Abstract
We investigate the dual geominimal surface area and volume forms of Busemann-Petty problems for the quasi intersection bodies and establish some new geometric inequalities. Our results provide a significant complement to the researches on Busemann-Petty problems for intersection bodies.
1. Introduction and Main Results
Let denote the unit sphere in Euclidean space . If is a compact star-shaped (about the origin) set in , then its radial function, , is defined by (see [1, 2]) If is positive and continuous, then will be called a star body (about the origin), and denotes the set of star bodies in . We will use and to denote the subset of star bodies in containing the origin in their interiors and origin-symmetric star bodies, respectively. Two star bodies and are said to be the dilation of one another if is independent of .
Let denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean . For , denotes the -dimensional subspace orthogonal to . We use to denote the -dimensional volume of a -dimensional compact convex set . Instead of we usually write . For the standard unit ball in , we write for its volume.
Busemann and Petty posed a problem [3]: let and be origin-symmetric convex bodies in . Is it true that, for any ,
A long list of authors contributed to the solution of this famous problem over a period of 40 years; see [4–18]. The question has a negative answer for and an affirmative answer for . For a detailed account of the interesting history of the Busemann-Petty problem, see the books by Gardner [1, Chapter 8] and Koldobsky [19, Chapter 5].
The crucial idea in the solution of the problem was to define new convex body which was called intersection body. This was done by Lutwak [15] whose work is considered the starting point of the solution of the Busemann-Petty problem in all dimensions. For , the intersection body, , of is a star body whose radial function in the direction is equal to the -dimensional volume of the section of by ; that is,
The intersection bodies have been intensively studied in recent years (see [20–28] and the books [19, 29]). From (3) and the fact that star bodies and satisfy if and only if , we see that the Busemann-Petty problem can be rephrased in the following way: for , is it true that
Lutwak [15] showed that the problem has an affirmative answer if the body is restricted to the class of intersection bodies. In addition, Lutwak proved that if is a sufficiently smooth origin-symmetric star body with positive radial function which is not an intersection body, then there exists an origin-symmetric star body such that but . Further, Busemann-Petty problems have been considered in the context of Brunn-Minkowski Theory (see [30–41]). In particular, Haberl and Ludwig [42] generalized the intersection body to form and introduced the notion of intersection body: let be a star body and nonzero . The intersection body of , , is the symmetric star body whose radial function is defined by
After that, associated with intersection bodies, Yuan and Cheung [41] gave an affirmative form of Busemann-Petty problem for the intersection bodies.
Theorem 1. Let be intersection body and be a star body in . If , then In both cases equality holds if and only if .
In 2007, Yu et al. [40] defined the quasi intersection body as follows: let be a star body and . The quasi intersection body, , of is defined by for , where denotes the dual mixed volume (see (22)).
Suppose that is a Borel function on . The spherical Radon transform [43] of is defined by for . Using the spherical Radon transform, the definition of is rewritten by for .
One aim of this paper is to establish the volume forms of the Busemann-Petty problems for the quasi intersection bodies. For convenience, let denote the set of quasi intersection bodies.
Theorem 2. Let and . If and , then If and , then And if and only if .
Remark 3. Theorem 2 can be found in [40]. However, what should be noted is that we give a new method of proof in this paper.
Theorem 4. For , if , then there exists such that but
Recall the definition of dual geominimal surface area, , of for in [44]:
Another aim of this paper is to give the dual geominimal surface area forms of Busemann-Petty problems for the quasi intersection bodies. If , then we rewrite the definition of dual geominimal surface area by
Theorem 5. If and , then implying And if and only if .
Theorem 6. For , if , then there exists such that but
2. Preliminaries
2.1. Dual Mixed Volume
For , , and (), the radial combination, , of and is defined by (see [40])
The polar coordinate formula for the volume of a body is
For , the dual mixed volume, , of was defined by (see [40])
From definition (22), the following integral representation of dual mixed volume was given (see [40]): if and , then Obviously,
The Minkowski inequality for the dual mixed volume was established in [40]: if , then, for , for , In every case, equality holds if and only if is a dilation of .
2.2. Dual Blaschke Body
For , , and (), the dual Blaschke combination, , of and is defined by (see [28])
Taking , in (27), the dual Blaschke body, , of is given by (see [28]) Obviously, the dual Blaschke body is origin-symmetric.
3. Proofs of Theorems 2–6
The proof of Theorem 2 needs the following Lemma.
Lemma 7 (see [40]). If , then, for ,
Proof of Theorem 2. For a star body with , it follows from Lemma 7 that Since we have From , we obtain that, for , that is,From the equality condition of (25), we know that if and only if .
From a star body with , By Lemma 7, we have Thus that is, For , it follows from that namely,
Lemma 8 (see [28]). If and (), then, for , with equality if and only if is a dilation of .
Let , in (40), the following is an immediate result of Lemma 8.
Corollary 9. If , then, for , with equality if and only if is origin-symmetric.
Lemma 10. If , then, for ,
Proof. From (9), (27), and (28), we have Since , we have that is,
Proof of Theorem 4. Since , Corollary 9 implies Let such that . Taking we have Combining with Lemma 10 we get
Proof of Theorem 5. From the condition , we have, for arbitrary , By Lemma 7, we obtain Let in (50). Together with (15), it follows that with equality if and only if .
Lemma 11. If and , then with equality if and only if is origin-symmetric.
Proof. From (14), (27), and (28), we get Note that ; thus we have . Together with (53), this yields Equality holds in (53) if and only if is a dilation of . This gives . Namely, is origin-symmetric. Hence, equality holds in (54) if and only if is origin-symmetric.
Proof of Theorem 6. Since , from Lemma 11 we have Choose such that . Taking we obtain It follows from Lemma 10 that
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the Natural Science Foundation of China (no. 11371239) and the Natural Science Foundation of Fujian Province of China (no. 2016J01023).