Journal of Function Spaces
Volume 2016 (2016), Article ID 1481793, 10 pages
http://dx.doi.org/10.1155/2016/1481793

Research Article

Dual -Mixed Geominimal Surface Area and Related Inequalities

Tongyi Ma and Yibin Feng

College of Mathematics and Statistics, Hexi University, Zhangye, Gansu 734000, China

Received 11 April 2016; Accepted 9 June 2016

Academic Editor: Carlo Bardaro

Copyright © 2016 Tongyi Ma and Yibin Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The integral formula of dual -geominimal surface area is given and the concept of dual -geominimal surface area is extended to dual -mixed geominimal surface area. Properties for the dual -mixed geominimal surface areas are established. Some inequalities, such as analogues of Alexandrov-Fenchel inequalities, Blaschke-Santaló inequalities, and affine isoperimetric inequalities for dual -mixed geominimal surface areas, are also obtained.

1. Introduction

The concept of geominimal surface area was introduced by Petty [1] about 40 years ago, and its -extension was first introduced by Lutwak [2, 3]. They have been proved to be key ingredients in connecting affine differential geometry, relative differential geometry, and Minkowski geometry. The basic theory concerning geominimal surface area is developed, and a close connection is established between this theory and affine differential geometry in [1]. The -geominimal surface area is now thought to be at the core of the rapidly developing -Brunn-Minkowski theory. Hence, it receives a lot of attention and motivates extensions of some known inequalities for geominimal surface areas to -geominimal surface areas. These new inequalities of -type are stronger than their classical counterparts.

However, finding an integral expression for the -geominimal surface area seems to be intractable. This also leads to a big obstacle on extending the -geominimal surface area. Until more recently, Zhu et al. [4] provided an integral formula for -geominimal surface area by -Petty body and introduced -mixed geominimal surface areas which extended the -geominimal surface area. Thereout, they established some new -affine isoperimetric inequalities.

Recently, Wang and Qi [5] introduced a concept of dual -geominimal surface area, which is a dual concept for -geominimal surface area and belongs to the dual -Brunn-Minkowski theory for star bodies also developed by Lutwak (see [6, 7]). The dual -Brunn-Minkowski theory for star bodies and a more extensive dual Orlicz-Brunn-Minkowski theory for star bodies received considerable attention (see, e.g., [8–21]), and they have been proved to be very powerful in solving many geometric problems, for instance, the Busemann-Petty problems (see, e.g., [6, 22–24]).

In this paper, we show that the infimum in the definition of dual -geominimal surface area is a minimum and provide an integral formula for dual -geominimal surface area by dual -Petty body. Moreover, we define the dual -mixed geominimal surface area and establish some new -affine isoperimetric inequalities for it.

Our paper is organized as follows. In Section 2, we provide the necessary background, such as definitions and known results which will be needed. Section 3 includes the basic theory of dual -geominimal surface area, such as theorem of existence and uniqueness for dual -geominimal surface area, as well as the integral definition of dual -geominimal surface area. In Section 4, we introduce the dual -mixed geominimal surface area and prove some important properties, such as affine invariant properties. We also obtain analogues of Alexandrov-Fenchel inequalities, Blaschke-Santaló inequalities, and affine isoperimetric inequalities for dual -mixed geominimal surface areas. Finally, we investigate the dual th -mixed geominimal surface areas and obtain analogues of Blaschke-Santaló and affine isoperimetric inequalities in Section 5.

2. Preliminaries and Notations

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 convex bodies whose centroids lie at the origin in , we write and , respectively. Let denote the -dimensional volume of a body , and let denote the standard Euclidean unit ball in and write for its volume, and let denote the unit sphere for .

For , its support function is defined by , , where is the standard inner product on . Associated with each , one can uniquely define its polar body by . It is easily verified that if .

For and (not both zero), the Minkowski linear combination is defined by

The classical Brunn-Minkowski inequality (see [25]) states that for convex bodies and real (not both zero), the volume of the bodies and the volume of their Minkowski linear combination are related by

with equality if and only if and are homothetic.

