),$
(36)

with equality if and only if
$\Gamma_{\mu,p}^{\tau,\ast}K=\Gamma_{\mu,p}^{\tau,\ast}L$. From these and
Proposition 3.2, it follows that

$\displaystyle\frac{1}{\mu(K)}\widetilde{V}_{\mu,-p}(K,\Gamma_{\mu,p}^{\tau,*}M)\leq\frac{1}{\mu(L)}\widetilde{V}_{\mu,-p}(L,\Gamma_{\mu,p}^{\tau,*}M).$

Let $\Gamma_{\mu,p}^{\tau,*}M=L$ in the above inequality, from Lemma 2.4, it
follows that

$\displaystyle
1\geq\frac{1}{\mu(K)}\widetilde{V}_{\mu,-p}(K,L)\geq\frac{1}{\mu(K)}\mu(K)^{1+sp}\mu(L)^{-sp},$

equivalently,

$\displaystyle\mu(K)\leq\mu(L).$

By the equality conditions of Lemma 2.4 and (36), we see that equality holds
in (b) if and only if $K=L$. ∎

Especially, let $p=1$ and $\tau=0$ in Theorem 3.6, we can get the following
corollary.

######  Corollary 3.7.

Let $\mu$ satisfy the homogeneity condition (I) and
$K\in\mathscr{S}^{n}_{\mathrm{o}}$.

(a) If $\Gamma_{\mu}K\subseteq\Gamma_{\mu}L$ and $L\in\mathcal{P}^{*}_{\mu}$,
then

$\displaystyle\mu(K)\leq\mu(L).$

(b) If $\Gamma_{\mu}^{\ast}K\supseteq\Gamma_{\mu}^{\ast}L$ and
$L\in\mathcal{C}^{*}_{\mu}$, then

$\displaystyle\mu(K)\leq\mu(L).$

Equality holds in (a) and (b) if and only if $K=L$. Here,
$\mathcal{P}^{*}_{\mu}$ and $\mathcal{C}^{*}_{\mu}$ denotes the set of polar
bodies of $\mu$-projection bodies and $\mu$-centroid bodies, respectively.

##  4\. Extremal Inequalities for general $L_{p}$ $\mu$-projection body and
$\mu$-centroid body

In this section, we will give the proof of Theorem 1.1 and Theorem 1.4, which
are Theorem 4.4 and Theorem 4.6. In order to prove the theorems, the following
properties are required.

######  Proposition 4.1.

Let $\mu$ satisfy the homogeneity condition (I), if $K\in\mathscr{K}^{n}_{o}$,
$L\in\mathscr{S}^{n}_{o}$, $p\geq 1$ and $\tau\in[-1,1]$, then

$\displaystyle\Pi_{\mu,p}^{-\tau}K=-\Pi_{\mu,p}^{\tau}K,$ (37)

and

$\displaystyle\Gamma_{\mu,p}^{-\tau}L=-\Gamma_{\mu,p}^{\tau}L.$ (38)

###### Proof.

From (28), (30) and (21), we have that

$\displaystyle\Pi_{\mu,p}^{-\tau}K$
$\displaystyle=g_{1}(-\tau)\cdot\Pi_{\mu,p}^{+}K+_{p}g_{2}(-\tau)\cdot\Pi_{\mu,p}^{-}K$
$\displaystyle=g_{2}(\tau)\cdot(-\Pi_{\mu,p}^{-}K)+_{p}g_{1}(\tau)\cdot(-\Pi_{\mu,p}^{+}K),$

from this, (4) and (28), for $u\in\mathbb{S}^{n-1}$, we have

$\displaystyle h(\Pi_{\mu,p}^{-\tau}K,u)^{p}$
$\displaystyle=g_{2}(\tau)h(-\Pi_{\mu,p}^{-}K,u)^{p}+g_{1}(\tau)h(-\Pi_{\mu,p}^{+}K,u)^{p}$
$\displaystyle=g_{2}(\tau)h(\Pi_{\mu,p}^{-}K,-u)^{p}+g_{1}(\tau)h(\Pi_{\mu,p}^{+}K,-u)^{p}$
$\displaystyle=h(g_{1}(\tau)\cdot\Pi_{\mu,p}^{+}K+_{p}g_{2}(\tau)\cdot\Pi_{\mu,p}^{-}K,-u)^{p}$
$\displaystyle=h(-\Pi_{\mu,p}^{\tau}K,u)^{p}.$

This obtain

$\displaystyle\Pi_{\mu,p}^{-\tau}K=-\Pi_{\mu,p}^{\tau}K.$

Similar to the above method, from (33), (30), (25), (4) and using (33) again,
we can easily verify that (38) holds. ∎

