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Some inequalities related to -type -mixed affine surface area and -mixed curvature image
Journal of Inequalities and Applications volume 2013, Article number: 470 (2013)
Abstract
In this article, we introduce two concepts: the -type -mixed affine surface area and -type -mixed affine surface area in the set of convex bodies such that -affine surface area by Lutwak et al. is proposed in its special cases. Besides, applying these concepts, we establish the extension results of the well-known -Petty affine projection inequality, -Busemann centroid inequality and its dual inequality.
MSC:52A40, 52A20.
1 Introduction
Let denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space . For the set of convex bodies, which contain the origin in their interiors, and the set of origin-symmetric convex bodies in , we write and , respectively. Let denote the set of star bodies (about the origin) in , and let denote the set of origin-symmetric star bodies in . Let denote the unit sphere in , and let denote the n-dimensional volume of body K. If K is the standard unit ball B in , then it is denoted by . Note that defines for all non-negative real n (not just the positive integers).
A body is said to have a continuous i th curvature function if and only if is absolutely continuous with respect to S and has the Radon-Nikodym derivative (see [1])
Let denote the subset of all bodies which have a positive continuous i th curvature function. Let , denote the set of all bodies in , , respectively, and both of them have a positive continuous i th curvature function.
A convex body is said to have an -curvature function if its -surface area measure is absolutely continuous with respect to the spherical Lebesgue measure S, and it has the Radon-Nikodym derivative (see [2])
For , , then -affine surface area of K by (see [3–5])
For each and , the -projection body, , of K is an origin-symmetric convex body whose support function is given by (see [6])
where . When , (1.4) is the notion of projection body (see [7]).
It is easy to show that if E is an ellipsoid which is centered at the origin, then (see [[8], p.105])
The well-known -Petty affine projection inequality is expressed as follows (see [8–10]).
Theorem A (-Petty affine projection inequality)
If , , then
with equality if and only if K is an ellipsoid which is centered at the origin.
Let , and let , then the -centroid body, , of K is the origin-symmetric convex body whose support function is given by (see [6, 11])
If E is an ellipsoid which is centered at the origin, then . In particular, .
The well-known -Busemann-Petty centroid inequality is as follows (see [6]).
Theorem B (-Busemann-Petty centroid inequality)
If and , then
with the equality if and only if K is an ellipsoid which is centered at the origin.
Lutwak et al. introduced the concept of dual -centroid bodies (see [12]). We give the concept of the unusual normalization of dual -centroid bodies such that : Let and real , then radial function of dual -centroid body, , of K is defined by
It is easy to show that if E is an ellipsoid which is centered at the origin, then
Combined with (1.5) and (1.10), we have that . In particular, .
Si Lin gives the following dual inequality of inequality (1.8) (see [[13], p.9, Theorem 5.4]).
Theorem C (Dual -Busemann-Petty centroid inequality)
If and , then
with equality if and only if K is an ellipsoid which is centered at the origin.
Liu et al. [14], Lu and Wang [15], Ma and Liu [16, 17] independently proposed the notion of -mixed curvature function: Let , , a convex body is said to have an -mixed curvature function , if its -mixed surface area measure is absolutely continuous with respect to spherical Lebesgue measure S and has the Radon-Nikodym derivative
If the mixed surface area measure is absolutely continuous with respect to spherical Lebesgue measure S, we have
According to the concept of -mixed curvature function of convex body, Lu and Wang [15] and Ma introduce the concept of -mixed curvature image of convex body as follows: For each () and real , define , the -mixed curvature image of K, by
If in (1.14), then . The unusual normalization of definition (1.14) is chosen so that for the unit ball B, we have . For , , ,
Let denote the set of -mixed curvature images of convex bodies in . That is,
Because the -mixed curvature image belongs to star bodies, thus, .
For each , real and , the -mixed projection body, , of K is an origin-symmetric convex body whose support function is given by (see [18])
where () is -mixed surface area measure. Obviously, , and for the standard unit ball B, we have . For , , and , then
Let , real and i be arbitrary real numbers, then the -mixed centroid body, , of K is the origin-symmetric convex body whose support function is given by (see [19])
Obviously, , and for the standard unit ball B, we have .
Ma introduced the concept of dual -mixed centroid body (see [19]). Further, we introduced the concept of the unusual normalization of dual -mixed centroid body as follows: Let , , , then the dual -mixed centroid bodies, , of K are defined by:
Obviously, , and for the standard unit ball B, we have .
In this article, we will first introduce the concept of -type -mixed affine surface area of convex body as follows.
Definition 1.1 For , and , the -type -mixed affine surface area, , of K is defined by
Next, we have established an extension of -Petty affine projection inequality (1.6) as follows.
