Open Access

Some Inequalities for the -Curvature Image

Journal of Inequalities and Applications20092009:320786

DOI: 10.1155/2009/320786

Received: 16 May 2009

Accepted: 14 October 2009

Published: 19 October 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])

(1.1)

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

(1.2)

For , and real , the -surface area measure, , of is defined by (see [2, 3])

(1.3)

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

(1.4)

If is a compact star shaped (about the origin) in , its radial function, , is defined by (see [1, page 18])

(1.5)

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])

(1.6)

From (1.6), we have that, for ,

(1.7)

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

(1.8)

Note that, for , this definition differs from the definition of classical curvature image (see [2]). For the study of classical curvature image [1, 47].

Further, he proved that if and , then

(1.9)

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
(1.10)

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
(1.11)

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
(1.12)

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
(1.13)

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])

(2.1)

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])

(2.2)

The Blaschke-Santaló inequality can be stated that (see [1] or [7]): If , then

(2.3)

with equality if and only if is an ellipsoid.

2.2. -Mixed Volume

For and , the Firey -combination is defined by (see [8])

(2.4)

where " " in denotes the Firey scalar multiplication.

If in , then for , the -mixed volume, , of the and is defined by (see [9])

(2.5)

Corresponding to each , there is a positive Borel measure, , on such that (see [9])

(2.6)

for each . The measure is just the -surface area measure of .

From the formula (2.6) and definition (1.3), we immediately get that

(2.7)

The -Minkowski inequality states that (see [9]) if and then

(2.8)

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])

(2.9)

Note that here " " and the Firey scalar multiplication " " are different.

If , for , the -dual mixed volume, , of the and is defined by (see [2])

(2.10)

The definition above and the polar coordinate formula for volume give the following integral representation of the -dual mixed volume of :

(2.11)

where the integration is with respect to spherical Lebesgue measure on .

From the formula (2.11), it follows immediately that, for each and ,

(2.12)

The Minkowski inequality for the -dual mixed volume is that if and (see [2]), then

(2.13)

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

(2.14)

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

(2.15)

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

(2.16)

for all . Here is just the -surface area measure of , and

(2.17)

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]

(2.18)

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

(2.19)

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
(3.1)

Proof.

For , from (1.3) and (1.4), then
(3.2)
this together with (1.7) and (1.8), and notice that for , we get that
(3.3)
that is,
(3.4)
and this together with formula (2.12), we have that
(3.5)
Hence, from (3.4), then
(3.6)

and this yields (3.1).

If , Lutwak (see [2]) proved that, for ,

(3.7)

where denotes the inverse of the transpose of .

Now we rewrite (3.1) as follows:

(3.8)

this together with (3.7) and the fact , we easily get the following result.

Proposition 3.2.

If , , then for ,
(3.9)

Lemma 3.3 (see [2]).

If , , then
(3.10)

for all .

Lemma 3.4.

If , , then for all ,
(3.11)

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
(3.12)
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
(3.13)
for all . Let , together with (2.7) and -Minkowski inequality (2.8), we have
(3.14)
Thus
(3.15)

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

(3.16)
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
(3.17)
hence
(3.18)
From (3.18) and (3.1), equality (3.16) can be rewritten as follows:
(3.19)

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
(3.20)

with equality in inequality (3.20) if and only if and are dilates.

From this, and using the Blaschke-Santaló inequality (2.3), then

(3.21)

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
(3.22)
From (3.22), and together with inequality (2.8), we have
(3.23)

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
(3.24)

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
(3.25)
for all . According to (2.19), we also have that, for all ,
(3.26)
But (2.17) gives hence from (3.25) and (3.26), we obtain
(3.27)

for all . Thus .

Lemma 3.6 ([10] ( -Busemann-Petty centroid inequality)).

If , , then
(3.28)

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.

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, China Three Gorges University
(2)
Department of Mathematics, Hubei Institute for Nationalities

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© Wang Weidong et al. 2009

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