######  Proposition 4.2.

Let $\mu$ satisfy the homogeneity condition (I), if $K\in\mathscr{K}^{n}_{o}$,
$L\in\mathscr{S}^{n}_{o}$, $p\geq 1$, $\tau\in[-1,1]$ and $\tau\neq 0$, then

$\displaystyle\Pi_{\mu,p}^{\tau}K=\Pi_{\mu,p}^{-\tau}K\quad\Longleftrightarrow\quad\Pi_{\mu,p}^{+}K=\Pi_{\mu,p}^{-}K;$

and

$\displaystyle\Gamma_{\mu,p}^{\tau}L=\Gamma_{\mu,p}^{-\tau}L\quad\Longleftrightarrow\quad\Gamma_{\mu,p}^{+}L=\Gamma_{\mu,p}^{-}L.$

###### Proof.

Since $K,L\in\mathscr{K}^{n}_{o}$, $p\geq 1$, $\tau\in[-1,1]$ and $\tau\neq
0$, then by (28) and (30), we deduce

$\displaystyle\Pi_{\mu,p}^{-\tau}K=g_{2}(\tau)\cdot\Pi_{\mu,p}^{+}K+_{p}g_{1}(\tau)\cdot\Pi_{\mu,p}^{-}K,$

from this and (4), for all $u\in\mathbb{S}^{n-1}$, we obtain

$\displaystyle
h(\Pi_{\mu,p}^{-\tau}K,u)^{p}=g_{2}(\tau)h(\Pi_{\mu,p}^{+}K,u)^{p}+g_{1}(\tau)h(\Pi_{\mu,p}^{-}K,u)^{p}.$
(39)

When $\Pi_{\mu,p}^{+}K=\Pi_{\mu,p}^{-}K$, by (39), (28) and (4), for all
$u\in\mathbb{S}^{n-1}$, then

$\displaystyle h(\Pi_{\mu,p}^{-\tau}K,u)^{p}=h(\Pi_{\mu,p}^{\tau}K,u)^{p},$

thus $\Pi_{\mu,p}^{\tau}K=\Pi_{\mu,p}^{-\tau}K$.

Conversely, if $\Pi_{\mu,p}^{\tau}K=\Pi_{\mu,p}^{-\tau}K$, from (28) and (4),
for all $u\in\mathbb{S}^{n-1}$, we have

$\displaystyle
h(\Pi_{\mu,p}^{\tau}K,u)^{p}=g_{1}(\tau)h(\Pi_{\mu,p}^{+}K,u)^{p}+g_{2}(\tau)h(\Pi_{\mu,p}^{-}K,u)^{p}.$
(40)

From (40) and (39), for all $u\in\mathbb{S}^{n-1}$, we obtain

$\displaystyle(g_{1}(\tau)-g_{2}(\tau))h(\Pi_{\mu,p}^{+}K,u)^{p}=(g_{1}(\tau)-g_{2}(\tau))h(\Pi_{\mu,p}^{-}K,u)^{p},$

Since $g_{1}(\tau)-g_{2}(\tau)\neq 0$ when $\tau\neq 0$, we get
$\Pi_{\mu,p}^{+}K=\Pi_{\mu,p}^{-}K$.

In addition, from (33), (30) and (4), we obtain

$\displaystyle
h(\Gamma_{\mu,p}^{\tau}L,u)^{p}=g_{1}(\tau)h(\Gamma_{\mu,p}^{+}L,u)^{p}+g_{2}(\tau)h(\Gamma_{\mu,p}^{-}L,u)^{p},$
(41)

and

$\displaystyle
h(\Gamma_{\mu,p}^{-\tau}L,u)^{p}=g_{2}(\tau)h(\Gamma_{\mu,p}^{+}L,u)^{p}+g_{1}(\tau)h(\Gamma_{\mu,p}^{-}L,u)^{p}.$
(42)

If $\Gamma_{\mu,p}^{+}L=\Gamma_{\mu,p}^{-}L$, by (42) and (33), we can obtain
$h(\Gamma_{\mu,p}^{\tau}L,u)^{p}=h(\Gamma_{\mu,p}^{-\tau}L,u)^{p}$ for all
$u\in\mathbb{S}^{n-1}$, i.e.,
$\Gamma_{\mu,p}^{\tau}L=\Gamma_{\mu,p}^{-\tau}L$.