Theorem 1.1 Let , and , then
with equality in inequality (1.21) for if and only if K is a ball which is centered at the origin; for if and only if K is an ellipsoid which is centered at the origin.
Further, we obtain the following generalized -Busemann-Petty centroid inequality.
Theorem 1.2 Suppose that , and , then
with equality in inequality (1.22) for if and only if K is a ball which is centered at the origin; for if and only if K is an ellipsoid which is centered at the origin.
Finally, we get the following dual inequality of the inequality (1.22).
Theorem 1.3 Suppose that . If , and , then
with equality in inequality (1.23) for if and only if K is a ball which is centered at the origin; for if and only if K is an ellipsoid which is centered at the origin.
2 Preliminaries
2.1 Support function, radial function and polar of convex body
If , then its support function, , is defined by (see [20, 21])
Obviously, if , λ is a positive constant, , then .
If K is a compact star-shaped (about the origin) in , its radial function, , is defined by (see [20, 21])
When is positive and continuous, K is called a star body (about the origin). Obviously, if , , , then and . Two star bodies K and L are said to be dilates (of one another) if is independent on .
For , the polar body, , of K is defined by (see [20, 21])
Obviously, we have . If , then
If , then the support and radial function of the polar body , of K are defined respectively by (see [20, 21])
for all .
2.2 The quermassintegrals, -mixed quermassintegrals and -mixed volume
For and , the quermassintegrals, , of K are defined by (see [20, 21])
From (2.3), we easily see that
For , and , the Firey -combination is defined by (see [22])
where ‘⋅’ in denotes the Firey scalar multiplication.
For , and real , the -mixed quermassintegrals, , of K and L () are defined by (see [1])
Obviously, for , is just the classical mixed quermassintegral . For , the -mixed quermassintegral is just the -mixed volume , namely,
For , and each , there exists a positive Borel measure on such that the -mixed quermassintegral has the following integral representation (see [1]):
for all . It turns out that the measure () on is absolutely continuous with respect to and has the Radon-Nikodym derivative
From (2.3) and (2.7), it follows immediately that for each ,
If , , by definition (1.12), then formula (2.7) of the -mixed quermassintegral can be rewritten as follows:
We shall require the Minkowski inequality for the -mixed quermassintegrals as follows (see [1]): For , and , , then
with equality for and if and only if K and L are homothetic; for and if and only if K and L are dilates. For and , inequality (2.11) is identical.
2.3 Dual quermassintegrals and -dual mixed quermassintegrals
For and any real i, the dual quermassintegrals, , of K are defined by (see [20, 21])
Obviously,
For , and , the -harmonic radial combination is defined by (see [2, 23, 24])
Note that here ‘’ is different from ‘’ in the Firey -combination.
For , , and real , the -dual mixed quermassintegrals, , of K and L are defined by [25]
If , we easily see that definition (2.14) is just the definition of -dual mixed volume, namely,
From definition (2.14), the integral representation of the -dual mixed quermassintegrals is given by (see [25]): If , , and real , , then
Together with (2.12) and (2.16), for , , and , we get
Analog of the Minkowski inequality for -dual mixed quermassintegrals is as follows (see [25]): If , , then for or ,
For , inequality (2.18) is reversed. With equality in every inequality if and only if K and L are dilates.
3 The -type -mixed affine surface area
In this section, we further propose the concept of -type -mixed affine surface area as follows.
Definition 3.1 For , , and , the -type -mixed affine surface area, , of K is defined by
Obviously, and .
Next, we introduce the concept of -type -mixed affine surface area of the convex bodies as follows.
Definition 3.2 For , , the -type -mixed affine surface area, , of is defined by
Let and (), we define with copies of K and j copies of L. From this, if j is any real number, we can define the following.
Definition 3.3 For , , , , the -type -mixed affine surface area, , of K, L is defined by
Specially for the case , we have that
Take in (3.4) and write
Because for , , , so by (2.8) and (1.13), we get . This together with (3.3) and (3.5) yields
and is called the -type -mixed affine surface area of . In particular, is called the j th -mixed affine surface area of (see [26]).
Next, we give some propositions of -mixed curvature image and -type -mixed affine surface area.
Proposition 3.1 Let , , . Then
In particular, take in (3.7), then
Proof From (3.6), (1.14) and (2.12), we have
□
Proposition 3.2 Let , and . Then
for each .
Proof For each , from (2.10), (1.14), (2.2) and (2.16), we have
□
Proposition 3.3 If , , then
for all with equality if and only if K and are dilates.
