© Wang Weidong et al. 2009
Received: 16 May 2009
Accepted: 14 October 2009
Published: 19 October 2009
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.
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])
A convex body is said to have a -curvature function (see ) , if its -surface area measure is absolutely continuous with respect to spherical Lebesgue measure , and
If is a compact star shaped (about the origin) in , its radial function, , is defined by (see [1, page 18])
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])
Lutwak in  showed the notion of -curvature image as follows. For each and real , define , the -curvature image of , by
2.1. Polar of Convex Body
If , the polar body of , , is defined by (see [1, page 20])
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 )
For and , the Firey -combination is defined by (see )
If in , then for , the -mixed volume, , of the and is defined by (see )
Corresponding to each , there is a positive Borel measure, , on such that (see )
From the formula (2.6) and definition (1.3), we immediately get that
The -Minkowski inequality states that (see ) if and then
For , and , the -harmonic radial combination is the star body whose radial function is defined by (see )
If , for , the -dual mixed volume, , of the and is defined by (see )
The Minkowski inequality for the -dual mixed volume is that if and (see ), then
Lutwak in  showed that for each and , the -affine surface area, , of can be defined by
For , is just classical affine surface area by Leichtweiβ (see ). Further, Lutwak proved that if and , then the -affine surface area of has the integral representation
The notion of -projection body is shown by Lutwak et al. (see ). For and , the -projection body, , of is the origin-symmetric convex body whose support function is given by
Lutwak and Zhang in  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 
3. The Proof of Theorems
In order to prove our theorems, the following lemmas are essential.
and this yields (3.1).
If , Lutwak (see ) proved that, for ,
Now we rewrite (3.1) as follows:
Lemma 3.3 (see ).
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.
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 .
Proof of Theorem 1.2.
From this, and using the Blaschke-Santaló inequality (2.3), then
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.
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.
Note that the proof of Lemma 3.5 can be found in . Here, for the sake of completeness, we present the proof as follows.
Lemma 3.6 ( ( -Busemann-Petty centroid inequality)).
Proof of Theorem 1.4.
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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