Journal of Function Spaces
Volume 2017 (2017), Article ID 3124285, 6 pages
https://doi.org/10.1155/2017/3124285

Research Article

Busemann-Petty Problems for Quasi Intersection Bodies

Yanping Zhou¹ and Shanhe Wu²

¹Department of Mathematics, Shanghai University, Shanghai 200444, China
²Department of Mathematics, Longyan University, Longyan 364012, China

Correspondence should be addressed to Shanhe Wu; shanhewu@gmail.com

Received 8 July 2017; Accepted 18 September 2017; Published 31 October 2017

Academic Editor: Alberto Fiorenza

Copyright © 2017 Yanping Zhou and Shanhe Wu. 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.

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).

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