Proof Let and each , then from (1.20), (2.2), (2.7), (2.12) and Hölder’s inequality, we have
From this, we immediately get (3.10).
According to the condition of equality to hold in the Hölder inequality, we know that equality holds in (3.10) if and only if
for any , where c is a constant. Combined with the definition of -mixed curvature image, for any , we have
this shows that K and are dilates. Therefore, the equality holds in inequality (3.10) if and only if K and are dilates. The proof is complete. □
Now, according to Proposition 3.3, we can give an expansion of the definition of the -type -mixed affine surface area of as follows.
Definition 3.4 If , , then the -type -mixed affine surface area, , of K is defined by
For , the definition is just the definition of -affine surface area by Lutwak proposed in [2].
4 Generalized -Petty affine projection inequality
In this section, we complete the proof of Theorem 1.1 in the introduction. In fact, we prove the following more general conclusion.
Theorem 4.1 Let , , , , then
with equality in inequality (4.1) for if and only if K and L are balls of dilates which are centered at the origin; for if and only if K and L are ellipsoids of dilates which are centered at the origin.
In order to prove the theorems above, we first give the following three lemmas.
Lemma 4.1 (See [27])
Suppose that , and , then
with the equality for if and only if K is a ball which is centered at the origin. If , then (4.2) is identical.
Lemma 4.2 (See [18])
Suppose that , and , i is a positive integer, then
with equality if and only if K is a ball which is centered at the origin.
Remark 4.1 The conditions of inequality (4.3) can be relaxed to and , while the conditions of the equality that holds can be given separately. For and , the inequality (4.3) is proved by Lutwak with the equality holding if and only if K is a ball (see [7]). For and , inequality (4.3) is proved by Lutwak et al. with the equality that holds if and only if K is an ellipsoid which is centered at the origin (see [6]). For and , then (4.3) is the famous Petty projection inequality (see [28]), with the equality that holds if and only if K is an ellipsoid.
Lemma 4.3 If , , , then
Proof From (1.16), (2.10) and the Fubini theorem, it is easy to prove Lemma 4.3. □
Proof of Theorem 4.1 For and any , by inequality (3.10) and Lemma 4.1, we have
with equality for if and only if and are centered balls of dilates; for if and only if and are dilates.
Take with in (4.5), then
with equality for if and only if and are centered balls of dilates; for if and only if and are dilates.
Combining with inequalities (4.3) and (4.6), we give
which implies that inequality (4.1) holds.
Next, we discuss the conditions of equality that holds in inequality (4.1).
According to the condition of the equality that holds in inequality (4.3) and inequality (4.6) with Remark 4.1, the four steps will be given.
-
(1)
For the case and , the equality holds in (4.1) if and only if and are balls of dilates which are centered at the origin, and K is a ball which is centered at the origin. Together with and , we know that K and L are balls of dilates which are centered at the origin.
-
(2)
For the case and , the equality holds in (4.1) if and only if and are balls of dilates which are centered at the origin, and K is a ball. By using and (1.17), it is obtained that () is a ball which is centered at the origin. Because and are balls of dilates which are centered at the origin, then is a ball of dilates which are centered at the origin, and together with and (1.15), L is a ball which is centered at the origin. However, K is a ball, so the equality holds in (4.1) if and only if K and L are balls of dilates which are centered at the origin.
-
(3)
For the case and , the equality holds in (4.1) if and only if and are dilates and K is an ellipsoid which is centered at the origin. Let be an ellipsoid which is centered at the origin, from (1.5), we know that is an ellipsoid which is centered at the origin. Other, from the literature [2], we know that L is an ellipsoid E which is centered at the origin if and only if are dilates of polar body of this E. So we know that the equality holds in (4.1) if and only if L and K are ellipsoids which are centered at the origin and both are dilates.
-
(4)
For the case and , the equality holds in (4.1) if and only if and are dilates, and K is an ellipsoid. Suppose that with , , and E is an ellipsoid which is centered at the origin, noting that (see [29]), this together with (1.5) is an ellipsoid which is centered at the origin. Because and are dilates, then is an ellipsoid which is centered at the origin. However, from [2], we know that L is an ellipsoid E which is centered at the origin if and only if are dilates of polar body of this E. Therefore, the equality holds in (4.1) if and only if K and L are ellipsoids of the dilates which are centered at the origin.
To sum up, the equality holds in (4.1) for and if and only if K and L are balls of the dilates which are centered at the origin; for and if and only if K and L are ellipsoids of the dilates which are centered at the origin. The proof is complete. □
Proof of Theorem 1.1 Exchange K and L in inequality (4.1), we have for , , ,
By using Lemmas 4.3 and (4.7), we have
Taking in the inequality above, we immediately obtain inequality (1.21). The proof is complete. □
Combining with Theorem 1.1 and (3.8), we immediately obtain the following.
