Equivalence of Some Affine Isoperimetric Inequalities

Abstract

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

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Recommended by Peter Pang

Institute of Management Decision & Innovation, Hangzhou Dianzi University, Zhejiang 310018, China

Received 24 May 2009; Accepted 10 September 2009

1. Introduction

In the recent years, the _Lp -analogs of the projection bodies and centroid bodies have received considerable attentions [1-7]. Lutwak et al. established the _Lp -analog of the Petty projection inequality [4]. It states that if K is a convex body in ^...n , then for 1≤p<∞, [figure omitted; refer to PDF] with an equality if and only if Kis an ellipsoid . Here, ^Πp* K=^{(_Πp K)*} is used to denote the polar body of the _Lp -projection body, _Πp K , of K, and write _ωn for V(_Bn ) , the n -dimensional volume of the unit ball _Bn .

They also established the _Lp -analog of the Busemann-Petty centroid inequality [4]. It states that if K is a star body (about the origin) in ^...n , then for 1≤p<∞, [figure omitted; refer to PDF] with an equality if and only if Kis a centroid ellipsoid at the origin . Here, _Γp K is the _Lp -centroid body of K . It is also shown in [4] that the _Lp -Busemann-Petty inequality (1.2) implies _Lp -Petty projection inequality (1.1). A quite different proof of the _Lp -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 _Lp -John ellipsoids, _Ep K , which can be associated with a fixed convex body K : if K contains the origin in its interior and p>0 , among all origin-centered ellipsoids E , the unique ellipsoid _Ep K solves the constrained maximization problem: [figure omitted; refer to PDF]

Corresponding to Lutwak et al.'s work, Yu et al. [9] proved that there is a family of dual _Lp -John ellipsoids, _E...p K , which can be associated with a fixed convex body K : if K contains the origin in its interior and p>0 , among all origin-centered ellipsoids E , the unique ellipsoid _E...p K solves the constrained maximization problem: [figure omitted; refer to PDF]

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

Theorem A.

If K is a convex body in ^...n that contains the origin in its interior, and 1≤p , then [figure omitted; refer to PDF] with an equality in the right inequality if and only if K is a centered ellipsoid and an equality in the left inequality if K 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 K is a convex body in ^...n that contains the origin in its interior, and 1≤p , then [figure omitted; refer to PDF] with an equality if and only if K is a centered ellipsoid. Here, V(_Γ-p K)≤V(K) is a dual form of _Lp -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 K is a convex body in ^...n that contains the origin in its interior, and 1≤p , then the following inequalities are equivalent: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] all above inequalities with an equality if and only if K is a centered ellipsoid.

Note that (1.7) is the _Lp -Busemann-Petty centroid inequality (1.2), (1.8) is the dual form of _Lp -Busemann-Petty centroid inequality in Theorem B, (1.9) is a "dual" form of _Lp -Petty projection inequality, and (1.10) is the _Lp -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 K is a convex body in ^...n that contains the origin in its interior, and 1≤p , then Theorem A is equivalent to Theorem A'.

Theorem A'

There exist an ellipsoid E and a parallelotope P such that [figure omitted; refer to PDF] where the left inclusion with an equality if and only if K is a centered ellipsoid and the right inclusion with an equality if and only if K is a parallelotope.

Some notation and background material contained in Section 2.

2. Notations and Background Materials

We will work in ^...n equipped with a fixed Euclidean structure and write |·| for the corresponding Euclidean norm. We denote the Euclidean unit ball and the unit sphere by _Bn and ^Sn-1 , respectively. The volume of appropriate dimension will be denoted by V(·) . The group of nonsingular affine transformations of ^...n is denoted by GL(n) . The group of special affine transformations is denoted by SL(n) , these are the members of GL(n) whose determinant is one. We will write _ωn for the volume of the Euclidean unit ball in ^...n . Note that [figure omitted; refer to PDF] defines _ωn for all nonnegative real n (not just the positive integers). For real p≥1 , define _cn,p =_ωn+p /_ω2ωnωp-1 .

If K is a convex body in ^...n that contains the origin in its interior, then we will use ^K* to denote the polar body of K , that is, [figure omitted; refer to PDF] From the definition of the polar body, we can easily find that for λ>0 , there is [figure omitted; refer to PDF]

If K is a convex body in ^...n , then its support function , _hK (·)=h(K,·):^...n [arrow right]R , is defined for x∈^...n by h(K,x)=max {x·y:y∈K}. A star body in ^...n is a nonempty compact set K satisfying [o,x]⊂K for all x∈K and such that the radial function _ρK (·)=ρ(K,·) , defined by ρ(K,x)=max {λ≥0:λx∈K}, is positive and continuous. Two star bodies K and L are said to be dilates if _ρK (u)/_ρL (u) is independent of u∈^Sn-1 .

