1. Introduction
The projection body was introduced in 1934 by Minkowski [1] . The research on the projection body has attracted much attention. The intersection operator and the class of intersection bodies were introduced in 1988 by Lutwak [2] , who found a close connection between those bodies and famous Busemann-Petty problem (See [3] - [6] ).
In [2] , Lutwak presented the mysterious duality between projection and intersection bodies.
For convex bodies K and
, let
and
denote the projection body of K and mixed projection body of K and
, respectively. In [7] , Lutwak established the following Brunn-Minkowski inequality for projection body and Minkowski inequality for mixed projection body:
Theorem A. Let K and
be convex bodies in
. Then
(1.1)
with equality if and only if K and
are homothetic.
Theorem B. Let K and
be convex bodies in
. Then
(1.2)
with equality if and only if K and
are homothetic.
In [8] , Theorem A and Theorem B were extended to volume differences:
Theorem C. Suppose that K, L, and
are convex bodies in
, and
,
,
is a homo- thetic copy of
. Then
(1.3)
with equality if and only if K and
are homothetic and
, where
is a constant.
Theorem D. Suppose that
, and
are convex bodies in
, and
,
,
is a ho- mothetic copy of
. Then
(1.4)
with equality if and only if K and L are homothetic.
For star bodies K and
, let
and
denote the intersection body of K and mixed intersection body of K and
, respectively. In [9] , Zhao et al. established the following dual Brunn-Minkowski inequality for intersection body and dual Minkowski inequality for mixed intersection body:
Theorem E. Let K and
be star bodies in
. Then
(1.5)
with equality if and only if
is a dilatate of K.
Theorem F. Let K and
be star bodies in
. Then
(1.6)
with equality if and only if
is a dilatate of K.
In this paper, we shall prove the dual forms of inequalities (1.3) and (1.4) for mixed intersection body. In this work new contributions that illustrate this mysterious duality will be presented. Our main results can be stated as follows:
Theorem 1.1. Let
and
are star bodies in
and
,
is a dilatation of
. Then
(1.7)
with equality if and only if
is a dilatate of K and
, where
is a constant.
Theorem 1.2. Let
and
are star bodies in
and
,
is a dilatation of
. Then
(1.8)
with equality if and only if
is a dilatate of K.
Please see the next section for related definitions and notations.
2. Definitions and Notations
In this section, we will recall some basic results for dual quermassintegrals of star bodies. The reader is referred to Gardner [10] , Lutwak [2] [11] and Thompson [12] for the Brunn-Minkowski theory with its dual theory.
As usual, let
denote the unit ball in Euclidean
-space,
. While its boundary is
and its volume is denoted by
. For a compact subset K of
, with
, star-shaped with respect to
, the radial function
, is defined by
(2.1)
If
is continuous and positive, K will be called a star body.
Two star bodies
are said to be dilatate (of each other) if
is independent of
.
The radial sum of two star bodies
is defined as the star body K satisfying
. This operation will be denoted by
, i.e., ![]()
For star bodies
, the dual mixed volume
is defined by (see e.g. [11] )
(2.2)
If
, and
, then the dual mixed volume
is called dual
-quermassintegral of
, and denoted by
and allow
. It is easily seen that
.
Let
and
be star bodies in
. If
, then the dual
-quermassintegral difference function of
and
, ![]()
, can be defined by
(2.3)
If
in (2.3), then we get the volume difference of star bodies
and
:
![]()
(See [13] for the concept of the volume difference of two compact domains).
The intersection body of a star body K,
, is the centrally symmetric body whose radial function on
is given by [2]
(2.4)
where
is
-dimensional volume.
Let
be star bodies in
. The mixed intersection body
of star bodies
is defined by
(2.5)
where
is
-dimensional dual mixed volume.
If
,
, then
will be denoted as
. If
, then
is called the intersection body of order
of
; it will often be written as
. Specially,
. This term was introduced by Zhang [14] .
3. Inequalities for Dual Quermassintegral Differences
In this section, we will establish two inequalities for dual quermassintegral differences of star bodies, which are generalizations of Theorem 1.1 and Theorem 1.2 presented in introduction.
Theorem 3.1. Suppose that
and
are star bodies in
,
is a dilatate of
. If
,
, then
![]()
with equality if and only if
is a dilatate of K and
where
is a constant.
Obviously, the case
in Theorem 3.1 is just Theorem 1.1. Furthermore, taking
and
to be two closed balls with radii
and
in Theorem 1.1, we infer
Corollary 3.2. Let
and
be the circumradii of star bodies K and L. If
,
, then
![]()
with equality if and only if
is a dilatate of K and
.
Theorem 3.3. Suppose that
and D1 are star bodies in
, D2 is a dilatate of D1. If
, and
. Then
![]()
with equality if and only if
is a dilatate of K and
where
is a constant.
Obviously, the case
in Theorem 3.3 is just Theorem 1.2.
We will require some additional notations and two analytic inequalities to prove Theorem 3.1 and Theorem 3.3. Firstly, we define a function
by
![]()
where
for p > 0. Note that
is an indefinite form of its variables.
Lemma 3.4. If
,
, then
, and
(3.1)
with equality holds if and only if the coordinates of
are proportional.
A proof of Lemma 3.4 can be found in [15] . The inequality (3.1) was first proved by Bellman [16] and is known as Bellman’s inequality.
Lemma 3.5. If
,
and
. Let
and
, then
![]()
with equality if and only if
.
Proof. Consider the following function
![]()
Let
![]()
We get
.
On the other hand, if
, then
; if
then
, and it follows that
![]()
This completes the proof.
Lemma 3.6. [15] Let
be star bodies in
. If
, then
(3.2)
and
(3.3)
with equality if and only if
is a dilatate of
.
Proof of Theorem 3.1. For star bodies
, applying inequality (3.2), we have
(3.4)
with equality if and only if
is a dilatate of
.
(3.5)
Since
, we get
![]()
From (3.4) and (3.5), we obtain that
![]()
Then by Lemma 3.4, we get
(3.6)
Note that the equality holds in (3.4) if and only if L is a dilatate of K. By Lemma 3.4 we know that the equality holds in (3.6) if and only if L is a dilatate of K. and
is proportional to ![]()
This completes the proof.
Proof of Theorem 3.3. Applying inequality (3.3), we have
![]()
with equality if and only if
is a dilatate of
.
![]()
Hence, by Lemma 3.5, we obtain that
![]()
The proof is complete.
Acknowledgments
We thank the Editor and the referee for their comments. The research is supported by National Natural Science Foundation of China (11101216), Qing Lan Project and the Nanjing Xiaozhuang University (2010KYQN24, 2010KYYB13).
NOTES
*Corresponding author.