On the other hand, If $\Gamma_{\mu,p}^{\tau}L=\Gamma_{\mu,p}^{-\tau}L$, by
(41) and (42), we obtain

$\displaystyle(g_{1}(\tau)-g_{2}(\tau))h(\Gamma_{\mu,p}^{+}L,u)^{p}=(g_{1}(\tau)-g_{2}(\tau))h(\Gamma_{\mu,p}^{-}L,u)^{p},$

where $u\in\mathbb{S}^{n-1}$. Since $\tau\neq 0$, then
$g_{1}(\tau)-g_{2}(\tau)\neq 0$. Hence,
$\Gamma_{\mu,p}^{+}L=\Gamma_{\mu,p}^{-}L$. ∎

######  Proposition 4.3.

Let $\mu$ satisfy the homogeneity condition (I), if $K\in\mathscr{K}^{n}_{o}$,
$L\in\mathscr{S}^{n}_{o}$, $p\geq 1$ and $\tau\in[-1,1]$, then

$\displaystyle\Pi_{\mu,p}^{\tau}K+_{p}\Pi_{\mu,p}^{-\tau}K=\Pi_{\mu,p}^{+}K+_{p}\Pi_{\mu,p}^{-}K,$
(43)

and

$\displaystyle\Gamma_{\mu,p}^{\tau}L+_{p}\Gamma_{\mu,p}^{-\tau}L=\Gamma_{\mu,p}^{+}L+_{p}\Gamma_{\mu,p}^{-}L.$
(44)

###### Proof.

By (40), (39) and (30), for all $u\in\mathbb{S}^{n-1}$, we have

$\displaystyle
h(\Pi_{\mu,p}^{\tau}K,u)^{p}+h(\Pi_{\mu,p}^{-\tau}K,u)^{p}=h(\Pi_{\mu,p}^{+}K,u)^{p}+h(\Pi_{\mu,p}^{-}K,u)^{p},$

from this and (4), we obtain

$\displaystyle
h(\Pi_{\mu,p}^{\tau}K+_{p}\Pi_{\mu,p}^{-\tau}K,u)^{p}=h(\Pi_{\mu,p}^{+}K+_{p}\Pi_{\mu,p}^{-}K,u)^{p},$

that is

$\displaystyle\Pi_{\mu,p}^{\tau}K+_{p}\Pi_{\mu,p}^{-\tau}K=\Pi_{\mu,p}^{+}K+_{p}\Pi_{\mu,p}^{-}K.$

This gives (43).

Along the same lines, with (41), (42), (30) and (4), we can easily show that
(44) holds. ∎

Next, we prove the extreme inequality of the general $L_{p}$ $\mu$-projection
body and the polar body of the general $L_{p}$ $\mu$-projection body.

######  Theorem 4.4.

Suppose that $\mu$ satisfies the homogeneity condition (I) and $\omega$ is an
even density function. If $K\in\mathscr{K}^{n}_{o}$, $p\geq 1$, and
$\tau\in[-1,1]$, then

$\displaystyle\mu(\Pi_{\mu,p}^{\ast}K)\leq\mu(\Pi_{\mu,p}^{\tau,\ast}K)\leq\mu(\Pi_{\mu,p}^{\pm,\ast}K).$
(45)

When $\tau\neq 0$, equality holds in the left inequality if and only if
$\Pi_{\mu,p}^{\tau}K$ is origin-symmetric, and when $\tau\neq\pm 1$, equality
holds in the right inequality if and only if $\Pi_{\mu,p}^{\pm}K$ is origin-
symmetric.

###### Proof.

Clearly, if $\tau=\pm 1$ in (45), the inequality on the right of (45) is an
identity. Let $-1<\tau<1$. From (28) and (10), we get

$\Pi_{\mu,p}^{\tau,*}K=g_{1}(\tau)\circ\Pi_{\mu,p}^{+,*}K\widetilde{+}_{-p}\,g_{2}(\tau)\circ\Pi_{\mu,p}^{-,*}K.$
(46)

From Lemma 2.5, we obtain

$\displaystyle\mu(\Pi_{\mu,p}^{\tau,*}K)^{-sp}=\mu(g_{1}(\tau)\circ\Pi_{\mu,p}^{+,*}K\widetilde{+}_{-p}\,g_{2}(\tau)\circ\Pi_{\mu,p}^{-,*}K)^{-sp}\geq
g_{1}(\tau)\mu(\Pi_{\mu,p}^{+,*}K)^{-sp}+g_{2}(\tau)\mu(\Pi_{\mu,p}^{-,*}K)^{-sp},$

that is,

$\mu(\Pi_{\mu,p}^{\tau,*}K)\leq\mu(\Pi_{\mu,p}^{\pm,*}K),$ (47)

equality holds if and only if $\Pi_{\mu,p}^{+}K$ and $\Pi_{\mu,p}^{-}K$ are
dilatates. This and (25) imply that the equality holds on the right of (45) if
and only if $\Pi_{\mu,p}^{+}K$ and $\Pi_{\mu,p}^{-}K$ are origin-symmetric.