Corollary 4.1 If , and , then
with the equality in inequality (4.8) for if and only if K is a ball which is centered at the origin; for if and only if K is an ellipsoid which is centered at the origin.
Further, we have established the following results.
Theorem 4.2 Let , , , then
with the equality in inequality (4.9) for if and only if K and L are balls of dilates which are centered at the origin; for if and only if K and L are ellipsoids of dilates which are centered at the origin.
Proof From inequality (4.1), we know that for , , , ,
Take in (4.9), and using Corollary 4.1 and Lemma 4.3, we have
Combining inequality (4.11) with (3.8), we immediately obtain inequality (4.8). According to the condition of the equality holding in inequalities (4.1) and (4.9), the condition of the equality that holds in inequality (4.8) is easily obtained. The proof is complete. □
5 Generalized -Busemann-Petty centroid inequality and dual inequality
In this section, we give the extension of the well-known -Busemann-Petty centroid inequality (1.8). Namely, we complete the proof of Theorem 1.2 and Theorem 1.3 (i.e., dual inequality of Theorem 1.2) in the introduction.
Lemma 5.1 If , , , then
Proof Using definition (1.16) of -mixed projection body and definition (1.14) of -mixed curvature image, it is easy to prove (5.1). □
Lemma 5.2 If , , , , then
Proof By (1.18), (1.19), (2.3), (2.7), (2.12) and (2.16), it is easy to prove (5.2). □
Proof of Theorem 1.2 For , using (5.1) and Corollary 4.1, we have
taking in the inequality above, we immediately get inequality (1.22).
According to the condition of the equality that holds in inequalities (4.8) and (1.8), and noting that and (1.15), we know with the equality in inequality (1.22) for and if and only if K is a ball which is centered at the origin; for and if and only if K is an ellipsoid which is centered at the origin. □
Proof of Theorem 1.3 Take in (5.2), we have
Using the Minkowski inequality (2.11) of the -mixed quermassintegrals, we have
Together with inequality (1.22), we can get
from this, we can get inequality (1.23).
According to the condition of the equality that holds in inequalities (2.11) and (1.22), we discuss the conditions of the equality that holds in (1.23) in the following four cases:
-
(1)
For the case and , the equality holds in (1.23) if and only if K and are dilates, and is a ball which is centered at the origin. Because , then is a ball which is centered at the origin if and only if K is a ball which is centered at the origin. Therefore, K is a ball which is centered at the origin. While the equality shows that is a ball which is centered at the origin if and only if is a ball which is centered at the origin. From this, for and the equality holds in inequality (1.23) if and only if K is a ball which is centered at the origin.
-
(2)
For the case and , the equality holds in (1.23) if and only if K and are dilates and is an ellipsoid which is centered at the origin. Because is an ellipsoid which is centered at the origin, and together with (1.10), K is an origin-symmetric ellipsoid E if and only if is an origin-symmetric ellipsoid. On the other hand, the literature [11] tells us that if E is an ellipsoid which is centered at the origin, then . From this, is an ellipsoid which is centered at the origin. Therefore, for and , the equality holds in inequality (1.23) if and only if K is an ellipsoid which is centered at the origin.
-
(3)
For the case and , the equality holds in (1.23) if and only if K and are homothetic, and is a ball which is centered at the origin. From , we know that , then K is a ball which is centered at the origin. This together with , then is a ball which is centered at the origin. Therefore, for and , the equality holds in inequality (1.23) if and only if K is a ball which is centered at the origin.
-
(4)
For the case and , the equality holds in (1.23) if K and are homothetic, and is an ellipsoid which is centered at the origin. Because is an origin-symmetric ellipsoid E if and only if K is an origin-symmetric ellipsoid. On the other hand, from , we know that is an ellipsoid which is centered at the origin if and only if is an ellipsoid which is centered at the origin. Therefore, for and , the equality holds in inequality (1.23) if and only if K is an ellipsoid which is centered at the origin.
To sum up, the equality holds in (1.23) for and if and only if K is a ball which is centered at the origin; for and if and only if K is an ellipsoid which is centered at the origin. The proof is complete. □
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The author is supported by the NNSF of China (11161019, 11371224).
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Ma, TY. Some inequalities related to -type -mixed affine surface area and -mixed curvature image. J Inequal Appl 2013, 470 (2013). https://doi.org/10.1186/1029-242X-2013-470
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DOI: https://doi.org/10.1186/1029-242X-2013-470