- Research Article
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Some Inequalities for the
-Curvature Image
Journal of Inequalities and Applications volume 2009, Article number: 320786 (2009)
Abstract
Lutwak introduced the notion of -curvature image and proved an inequality for the volumes of convex body and its
-curvature image. In this paper, we first give an monotonic property of
-curvature image. Further, we establish two inequalities for the
-curvature image and its polar, respectively. Finally, an inequality for the volumes of
-projection body and
-curvature image is obtained.
1. Introduction
Let denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space
, for the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in
, we, respectively, write
and
. Let
denote the unit sphere in
, and denote by
the
-dimensional volume of body
, for the standard unit ball
in
, and denote
. The groups of nonsingular linear transformations and the group of special linear transformations are denoted by
and
, respectively.
Suppose that is the set of real numbers. If
, then its support function,
:
, is defined by (see [1, page 16])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ1_HTML.gif)
where denotes the standard inner product of
and
.
A convex body is said to have a curvature function
, if its surface area measure
is absolutely continuous with respect to spherical Lebesgue measure
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ2_HTML.gif)
For , and real
, the
-surface area measure,
, of
is defined by (see [2, 3])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ3_HTML.gif)
Equation (1.3) is also called Radon-Nikodym derivative, and the measure is absolutely continuous with respect to surface area measure
.
A convex body is said to have a
-curvature function (see [2])
, if its
-surface area measure
is absolutely continuous with respect to spherical Lebesgue measure
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ4_HTML.gif)
If is a compact star shaped (about the origin) in
, its radial function,
, is defined by (see [1, page 18])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ5_HTML.gif)
If is positive and continuous,
will be called a star body (about the origin). Let
denote the set of star bodies (about the origin) in
. Two star bodies
and
are said to be dilates (of one another) if
is independent of
.
For the radial function, if , then (see [1, page 18])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ6_HTML.gif)
From (1.6), we have that, for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ7_HTML.gif)
Let ,
denote the set of all bodies in
,
, respectively, that have a positive continuous curvature function.
Lutwak in [2] showed the notion of -curvature image as follows. For each
and real
, define
, the
-curvature image of
, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ8_HTML.gif)
Note that, for , this definition differs from the definition of classical curvature image (see [2]). For the study of classical curvature image [1, 4–7].
Further, he proved that if and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ9_HTML.gif)
with equality if and only if
is an ellipsoid centered at the origin.
In this paper, we continuously study the -curvature image for convex bodies. First, we give a monotonic property of
-curvature image as follows.
Theorem 1.1.
If ,
, and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ10_HTML.gif)
with equality for if and only if
and
are dilates, for
if and only if
, and for
if and only if
and
are translation.
Next, we establish an inequality for the -curvature image as follows.
Theorem 1.2.
If , and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ11_HTML.gif)
with equality if and only if is an ellipsoid.
Further, we get the following inequality for the polar of the -curvature image.
Theorem 1.3.
If ,
, and
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ12_HTML.gif)
with equality for if and only if
and
are dilates, and for
if and only if
and
are homothetic.
Here denote the polar of
, rather than
. Compare with inequality (1.9), we see that inequality (1.12) may be regarded as a dual form of inequality (1.9).
Finally, we obtain an interesting inequality for the -curvature image and
-projection body
as follows.
Theorem 1.4.
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ13_HTML.gif)
with equality if and only if is an ellipsoid centered at the origin.
2. Preliminaries
2.1. Polar of Convex Body
If , the polar body of
,
, is defined by (see [1, page 20])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ14_HTML.gif)
From the definition (2.1), we know that if , then the support and radial functions of
, the polar body of
, are defined, respectively, by (see [1])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ15_HTML.gif)
The Blaschke-Santaló inequality can be stated that (see [1] or [7]): If, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ16_HTML.gif)
with equality if and only ifis an ellipsoid.
2.2.
-Mixed Volume
For and
, the Firey
-combination
is defined by (see [8])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ17_HTML.gif)
where "" in
denotes the Firey scalar multiplication.
If in
, then for
, the
-mixed volume,
, of the
and
is defined by (see [9])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ18_HTML.gif)
Corresponding to each , there is a positive Borel measure,
, on
such that (see [9])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ19_HTML.gif)
for each . The measure
is just the
-surface area measure of
.
From the formula (2.6) and definition (1.3), we immediately get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ20_HTML.gif)
The -Minkowski inequality states that (see [9]) if
and
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ21_HTML.gif)
with equality for if and only if
and
are dilates, and for
if and only if
and
are homothetic.
2.3.
-Dual Mixed Volume
For , and
, the
-harmonic radial combination
is the star body whose radial function is defined by (see [2])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ22_HTML.gif)
Note that here "" and the Firey scalar multiplication "
" are different.
If , for
, the
-dual mixed volume,
, of the
and
is defined by (see [2])
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ23_HTML.gif)
The definition above and the polar coordinate formula for volume give the following integral representation of the -dual mixed volume
of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ24_HTML.gif)
where the integration is with respect to spherical Lebesgue measure on
.
From the formula (2.11), it follows immediately that, for each and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ25_HTML.gif)
The Minkowski inequality for the -dual mixed volume
is that if
and
(see [2]), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ26_HTML.gif)
with equality if and only if and
are dilates.
2.4.
-Affine Surface Area
Lutwak in [2] showed that for each and
, the
-affine surface area,
, of
can be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ27_HTML.gif)
For ,
is just classical affine surface area
by Leichtweiβ (see [4]). Further, Lutwak proved that if
and
, then the
-affine surface area of
has the integral representation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ28_HTML.gif)
2.5.