Next, let’s prove the other part of this theorem. If $\tau=0$, the inequality
on the left of (45) is an identity. Let $\tau\neq 0$. By (9), (46) and (30),
we deduce

$\displaystyle\rho(\Pi_{\mu,p}^{\tau,*}K,u)^{-p}=g_{1}(\tau)\rho(\Pi_{\mu,p}^{+,*}K,u)^{-p}+\,g_{2}(\tau)\rho(\Pi_{\mu,p}^{-,*}K,u)^{-p}$
(48)

and

$\displaystyle\rho(\Pi_{\mu,p}^{-\tau,*}K,u)^{-p}=g_{2}(\tau)\rho(\Pi_{\mu,p}^{+,*}K,u)^{-p}+\,g_{1}(\tau)\rho(\Pi_{\mu,p}^{-,*}K,u)^{-p}.$
(49)

Add (48) and (49) together, use (30) again to get

$\displaystyle\frac{1}{2}\rho(\Pi_{\mu,p}^{-\tau,*}K,u)^{-p}+\frac{1}{2}\rho(\Pi_{\mu,p}^{\tau,*}K,u)^{-p}=\frac{1}{2}\rho(\Pi_{\mu,p}^{+,*}K,u)^{-p}+\,\frac{1}{2}\rho(\Pi_{\mu,p}^{-,*}K,u)^{-p},$
(50)

combining the above formula with (9), (8), (4), (31) and using (8) again, we
can deduce

$\displaystyle\rho\left(\frac{1}{2}\circ\Pi_{\mu,p}^{-\tau,*}K\widetilde{+}_{-p}\,\frac{1}{2}\circ\Pi_{\mu,p}^{\tau,*}K,u\right)^{-p}=\rho(\Pi_{\mu,p}^{*}K,u)^{-p},$

hence,

$\displaystyle\Pi_{\mu,p}^{*}K=\frac{1}{2}\circ\Pi_{\mu,p}^{-\tau,*}K\widetilde{+}_{-p}\,\frac{1}{2}\circ\Pi_{\mu,p}^{\tau,*}K.$

From this and Lemma 2.5, we get

$\displaystyle\mu(\Pi_{\mu,p}^{*}K)^{-sp}\geq\frac{1}{2}\mu(\Pi_{\mu,p}^{-\tau,*}K)^{-sp}+\frac{1}{2}\mu(\Pi_{\mu,p}^{\tau,*}K)^{-sp},$

by (37), we obtain

$\displaystyle\mu(\Pi_{\mu,p}^{*}K)\leq\mu(\Pi_{\mu,p}^{\tau,*}K),$

with equality if and only if $\Pi_{\mu,p}^{\tau}K$ and $\Pi_{\mu,p}^{-\tau}K$
are dilatates. By (37), we know that the equality holds on the left of the
(45) if and only if $\Pi_{\mu,p}^{\tau}K$ is origin-symmetric. ∎

######  Theorem 4.5.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II) and $\omega$ is an even density function. If
$K\in\mathscr{K}^{n}_{o}$, $p\geq 1$, and $\tau\in[-1,1]$, then

$\displaystyle\mu(\Pi_{\mu,p}K)\geq\mu(\Pi_{\mu,p}^{\tau}K)\geq\mu(\Pi_{\mu,p}^{\pm}K).$

###### Proof.

From Lemma 2.1, (31), (21) and (30), we get

$\displaystyle\frac{1+(g_{1}(\tau)+g_{2}(\tau))(p-1)}{p}\mu(\Pi_{\mu,p}^{\tau}K)^{s}$
$\displaystyle\geq
g_{1}(\tau)\mu(\Pi_{\mu,p}^{+}K)^{s}+g_{2}(\tau)\mu(\Pi_{\mu,p}^{-}K)^{s}$
$\displaystyle=\mu(\Pi_{\mu,p}^{\pm}K)^{s},$

i.e.,

$\displaystyle\mu(\Pi_{\mu,p}^{\tau}K)\geq\mu(\Pi_{\mu,p}^{\pm}K).$

In order to show that the inequality on the left holds, by (43) and (31), it
follows that

$\displaystyle\Pi_{\mu,p}K=\frac{1}{2}\cdot\Pi_{\mu,p}^{\tau}K+_{p}\frac{1}{2}\cdot\Pi_{\mu,p}^{-\tau}K,$

from this, and using Lemma 2.1 and (37), we can immediately obtain

$\displaystyle\mu(\Pi_{\mu,p}K)\geq\mu(\Pi_{\mu,p}^{\tau}K).$

This proves our desire inequality. ∎

The proof of the extreme inequality of the general $L_{p}$ $\mu$-centroid body
is similar to that of the general $L_{p}$ $\mu$-projection body to a great
extent. We only give the sketch of proof of the following theorems.