For real , , and (not both zero), the Firey linear combination, , is defined by (see [26])

For the Firey linear combination , Firey [26] also established the -Brunn-Minkowski inequality (an inequality that is also known as the Brunn-Minkowski-Firey inequality, see [14]). If , (not both zero), and , then

with equality if and only if and are dilates.

A set in is star-shaped at if and for each , the intersection is a (possibly degenerate) compact line segment. If is star-shaped at the origin , we define its radial function for by . If is positive and continuous, then is called a star body about the origin. denotes the set of star bodies (about the origin) in . Two star bodies and are dilates of one another if is independent of . Note that can be uniquely determined by its radial function and vice versa. If , we have

More generally, from the definition of the radial function, it follows immediately that for the radial function of the image of is given by (see [27])

Obviously, for ,

The radial Hausdorff metric between the star bodies and is

A sequence of star bodies is said to be convergent to if

Therefore, a sequence of star bodies converges to if and only if the sequence of radial functions converges uniformly to (see [28, Theorem  7.9]).

According to the definitions of the polar body for convex body, support function, and radial function, it follows that for

One of the most important inequalities in convex geometry is the Blaschke-Santaló inequality about polar body (cf. [1, 27, 29]): If , then

where the equality holds if and only if is an ellipsoid.

If and (not both zero), then, for , the radial harmonic -combination, , is defined by (see [3])

For and , the dual harmonic -mixed volume, , is defined by

Let and . Then, the integral representation of dual harmonic -mixed volume of and , , is given (see [3]):

With (5) and (14) taken together, we obtain for

In [3], Lutwak proved the following: For and , if , then, for ,

Since , for all and , the following follows from (6), (14), and (15).

Proposition 1. If and , then, for ,

The case of Proposition 1 reduces to the following formula:

This integral representation of , with Hölder’s inequality (see [30, p. ]) together with the polar coordinate formula, immediately gives the following:

with equality if and only if and are dilates.

The following result is an immediate consequence of (19).

Lemma 2. Suppose that and such that . If for all

then .

The continuity of the dual harmonic -mixed volume is contained.

Lemma 3. Suppose that sequences and , . If , then .

Proof. Since and are equivalent to and , uniformly on , and , are positively continuous on , then and are uniformly bounded on (see [28, Theorem  7.9]). Hence,

Hence,

Namely, .

The volume-normalized dual conical measure of is defined by , where is Lebesgue measure on . We shall make use of the fact that the volume-normalized dual conical measure is a probability measure on .

The following lemma will be needed.

Lemma 4 (see [3]). Let denote the set of compact convex subsets of Euclidean -space , and suppose such that . If the sequence is bounded, then .

3. The Dual -Geominimal Surface Area

Based on the notion of dual -mixed volumes, Wang and Qi [5] defined the dual -geominimal surface area as follows: For , the dual -geominimal surface area, , of is defined by

For this notion of -dual geominimal surface area, Wang and Qi in [5] established the following affine isoperimetric inequality and Blaschke-Santaló type inequality: For and ,

with equality if and only if is an ellipsoid centred at the origin.

If and , then

with equality if and only if is an ellipsoid.

By the homogeneity of volume and dual -mixed volume, the dual -geominimal surface area could also be defined by

It will be shown that the infimum in the above definition is attained.

Theorem 5. If and , then there exists a unique body such that

Proof. From the definition of , there exists a sequence such that , with , for all , and . To see that the , are uniformly bounded, let

where is any of the points in at which this maximum is attained. Let . Then, . From definition (14) of dual harmonic -mixed volume and Jensen’s inequality, it follows that

Namely,

for a fixed ; then, the sequence is uniformly bounded.
Since the sequence is uniformly bounded, the Blaschke selection theorem guarantees the existence of a subsequence of , which will also be denoted by , and a compact convex , such that . Since , Lemma 4 gives . Now, implies that , and since , it follows that . Lemma 3 can now be used to conclude that will serve as the desired body .
The uniqueness of the minimizing body is easily demonstrated as follows. Suppose and , such that , and

Define by

Since, obviously,

and , it follows from Brunn-Minkowski inequality (2) that

with equality if and only if .
By formula (14) of dual -mixed volume, together with the convexity of , we have

with equality if and only if . Thus,

is the contradiction that would arise if it were the case that . This completes the proof.

