1. Introduction
Recently, Convex Geometry Analysis has made great achievement in Orlicz space (see [1] - [14] ). Zhu, Zhou and Xu [12] defined the Orlicz radial sum and dual Orlicz mixed volumes. Let
be the set of convex and strictly decreasing functions
such that
,
and
.
Let K and L be two star bodies about the origin in
and
; the Orlicz radial sum
was defined by [13]
(1.1)
The case
of the Orlicz radial sum is the
harmonic radial sum, which was defined by Lutwak (see [15] ).
Let
denote the right derivative of a real-valued function
. For
, there is
because
is convex and strictly decreasing. The dual Orlicz mixed volume
is defined by
(1.2)
In this paper, we will define the dual Orlicz mixed quermassintegral
by
(1.3)
The main purpose of this paper is to establish the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality for dual Orlicz mixed quermassintegrals.
Theorem 1.1 Let K and L be two star bodies about the origin in
and
. If
, then
(1.4)
with equality if and only if K and L are dilates of each other.
Theorem 1.2 Let K and L be two star bodies about the origin in
and
. If
, then
(1.5)
with equality if and only if K and L are dilates of each other.
This paper is organized as follows: In Section 2 we introduce above interrelated notations and their background materials. Section 3 contains the proofs of our main results.
2. Notation and Background Material
The radial function
of a compact star-shaped about the origin
is defined, for
, by
(2.1)
If
is positive and continuous, then K is called a star body about the origin. The set of star bodies about the origin in
is denoted by
. Obviously, for
,
(2.2)
If
is independent of
, then we say star bodies K and L are dilates of
each other.
If
and
are nonnegative real numbers, then the volume of
is a homogeneous polynomial of degree n in
given by
![]()
where the sum is taken over all n-tuples
of positive integers not exceeding m. The coefficient
depends only on the bodies
, and is uniquely determined by the above identity, it is called the dual mixed volume of
. More explicitly, the dual mixed volume
has the following integral representation [16] :
(2.3)
where S is the Lebesgue measure on ![]()
The coefficients
are nonnegative, symmetric and monotone (with respect to set inclusion). They are also multilinear with respect to the radial sum and
. Let
and
, then the dual mixed volume
is usually written as
. If L = B, then
is the dual quermassintegral
. For
, the dual mixed quermassinte-
gral
denotes the dual mixed volume
. For
,
then
.
The dual mixed quermassintegral
has the following integral representation:
(2.4)
where S is the Lebesgue measure on
.
By using the Minkowski’s integral inequality, we can obtain the dual Minkowski inequality for dual mixed quermassintegrals: If
, and
, then
(2.5)
equality holds if and only if K and L are dilates of each other.
Suppose that m is a probability measure on a space X and
is a m- intergrable function, where I is a possibly infinite interval. Jessen’s inequality states that if
is a convex function, then
(2.6)
If
is strictly convex, equality holds if and only if
is a constant for m-almost all
(see [17] ).
3. Main Results
Let
and
. For
, the dual Orlicz mixed quermassintegral
is defined by
(3.1)
For
, then
. The case
of the dual Orlicz mixed quermassintegral
is the dual Orlicz mixed volume
, which was defined by Zhu, Zhou and Xu [12] .
Corollary 3.1 The dual Orlicz mixed quermassintegral
is monotone with respect to set inclusion.
Proof. Let
and
. By (3.1), (2.2) and the fact that
is strictly decreasing on
, we have
![]()
Lemma 3.1 [12] Let
and
. If
, then
![]()
if and only if
![]()
Lemma 3.2 [12] Let
and
. Then
(3.2)
uniformly for all
.
Theorem 3.1 Let
and
. For
, then
![]()
Proof. Suppose
, and
. Note that
as
(see [12] ). By Lemma 3.2, it follows that
![]()
uniformly on
.
Hence
![]()
We complete the proof of Theorem 3.1. ,
From (3.1) and Theorem 3.1, we have
(3.3)
For
, since
, then
is a probabil-
ity measure on
.
Proof of Theorem 1.1
By (3.1), (2.6), (2.5) and the fact that
is decreasing on
, we obtain
![]()
This gives the desired inequality. Since
is strictly decreasing, from the equality condition of the dual Minkowski inequality (2.5), we have that K and L are dilates of each other.
Conversely, when
, by (3.1), we have
,
The following uniqueness is a direct consequence of the dual Orlicz-Minkowski inequality (1.4).
Corollary 3.2 Suppose
, and
such that
. For
, if
(3.4)
or
(3.5)
then
.
Proof. Suppose (3.4) holds. If we take K for M, then from (3.1), we obtain
![]()
Hence, from the dual Orlicz-Minkowski inequality (1.4), we have
![]()
with equality if and only if K and L are dilates of each other. Since
is strictly decreasing on
, we have
![]()
with equality if and only if K and L are dilates of each other. If we take L for M, we similarly have
. Hence,
and from the equality condition we can conclude that K and L are dilates of each other. However, since they have the same volume they must be equal.
Next, suppose (3.5) holds. If we take K for M, then from (3.1), we obtain
![]()
Then, from the dual Orlicz-Minkowski inequality (1.4), we have
![]()
with equality if and only if K and L are dilates of each other. Since
is strictly decreasing on
, we have
![]()
with equality if and only if K and L are dilates of each other. If we take L for M, we similarly have
. Hence,
and from the equality condition we can conclude that K and L are dilates of each other. However, since they have the same volume they must be equal.
From the dual Orlicz-Minkowski inequality, we will prove the following dual Orlicz-Brunn-Minkowski inequality which is more general than Theorem 1.2.
Theorem 3.2 Let
,
and
. If
, then
![]()
with equality if and only if K and L are dilates of each other.
Proof. Let
. From (2.3), Lemma 3.1 and (1.4), it follows that
![]()
By the equality condition of the dual Orlicz-Minkowski inequality (1.4), equality in (3.6) holds if and only if K and L are dilates of each other.
Indeed, we also can prove the dual Orilcz-Minkowski inequality by the dual Orilcz- Brunn-Minkowski inequality.
Proof. For
, let
. Note that
as
. By the dual Orlicz-Brunn-Minkowski inequality, the following function
![]()
is non-positive. Obviously,
. Thus
(3.7)
On the other hand, we have
(3.8)
Let
and note that
as
. Consequently,
(3.9)
By (3.3), we have
(3.10)
From (3.8), (3.9), and (3,10), it follows that
(3.11)
Combing (3.7) and (3.11), we have
(3.12)
Therefore, the equality in (3.12) holds if and only if
, this implies that K and L are dilates of each other.
Remark 3.1 The case
of Theorem 1.1 and Theorem 1.2 were established by Zhu, Zhou and Xu [12] . The dual forms of Theorem 1.1 and Theorem 1.2 were established by Xiong and Zou [11] .