######  Theorem 4.6.

Suppose that $\mu$ satisfies the homogeneity condition (I) and $\omega$ is an
even density function. If $K\in\mathscr{S}^{n}_{o}$, $p\geq 1$, and
$\tau\in[-1,1]$, then

$\displaystyle\mu(\Gamma_{\mu,p}^{\ast}K)\leq\mu(\Gamma_{\mu,p}^{\tau,\ast}K)\leq\mu(\Gamma_{\mu,p}^{\pm,\ast}K).$
(51)

When $\tau\neq 0$, equality holds in the left inequality if and only if
$\Gamma_{\mu,p}^{\tau}K$ is origin-symmetric, and when $\tau\neq\pm 1$,
equality holds in the right inequality if and only if $\Gamma_{\mu,p}^{\pm}K$
is origin-symmetric.

###### Proof.

Obviously, if $\tau=\pm 1$ in (51), the inequality on the right of (51) is an
identity. Let $-1<\tau<1$. From (33) and (10), it shows that

$\Gamma_{\mu,p}^{\tau,*}K=g_{1}(\tau)\circ\Gamma_{\mu,p}^{+,*}K\widetilde{+}_{-p}\,g_{2}(\tau)\circ\Gamma_{\mu,p}^{-,*}K,$
(52)

from this, with Lemma 2.5 and (10), we deduce

$\mu(\Gamma_{\mu,p}^{\pm,*}K)\geq\mu(\Gamma_{\mu,p}^{\tau,*}K),$

equality holds if and only if $\Gamma_{\mu,p}^{+}K$ and $\Gamma_{\mu,p}^{-}K$
are dilatates. This and (21) imply that the equality holds on the right of the
(51) if and only if $\Gamma_{\mu,p}^{+}K$ and $\Gamma_{\mu,p}^{-}K$ are
origin-symmetric.

Next, let’s prove the left part of (51). If $\tau=0$, the inequality on the
left of (51) is an identity. Let $\tau\neq 0$. By (9), (52) and (30), we
deduce

$\displaystyle\rho(\Gamma_{\mu,p}^{\tau,*}K,u)^{-p}=g_{1}(\tau)\rho(\Gamma_{\mu,p}^{+,*}K,u)^{-p}+\,g_{2}(\tau)\rho(\Gamma_{\mu,p}^{-,*}K,u)^{-p}$
(53)

and

$\displaystyle\rho(\Gamma_{\mu,p}^{-\tau,*}K,u)^{-p}=g_{2}(\tau)\rho(\Gamma_{\mu,p}^{+,*}K,u)^{-p}+\,g_{1}(\tau)\rho(\Gamma_{\mu,p}^{-,*}K,u)^{-p}.$
(54)

By (53) and (54), useing (30) again, combining with (9), (8), (4), (34) and
using (8) again, we can deduce

$\displaystyle\rho\left(\frac{1}{2}\circ\Gamma_{\mu,p}^{-\tau,*}K\widetilde{+}_{-p}\,\frac{1}{2}\circ\Gamma_{\mu,p}^{\tau,*}K,u\right)^{-p}$
$\displaystyle=\frac{1}{2}\rho(\Gamma_{\mu,p}^{+,*}K,u)^{-p}+\,\frac{1}{2}\rho(\Gamma_{\mu,p}^{-,*}K,u)^{-p}$
$\displaystyle=\rho(\Gamma_{\mu,p}^{*}K,u)^{-p},$

therefore,

$\displaystyle\Gamma_{\mu,p}^{*}K=\frac{1}{2}\circ\Gamma_{\mu,p}^{-\tau,*}K\widetilde{+}_{-p}\,\frac{1}{2}\circ\Gamma_{\mu,p}^{\tau,*}K.$

From this, with Lemma 2.5 and (38), we deduce

$\displaystyle\mu(\Gamma_{\mu,p}^{\tau,*}K)\leq\mu(\Gamma_{\mu,p}^{*}K).$

with equality if and only if $\Gamma_{\mu,p}^{\tau}K$ and
$\Gamma_{\mu,p}^{-\tau}K$ are dilatates. By (38), we know that the equality
holds on the left of the (51) if and only if $\Gamma_{\mu,p}^{\tau}K$ is
origin-symmetric. ∎