The unique body whose existence is guaranteed by Theorem 5 will be denoted by and will be called the dual -Petty body of . The polar body of will be denoted by rather than . Thus, for , the body is defined by

For , there exists a unique point in the interior of , called the Santaló point of , such that (see [3])

or, for the unique , this is equivalent to

Let denote the set of convex bodies having their Santaló point at the origin in . Thus, we have (see [3])

Let

The next result is an immediate consequence of Theorem 5.

Theorem 6. For each , there exists a unique body with .

The unique body is called the dual -Petty body of .

By Theorem 6 and the integral representation (14) of dual harmonic -mixed volume, we have the following integral formula of .

Proposition 7. For each , there exists a unique convex body with

4. The Dual -Mixed Geominimal Surface Area

Motivated by the definition of -mixed geominimal surface area of Zhu et al. (see [4]), we now define the dual -mixed geominimal surface area as follow: For each , , and , there exists a unique convex body (dual -Petty body of ) with

will be called the dual -mixed geominimal surface area of .

Let . Then, can be written as follows:

The following propositions will provide that the dual -mixed geominimal surface area is affine invariant.

Proposition 8. If and every , then

Proof. From definition (23) of dual -geominimal surface area and (18), for , we have

On the other hand, for , it follows from (23) that

Therefore, for every , we have

Proposition 9. If , then, for ,

Proof. From the definition of and Proposition 8, it follows that

From the definition of , Proposition 9, and (15),

Namely, from Lemma 2, for each ,

Proposition 10. If and , then, for ,

In particular, if , then is affine invariant; that is,

Proof. Since , for and any , we have

where . Therefore, for , we have

The dual mixed volume of sets is defined by

The classical dual Alexandrov-Fenchel inequalities for dual mixed volumes (cf. [27, 31, 32]) can be written as

with equality if are dilates of each other. If , equality holds trivially.

In particular, taking in the above inequality and noticing that , we have

with equality if and only if are dilates.

Take in , and

where is called the th dual quermassintegral of .

The following inequalities are the analogous of dual Alexandrov-Fenchel inequalities for dual -mixed geominimal surface area.

Theorem 11. If and , then, for ,

with equality if are dilates of each other. If , equality holds trivially.
In particular, if in the above inequality, then

with equality if are dilates of each other.

Proof. Let and for . By Hölder’s inequality (cf. [30]), we have

The equality in Hölder’s inequality holds if and only if for some and all . This is equivalent to . From Proposition 9, for constant . Thus, the equality holds if and are dilates of each other.

A lemma of the following type will be needed.

Lemma 12. If and , then

with equality if and only if is a ball centred at the origin.

Proof. From definition (23) and inequality (19), we have

Since , taking , it follows from inequalities (65) and (11) that

Namely,

By the equality condition of (19) and (65), we see that equality holds in (64) if and only if is a ball centred at the origin.

Now, we prove the affine isoperimetric inequalities for dual -mixed geominimal surface areas.

Theorem 13. Let and ; then,

with equality if and only if are balls centred at the origin that are dilates of each other.

Proof. From (77) in Section 5, it follows that . Then, , , and . By inequalities (62), (64), and (59), we have

By the equality condition of (62), (64), and (59), we see that equality holds in (68) if and only if is a ball centred at the origin.

Corollary 14. Let and ; then,

with equality if and only if are balls centred at the origin that are dilates of each other.

Take in (70), and we write

Corollary 15. Let and ; then, for ,

with equality if and only if is a ball centred at the origin.

5. The Dual th -Mixed Geominimal Surface Area

This section is mainly dedicated to investigating the dual th -mixed geominimal surface area.

