Open Access

Equivalence of Some Affine Isoperimetric Inequalities

Journal of Inequalities and Applications20092009:981258

https://doi.org/10.1155/2009/981258

Received: 24 May 2009

Accepted: 10 September 2009

Published: 28 September 2009

Abstract

We establish the equivalence of some affine isoperimetric inequalities which include the -Petty projection inequality, the -Busemann-Petty centroid inequality, the "dual" -Petty projection inequality, and the "dual" -Busemann-Petty inequality. We also establish the equivalence of an affine isoperimetric inequality and its inclusion version for -John ellipsoids.

1. Introduction

In the recent years, the -analogs of the projection bodies and centroid bodies have received considerable attentions [17]. Lutwak et al. established the -analog of the Petty projection inequality [4]. It states that if is a convex body in , then for ,
(1.1)

with an equality if and only if is an ellipsoid. Here, is used to denote the polar body of the -projection body, , of and write for , the -dimensional volume of the unit ball .

They also established the -analog of the Busemann-Petty centroid inequality [4]. It states that if is a star body (about the origin) in , then for ,
(1.2)

with an equality if and only if is a centroid ellipsoid at the origin. Here, is the -centroid body of . It is also shown in [4] that the -Busemann-Petty inequality (1.2) implies -Petty projection inequality (1.1). A quite different proof of the -analog of the Busemann-Petty centroid inequality is obtained by Campi and Gronchi [1].

Recently, Lutwak et al. [8] proved that there is a family of -John ellipsoids, , which can be associated with a fixed convex body : if contains the origin in its interior and , among all origin-centered ellipsoids , the unique ellipsoid solves the constrained maximization problem:
(1.3)
Corresponding to Lutwak et al.'s work, Yu et al. [9] proved that there is a family of dual -John ellipsoids, , which can be associated with a fixed convex body : if contains the origin in its interior and , among all origin-centered ellipsoids , the unique ellipsoid solves the constrained maximization problem:
(1.4)

Lutwak et al. [8] showed that the following results hold.

Theorem A.

If is a convex body in that contains the origin in its interior, and , then
(1.5)

with an equality in the right inequality if and only if is a centered ellipsoid and an equality in the left inequality if is a parallelotope.

Yu et al. [9] showed a theorem similar to Theorem A, and recently, Lu and Leng [10] gave a strengthened inequality as follows.

Theorem B.

If is a convex body in that contains the origin in its interior, and , then
(1.6)

with an equality if and only if is a centered ellipsoid. Here, is a dual form of -Busemann-Petty centroid inequality (1.2).

One purpose of this paper is to establish the equivalence of some affine isopermetric inequalities as follows.

Theorem 1.1.

If is a convex body in that contains the origin in its interior, and , then the following inequalities are equivalent:
(1.7)
(1.8)
(1.9)
(1.10)

all above inequalities with an equality if and only if is a centered ellipsoid.

Note that (1.7) is the -Busemann-Petty centroid inequality (1.2), (1.8) is the dual form of -Busemann-Petty centroid inequality in Theorem B, (1.9) is a "dual" form of -Petty projection inequality, and (1.10) is the -Petty projection inequality (1.1).

Another purpose of this paper is to establish the follow equivalence of Theorem A and its inclusion version Theorem A'.

Theorem 1.2.

If is a convex body in that contains the origin in its interior, and , then Theorem A is equivalent to Theorem A'.

Theorem A'

There exist an ellipsoid and a parallelotope such that
(1.11)

where the left inclusion with an equality if and only if is a centered ellipsoid and the right inclusion with an equality if and only if is a parallelotope.

Some notation and background material contained in Section 2.

2. Notations and Background Materials

We will work in equipped with a fixed Euclidean structure and write for the corresponding Euclidean norm. We denote the Euclidean unit ball and the unit sphere by and , respectively. The volume of appropriate dimension will be denoted by . The group of nonsingular affine transformations of is denoted by . The group of special affine transformations is denoted by , these are the members of whose determinant is one. We will write for the volume of the Euclidean unit ball in . Note that
(2.1)

defines for all nonnegative real (not just the positive integers). For real , define .

If is a convex body in that contains the origin in its interior, then we will use to denote the polar body of , that is,
(2.2)
From the definition of the polar body, we can easily find that for , there is
(2.3)

If is a convex body in , then its support function, , is defined for by A star body in is a nonempty compact set satisfying for all and such that the radial function , defined by is positive and continuous. Two star bodies and are said to be dilates if is independent of .

If is a centered (i.e., symmetric about the origin) convex body, then it follows from the definitions of support and radial functions, and the definition of polar body, that
(2.4)

For -mixed and dual mixed volumes, those formulae are directly given as follows.

It was shown in [11] that corresponding to each convex body that is containing the origin in its interior, there is a positive Borel measure, , on , such that
(2.5)

for each convex body .