######  Theorem 4.7.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II) and $\omega$ is an even density function. If
$K\in\mathscr{S}^{n}_{o}$, $p\geq 1$ and $\tau\in[-1,1]$, then

$\displaystyle\mu(\Gamma_{\mu,p}K)\geq\mu(\Gamma_{\mu,p}^{\tau}K)\geq\mu(\Gamma_{\mu,p}^{\pm}K).$

###### Proof.

By Lemma 2.1, (33), (25) and (30), we get

$\displaystyle\mu(\Gamma_{\mu,p}^{\tau}K)\geq\mu(\Gamma_{\mu,p}^{\pm}K).$

To prove the other part of the inequality, by (44) and (34), it follows that

$\displaystyle\Gamma_{\mu,p}K=\frac{1}{2}\cdot\Gamma_{\mu,p}^{\tau}K+_{p}\frac{1}{2}\cdot\Gamma_{\mu,p}^{-\tau}K,$

from this, and using Lemma 2.1, (38) and (30), we obtain

$\displaystyle\mu(\Gamma_{\mu,p}K)\geq\mu(\Gamma_{\mu,p}^{\tau}K).$

This proves our desire inequality. ∎

##  5\. Brunn-Minkowski Type Inequalities and Monotone inequalities

The $L_{p}$-Blaschke additions of measure is defined by Wu [40]. Suppose that
$\mu$ satisfies the homogeneity condition (I) and concavity condition (II) and
$\omega$ is an even density function. Let $p\geq 1$ and $p\neq 1/s$, for
$K,L\in\mathscr{K}_{o}^{n}$, the $L_{p}$-Blaschke addition of measures,
$K\\#_{\mu,p}L$ is a convex body, defined by

$\displaystyle
S_{\mu,p}\left(K\\#_{\mu,p}L,\cdot\right)=S_{\mu,p}(K,\cdot)+S_{\mu,p}(L,\cdot).$
(55)

Next, we prove Brunn-Minkowski type inequalities of the general $L_{p}$
$\mu$-projection body and its polar body with respect to $L_{p}$-Blaschke
addition of measures.

######  Theorem 5.1.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II) and $\omega$ is an even density function. Let $p\geq 1$ and
$p\neq 1/s$. If $K,L\in\mathscr{K}_{o}^{n}$ and $\tau\in[-1,1]$, then

$\displaystyle(2-1/p)\mu(\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L))^{s}\geq\mu(\Pi_{\mu,p}^{\tau}K)^{s}+\mu(\Pi_{\mu,p}^{\tau}L)^{s}.$

###### Proof.

From (55) and (26), we have

$\displaystyle
h(\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L),u)^{p}=h(\Pi_{\mu,p}^{\tau}K,u)^{p}+h(\Pi_{\mu,p}^{\tau}L,u)^{p},\qquad
u\in\mathbb{S}^{n-1},$ (56)

from this, (14) and (17), for $N\in\mathscr{K}_{o}^{n}$, we have

$\displaystyle V_{\mu,p}(N,\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L))$
$\displaystyle=$ $\displaystyle
s\int_{\mathbb{S}^{n-1}}h(\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L),u)^{p}dS_{\mu,p}(N,u)$
$\displaystyle=$ $\displaystyle
V_{\mu,p}(N,\Pi_{\mu,p}^{\tau}K)+V_{\mu,p}(N,\Pi_{\mu,p}^{\tau}L)$
$\displaystyle\geq$ $\displaystyle
p\mu(N)^{1-s}\mu(\Pi_{\mu,p}^{\tau}K)^{s}+(1-p)\mu(N)+p\mu(N)^{1-s}\mu(\Pi_{\mu,p}^{\tau}L)^{s}+(1-p)\mu(N).$

Let $N=\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L)$, then

$\displaystyle(2-1/p)\mu(\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L))^{s}\geq\mu(\Pi_{\mu,p}^{\tau}K)^{s}+\mu(\Pi_{\mu,p}^{\tau}L)^{s}.$

∎

Particularly, if $\tau=0$ and $p=1$ in Theorem 5.1, then the following result
is obvious.