For , , and , we define dual th -mixed geominimal surface area, , of , as

and write

By Theorem 6, we have

and, obviously,

Thus, the above two equations and the uniqueness part of Theorem 6 show that

Noticing that for , then

By (44), (73), and (74), we have

The following cyclic inequality for the dual th -mixed geominimal surface area will be established.

Theorem 16. For , , , and , we have

with equality if and are dilates of each other.

Proof. From definition (73) and Hölder’s inequality, it follows that, for ,

That is,

We obtain inequality (81). By the condition of equality in Hölder’s inequality, the equality holds in (81) if and only if, for any ,

is a constant; that is, is a constant for any . By the same argument in the proof of Theorem 11, we conclude that equality holds if and are dilates of each other.

Taking in Theorem 16 and using (74), we immediately obtain the following.

Corollary 17. For , , , and , then

with equality if is a ball centered at the origin.

Then, the following Minkowski inequality for the dual th -mixed geominimal surface area will be obtained.

Theorem 18. For , , and and then for or ,

and for ,

Each inequality holds as an equality if and are dilates of each other. For or , (86) (or (87)) is identical.

Proof. (i) For , let in Theorem 16; we obtain

with equality if and are dilates of each other. From (80), we can get

with equality if and are dilates of each other.
(ii) For , let in Theorem 16; we obtain

with equality if and are dilates of each other.
From (80), we can also get inequality (86).
(iii) For , let in Theorem 16; we obtain

with equality if and are dilates of each other.
From (80), we can get inequality (87).
(iv) For (or ), by (80), one can see (86) (or (87)) is identical.

Let in Theorem 18, , and (74) will lead to the following.

Corollary 19. For , , and and then for or ,

and for ,

Each inequality holds as an equality if is a ball centered at the origin. For or , (92) (or (93)) is identical.

Now we will give an extended form of inequality (24) as follows.

Theorem 20. If , , , and , then

with equality if and only if is an ellipsoid centred at the origin.

Proof. By inequalities (92) and (24), we can immediately obtain inequality (94).

As the extension of inequality (25), we obtain an analogue of Blaschke-Santaló inequality for the dual th -mixed geominimal surface area.

Theorem 21. If , , , and , then

and equality holds for if and are dilated ellipsoids of each other centered at the origin. The inequality holds as an equality for (or ) if (or ) is an ellipsoid centered at the origin.

Proof. Given (87) together with (25), it follows that

The equality holds for if and are dilated ellipsoids of each other. The inequality holds as an equality for (or ) if (or ) is an ellipsoid.

Recall Ye’s isoperimetric inequality (see [33]): If , , and the dual -surface area , then

On the other hand, if , then

The equality in every inequality holds if and only if is an origin-symmetric Euclidean ball.

We now establish generalized isoperimetric inequalities for .

Theorem 22. If , , and , then we have the following.
(i) If ,

with equality if is a ball centered at the origin.
(ii) If ,

with equality if is a ball centered at the origin.

Proof. (i) For , by (79) and (24), it follows that

This is Wang’s inequality (24).
For , by (74), (79), and (80), the equality holds trivially in (100).
For , since , by (24), (92), and (94), we have

Hence, for and , the -affine isoperimetric inequalities (92) and (24) imply that

with equality if is a ball centered at the origin.
(ii) For , by (74), (79), and (80), the equality holds trivially in (100). We now prove the case . Inequality (93) and the definition of dual -geominimal surface area give the following:

with equality if is a ball centered at the origin.

The following results are interesting.

Theorem 23. Let , , , and ; then,

with equality if is a ball centered at the origin.

Proof. Inequality (93) and the definition of dual -geominimal surface area give the following:

with equality if is a ball centered at the origin.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11561020), the Science and Technology Plan of Gansu Province (Grant no. 145RJZG227), the Young Foundation of Hexi University (Grant no. QN2015-02), and partly the National Natural Science Foundation of China (Grant no. 11371224).

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