-Projection Body
The notion of -projection body is shown by Lutwak et al. (see [10]). For
and
, the
-projection body,
, of
is the origin-symmetric convex body whose support function is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ29_HTML.gif)
for all . Here
is just the
-surface area measure of
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ30_HTML.gif)
2.6.
-Centroid Body
Lutwak and Zhang in [11] introduced the notion of -centroid body. For each compact star-shaped body about the origin
and for real number
, the polar of
-centroid body,
(rather than
), of
is the origin-symmetric convex body, whose radial function is defined by [11]
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ31_HTML.gif)
for all , where
satisfy (2.17).
From definition (2.18) and equality (2.2), if , then the
-centroid body
of
is the origin-symmetric convex body whose support function is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ32_HTML.gif)
for all .
3. The Proof of Theorems
In order to prove our theorems, the following lemmas are essential.
Lemma 3.1.
If ,
and the constant
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ33_HTML.gif)
Proof.
For , from (1.3) and (1.4), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ34_HTML.gif)
this together with (1.7) and (1.8), and notice that for
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ35_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ36_HTML.gif)
and this together with formula (2.12), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ37_HTML.gif)
Hence, from (3.4), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ38_HTML.gif)
and this yields (3.1).
If , Lutwak (see [2]) proved that, for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ39_HTML.gif)
where denotes the inverse of the transpose of
.
Now we rewrite (3.1) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ40_HTML.gif)
this together with (3.7) and the fact , we easily get the following result.
Proposition 3.2.
If ,
, then for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ41_HTML.gif)
Lemma 3.3 (see [2]).
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ42_HTML.gif)
for all .
Lemma 3.4.
If ,
, then for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ43_HTML.gif)
Proof.
Taking in (3.11), and using (2.12), we have that
. Now inequality (2.13) gives
, with equality if and only if
and
are dilates. Let
in (3.11), and get
. Hence
, and
and
must be dilates. Thus
. In turn, when
the result obviously is true.
Proof of Theorem 1.1.
Since , then from formula (2.11), we know
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ44_HTML.gif)
for all , with equality in (3.12) if and only if
by (3.11). Using equality (3.10), then inequality (3.12) can be rewritten
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ45_HTML.gif)
for all . Let
, together with (2.7) and
-Minkowski inequality (2.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ46_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ47_HTML.gif)
and this is just inequality (1.10).
According to the conditions of equality that hold in inequalities (3.12) and (2.8), we know that equality holds in inequality (1.10) for if and only if
and
are dilates and
, and for
if and only if
and
are homothetic and
.
For the case of equality that holds in (1.10), we may suppose
(
), and together with
, then
. Thus, from (3.1), we have
. Hence
when
, this means that if
then
. For
, we easily see that
and
are dilates that impliy
. So we know that equality holds in inequality (1.10) for
if and only if
and
are dilates, and for
if and only if
.
For the case of equality that holds in (1.10), we may take
(
), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ48_HTML.gif)
But , then
by (1.2). By this together with (2.15) and (3.10), we have
and
, respectively. Thus, from the definition (1.8), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ49_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ50_HTML.gif)
From (3.18) and (3.1), equality (3.16) can be rewritten as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ51_HTML.gif)
and this gives , that is,
when
. Therefore, we see that equality holds in inequality (1.10) for
if and only if
and
are translation.
Proof of Theorem 1.2.
Let in (3.10), together with (2.7) and (2.13), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ52_HTML.gif)
with equality in inequality (3.20) if and only if and
are dilates.
From this, and using the Blaschke-Santaló inequality (2.3), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ53_HTML.gif)
and equality holds in second inequality of (3.21) if and only if is an ellipsoid.
From (3.21), we immediately obtain inequality (1.11). According to the conditions of equality that hold in (3.20) and second inequality of (3.21), we get equality in (1.11) if and only if is an ellipsoid.
Proof of Theorem 1.3.
Taking in (3.10), and using (2.12), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ54_HTML.gif)
From (3.22), and together with inequality (2.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ55_HTML.gif)
this inequality immediately gives (1.12). According to equality conditions of inequality (2.8), we get equality in (1.12) for if and only if
and
are dilates, and for
if and only if
and
are homothetic.
The proof of Theorem 1.4 requires the following two lemmas.
Lemma 3.5.
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ56_HTML.gif)
Note that the proof of Lemma 3.5 can be found in [12]. Here, for the sake of completeness, we present the proof as follows.
Proof.
Using the definitions (2.16), (1.4), and (1.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ57_HTML.gif)
for all . According to (2.19), we also have that, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ58_HTML.gif)
But (2.17) gives hence from (3.25) and (3.26), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ59_HTML.gif)
for all . Thus
.
Lemma 3.6 ([10] (-Busemann-Petty centroid inequality)).
If ,
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F320786/MediaObjects/13660_2009_Article_1935_Equ60_HTML.gif)
with equality if and only if is an ellipsoid centered at the origin.
Proof of Theorem 1.4.
From (3.28) and (3.24), we immediately get inequality (1.13). According to the case of equality that holds in (3.28), we see equality in (1.13) if and only if is an ellipsoid centered at the origin.
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Acknowledgment
This research is supported in part by the Natural Science Foundation of China (Grant no. 10671117), Academic Mainstay Foundation of Hubei Province of China (Grant no. D200729002), and Science Foundation of China Three Gorges University. The authors wish to thank the referees for their very helpful comments and suggestions on this paper.
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Weidong, W., Daijun, W. & Yu, X. Some Inequalities for the -Curvature Image.
J Inequal Appl 2009, 320786 (2009). https://doi.org/10.1155/2009/320786
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DOI: https://doi.org/10.1155/2009/320786