######  Corollary 5.2.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II) and $\omega$ is an even density function. If
$K,L\in\mathscr{K}_{o}^{n}$, then

$\displaystyle\mu(\Pi_{\mu}(K\\#_{\mu}L))^{s}\geq\mu(\Pi_{\mu}K)^{s}+\mu(\Pi_{\mu}L)^{s}.$

######  Theorem 5.3.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II) and $\omega$ is an even density function. Let $p\geq 1$ and
$p\neq 1/s$. If $K,L\in\mathscr{K}_{o}^{n}$ and $\tau\in[-1,1]$, then

$\displaystyle\mu(\Pi_{\mu,p}^{\tau,*}(K\\#_{\mu,p}L))^{-sp}\geq\mu(\Pi_{\mu,p}^{\tau,*}K)^{-sp}+\mu(\Pi_{\mu,p}^{\tau,*}L)^{-sp}.$
(57)

###### Proof.

From (18), (8), (56), and Lemma 2.4, for $Q\in\mathscr{K}_{o}^{n}$, we get

$\displaystyle\widetilde{V}_{\mu,-p}(Q,\Pi_{\mu,p}^{\tau,*}(K\\#_{\mu,p}L))=$
$\displaystyle
s\int_{\mathbb{S}^{n-1}}\rho(Q,u)^{\frac{1+sp}{s}}\rho(\Pi_{\mu,p}^{\tau,*}(K\\#_{\mu,p}L),u)^{-p}d\mu(u)$
$\displaystyle=$ $\displaystyle
s\int_{\mathbb{S}^{n-1}}\rho(Q,u)^{\frac{1+sp}{s}}h(\Pi_{\mu,p}^{\tau}(K\\#_{\mu,p}L),u)^{p}d\mu(u)$
$\displaystyle=$ $\displaystyle
s\int_{\mathbb{S}^{n-1}}\rho(Q,u)^{\frac{1+sp}{s}}(h(\Pi_{\mu,p}^{\tau}K,u)^{p}+h(\Pi_{\mu,p}^{\tau}L,u)^{p})d\mu(u)$
$\displaystyle=$
$\displaystyle\widetilde{V}_{\mu,-p}(Q,\Pi_{\mu,p}^{\tau,*}K)+\widetilde{V}_{\mu,-p}(Q,\Pi_{\mu,p}^{\tau,*}L)$
$\displaystyle\geq$
$\displaystyle\mu(Q)^{1+sp}\mu(\Pi_{\mu,p}^{\tau,*}L)^{-sp}+\mu(Q)^{1+sp}\mu(\Pi_{\mu,p}^{\tau,*}K)^{-sp}$

Let $Q=\Pi_{\mu,p}^{\tau,*}(K\\#_{\mu,p}L)$, we obtain

$\displaystyle\mu(\Pi_{\mu,p}^{\tau,*}(K\\#_{\mu,p}L))^{-sp}\geq\mu(\Pi_{\mu,p}^{\tau,*}L)^{-sp}+\mu(\Pi_{\mu,p}^{\tau,*}K)^{-sp}.$

∎

Clearly, if $\tau=0$ and $p=1$ in Theorem 5.3, then we can obtain

######  Corollary 5.4.

Suppose that $\mu$ satisfies the homogeneity condition (I) and concavity
condition (II) and $\omega$ is an even density function. If
$K,L\in\mathscr{K}_{o}^{n}$, then

$\displaystyle\mu(\Pi_{\mu}^{\ast}(K\\#_{\mu}L))^{-s}\geq\mu(\Pi_{\mu}^{\ast}K)^{-s}+\mu(\Pi_{\mu}^{\ast}L)^{-s}.$

Finally, we prove monotone inequalities for the general $L_{p}$ $\mu$-centroid
body and its polar.

######  Theorem 5.5.

(a) Let $\mu$ satisfy the homogeneity condition (I) and concavity condition
(II), if $p\geq 1$ and $\tau\in[-1,1]$, $K,L\in\mathscr{S}^{n}_{\mathrm{o}}$
and $K\subseteq L$, then

$\displaystyle
p\mu(K)\mu(\Gamma_{\mu,p}^{\tau}K)^{s}\leq(\mu(L)+(p-1)\mu(K))\mu(\Gamma_{\mu,p}^{\tau}L)^{s}.$
(58)

(b) Let $\mu$ satisfy the homogeneity condition (I), if $p\geq 1$ and
$\tau\in[-1,1]$, $K,L\in\mathscr{S}^{n}_{\mathrm{o}}$ and $K\subseteq L$, then

$\displaystyle\frac{\mu(\Gamma_{\mu,p}^{\tau,\ast}K)^{sp}}{\mu(K)}\geq\frac{\mu(\Gamma_{\mu,p}^{\tau,\ast}L)^{sp}}{\mu(L)},$
(59)

equality holds in (59) if and only if $K=L$.