If are star bodies in , then for , the dual mixed volume, , of and was defined by [4]
(2.6)

where the integration is with respect to spherical Lebesgue measure on .

From the integral representation (2.5), it follows immediately that for each convex body ,
(2.7)
From (2.6), of the dual -mixed volume, it follows immediately the for each star body ,
(2.8)
We will require two basic inequalities for the -mixed volume and the dual -mixed volume . The -Minkowski inequality states that for convex bodies [3],
(2.9)
with an equality if and only if and are dilates [11]. The dual -Minkowski inequality states that for star bodies [4],
(2.10)

with an equality if and only if and are dilates.

The -projection bodies was first introduced by Lutwak et al. in [4], and is defined as the body whose support function, for is given by
(2.11)
If is a star body about the origin in , and , the -centroid body of is the origin-symmetric convex body whose support function is given by [4]
(2.12)
The normalized polar projection body of , , for is defined as the body whose radial function, for is given by [8]
(2.13)
Here, we introduce a new convex body of , , for , defined as the body whose radial function, for that is given by
(2.14)

Noting that the normalization is chosen for the standard unit ball in , we have . For general reference the reader may wish to consult the books of Gardner [12] and Schneider [13].

3. Proof of the Results

Lemma 3.1.

If is a convex body in that contains the origin in its interior, then
(3.1)
(3.2)

Proof.

From the definition (2.11) and (2.13) combined with (2.4), for , we have
(3.3)
So we get
(3.4)
From the definition (2.12) and (2.14) combined with (2.4), for , we have
(3.5)
So we get
(3.6)

Corollary 3.2.

If is a convex body in that contains the origin in its interior, let , then for ,
(3.7)

Proof.

Since for (see [4]), combined with (3.2) and we know that for ,
(3.8)

From Corollary 3.2, we know that (1.9) is an affine isoperimetric inequality.

Lemma 3.3.

If are convex bodies in that contain the origin in their interior, then the following equalities are equivalent:
(3.9)
(3.10)
(3.11)
(3.12)

Proof.

First, from Lemma 3.1, we know that
(3.13)
From (2.5) and (2.6), we have for ,
(3.14)
(3.15)

Substitute (3.13) in (3.9) and combine (3.15) to just get (3.10); substitute (3.2) in (3.10) and combine (3.14) to just get (3.11); substitute (3.13) in (3.11) and combine (3.15) to just get (3.12); substitute (3.2) in (3.12) and combine (3.14) to just get (3.9).

Note.

Equation (3.9) is proved in [4] and (3.10) is proved in [10].

Proof of Theorem 1.1.

(1.7) (1.8): substituting in (3.10), followed by (2.8), (2.9), and (1.7), we have for each convex body that contains the origin in its interior,
(3.16)
(1.8) (1.9): substituting in (3.11), followed by (2.7), (2.9), and (1.8), we have
(3.17)
(1.9) (1.10): substituting in (3.12), followed by (2.9), we get
(3.18)
that is,
(3.19)
So, we have
(3.20)
(1.10) (1.7): substituting in (3.9), followed by (2.7), (2.10), we have
(3.21)
that is,
(3.22)
Combined with (1.10), we get
(3.23)
that is,
(3.24)

Lemma 3.4 (see [8]).

If is a convex body in that contains the origin in its interior, and , then for ,
(3.25)

Proof of Theorem 1.2.

Firstly, we prove that Theorem A implies Theorem A'.

From , taking , since for , we know that and followed by Lemma 3.4,
(3.26)

where the inclusion with an equality if and only if is a centered ellipsoid.

Suppose that for some , then
(3.27)
Take , here is the unit cube . Since Lutwak et al. [8] proved that the -John ellipsoid of the unit cube is , that is, , so we have by the fact . Following Lemma 3.4, , (3.27) and the left inequality of Theorem A, we have
(3.28)

where the inclusion with an equality if and only if is a parallelotope. By (3.26) and (3.28), we know that Theorem A implies Theorem A'.

Secondly, we prove that Theorem A' implies Theorem A.

On the one hand, since and by Lemma 3.4, we have
(3.29)
with an equality holds if and only if is a centered ellipsoid. On the other hand, suppose that for some , then , so . Following Theorem A' and Lemma 3.4, we have
(3.30)
that is,
(3.31)

with an equality if and only if is a parallelotope. By (3.29) and (3.31), we know that Theorem A' implies Theorem A.

Declarations

Acknowledgments

The author thanks the referee for careful reading and useful comments. This article is supported by National Natural Sciences Foundation of China (10671117).

Authors’ Affiliations

(1)
Institute of Management Decision & Innovation, Hangzhou Dianzi University

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Copyright

© Wuyang Yu. 2009

This article is published under license to BioMed Central Ltd. 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.