###### Proof.

Since $K,L\in\mathscr{S}^{n}_{o}$ and $K\subseteq L$, by (6), (18), for
$Q\in\mathscr{K}^{n}_{o}$, we obtain

$\displaystyle\widetilde{V}_{\mu,-p}(K,Q)\leq\widetilde{V}_{\mu,-p}(L,Q),$
(60)

equality holds if and only if $K=L$.

For (a), taking $Q=\Pi_{\mu,p}^{\tau,\ast}M$ and $M\in\mathscr{K}^{n}_{o}$ in
(60), we obtain

$\displaystyle\widetilde{V}_{\mu,-p}(K,\Pi_{\mu,p}^{\tau,\ast}M)\leq\widetilde{V}_{\mu,-p}(L,\Pi_{\mu,p}^{\tau,\ast}M),$

From this and Proposition 3.2, we get

$\displaystyle\mu(K)V_{\mu,p}(M,\Gamma_{\mu,p}^{\tau}K)\leq\mu(L)V_{\mu,p}(M,\Gamma_{\mu,p}^{\tau}L),$

in the above inequality, let $M=\Gamma_{\mu,p}^{\tau}L$, and from (17), we
have

$\displaystyle\mu(L)\mu(\Gamma_{\mu,p}^{\tau}L)$
$\displaystyle\geq\mu(K)V_{\mu,p}(\Gamma_{\mu,p}^{\tau}L,\Gamma_{\mu,p}^{\tau}K)$
$\displaystyle\geq
p\mu(K)(\mu(\Gamma_{\mu,p}^{\tau}L)^{1-s}\mu(\Gamma_{\mu,p}^{\tau}K)^{s}+(1-p)\mu(\Gamma_{\mu,p}^{\tau}L)).$

This immediately gives inequality (58).

For (b), let $Q=\Gamma_{\mu,p}^{\tau,\ast}N$ and $N\in\mathscr{K}^{n}_{o}$ in
(60), we obtain

$\displaystyle\widetilde{V}_{\mu,-p}(K,\Gamma_{\mu,p}^{\tau,\ast}N)\leq\widetilde{V}_{\mu,-p}(L,\Gamma_{\mu,p}^{\tau,\ast}N),$

with equality if and only if $K=L$. From this and using Proposition 3.5, we
obtain

$\displaystyle\mu(K)\widetilde{V}_{\mu,-p}(N,\Gamma_{\mu,p}^{\tau,\ast}K)\leq\mu(L)\widetilde{V}_{\mu,-p}(N,\Gamma_{\mu,p}^{\tau,\ast}L).$
(61)

Taking $N=\Gamma_{\mu,p}^{\tau,\ast}L$ in (61), and using Lemma 2.4, we deduce

$\displaystyle\mu(L)\mu(\Gamma_{\mu,p}^{\tau,\ast}L)\geq$
$\displaystyle\mu(K)\widetilde{V}_{\mu,-p}(\Gamma_{\mu,p}^{\tau,\ast}L,\Gamma_{\mu,p}^{\tau,\ast}K)$
$\displaystyle\geq$
$\displaystyle\mu(K)\mu(\Gamma_{\mu,p}^{\tau,\ast}L)^{1+sp}\mu(\Gamma_{\mu,p}^{\tau,\ast}K)^{-sp}.$
(62)

with equality if and only if $\Gamma_{\mu,p}^{\tau}K$ and
$\Gamma_{\mu,p}^{\tau}L$ are dilatates.

(5) yields inequality (59). By the equality conditions of inequalities (60)
and (5), we see that equality holds in (59) if and only if $K=L$. ∎

In particular, if $\tau=0$ and $p=1$ in Theorem 5.5, then we obtain the
following

######  Corollary 5.6.

(a) Let $\mu$ satisfy the homogeneity condition (I) and concavity condition
(II), if $K,L\in\mathscr{S}^{n}_{\mathrm{o}}$ and $K\subseteq L$, then

$\displaystyle\mu(K)\mu(\Gamma_{\mu}K)^{s}\leq\mu(L)\mu(\Gamma_{\mu}L)^{s}.$

(b) Let $\mu$ satisfy the homogeneity condition (I), if
$K,L\in\mathscr{S}^{n}_{\mathrm{o}}$ and $K\subseteq L$, then

$\displaystyle\frac{\mu(\Gamma_{\mu}^{\ast}K)^{s}}{\mu(K)}\geq\frac{\mu(\Gamma_{\mu}^{\ast}L)^{s}}{\mu(L)},$

equality holds if and only if $K=L$.

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