If K 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 [figure omitted; refer to PDF]

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

It was shown in [11] that corresponding to each convex body K∈^...n that is containing the origin in its interior, there is a positive Borel measure, _Sp (K,·) , on ^Sn-1 , such that [figure omitted; refer to PDF] for each convex body Q .

If K,L are star bodies in ^...n , then for p≥1 , the dual _Lp mixed volume, _V...-p (K,L) , of K and L was defined by [4] [figure omitted; refer to PDF] where the integration is with respect to spherical Lebesgue measure S on ^Sn-1 .

From the integral representation (2.5), it follows immediately that for each convex body K , [figure omitted; refer to PDF] From (2.6), of the dual _Lp -mixed volume, it follows immediately the for each star body K , [figure omitted; refer to PDF]

We will require two basic inequalities for the _Lp -mixed volume _Vp and the dual _Lp -mixed volume _V...-p . The _Lp -Minkowski inequality states that for convex bodies K,L [3], [figure omitted; refer to PDF] with an equality if and only if K and L are dilates [11]. The dual _Lp -Minkowski inequality states that for star bodies K,L [4], [figure omitted; refer to PDF] with an equality if and only if K and L are dilates.

The _Lp -projection bodies was first introduced by Lutwak et al. in [4], and is defined as the body whose support function, for u∈^Sn-1 , is given by [figure omitted; refer to PDF]

If K is a star body about the origin in ^Rn , and p≥1 , the _Lp -centroid body _Γp K of K is the origin-symmetric convex body whose support function is given by [4] [figure omitted; refer to PDF]

The normalized _Lp polar projection body of K , _Γ-p K , for p>0, is defined as the body whose radial function, for u∈^Sn-1 , is given by [8] [figure omitted; refer to PDF]

Here, we introduce a new convex body of K , _Π-p K , for p>0 , defined as the body whose radial function, for u∈^Sn-1 , that is given by [figure omitted; refer to PDF]

Noting that the normalization is chosen for the standard unit ball _Bn in ^Rn , we have _ΠpBn =_ΓpBn =_Γ-pBn =_Π-pBn =_Bn . 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 K is a convex body in ^...n that contains the origin in its interior, then [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Proof.

From the definition (2.11) and (2.13) combined with (2.4), for u∈^Sn-1 , we have [figure omitted; refer to PDF] So we get [figure omitted; refer to PDF]

From the definition (2.12) and (2.14) combined with (2.4), for u∈^Sn-1 , we have [figure omitted; refer to PDF] So we get [figure omitted; refer to PDF]

Corollary 3.2.

If K is a convex body in ^...n that contains the origin in its interior, let p(K)=V^{(Π-p* K)-1} V^(K)(n+p)/p , then for [varphi]∈GL(n) , [figure omitted; refer to PDF]

Proof.

Since for [varphi]∈GL(n), _Γp ([varphi]K)=[varphi]_Γp K (see [4]), combined with (3.2) and V([varphi]K)=|det [varphi]|V(K), we know that for [varphi]∈GL(n) , [figure omitted; refer to PDF] From Corollary 3.2, we know that (1.9) is an affine isoperimetric inequality.

Lemma 3.3.

If K, L are convex bodies in ^...n that contain the origin in their interior, then the following equalities are equivalent: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Proof.

First, from Lemma 3.1, we know that [figure omitted; refer to PDF] From (2.5) and (2.6), we have for λ>0 , [figure omitted; refer to PDF] [figure omitted; refer to PDF]

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)[implies] (1.8): substitutingK=_Γ-p L in (3.10), followed by (2.8), (2.9), and (1.7), we have for each convex body L that contains the origin in its interior, [figure omitted; refer to PDF]

(1.8)[implies] (1.9): substitutingL=^Π-p* K in (3.11), followed by (2.7), (2.9), and (1.8), we have [figure omitted; refer to PDF]

(1.9)[implies] (1.10): substitutingK=^Πp* L in (3.12), followed by (2.9), we get [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] So, we have [figure omitted; refer to PDF]

(1.10)[implies] (1.7): substitutingL=_Γp K in (3.9), followed by (2.7), (2.10), we have [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Combined with (1.10), we get [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF]

Lemma 3.4 (see [8]).

If K is a convex body in ^...n that contains the origin in its interior, and p>0 , then for [varphi]∈GL(n) , [figure omitted; refer to PDF]

Proof of Theorem 1.2.

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

From V(_Ep K)≤V(K) , taking E=^{(V(K)/V(_Ep K))1/n}_Ep K , since V(λK)=^λn V(K) for λ>0 , we know that V(E)=V(K) and followed by Lemma 3.4, [figure omitted; refer to PDF] where the inclusion with an equality if and only if K is a centered ellipsoid.

Suppose that _Ep K=[varphi]..._Bn , for some [varphi]...∈GL(n) , then [figure omitted; refer to PDF] Take P=([varphi].../^{|det [varphi]...|1/n} )(V^(K)1/n /2)Q , here Q is the unit cube ^[-1,1]n . Since Lutwak et al. [8] proved that the _Lp -John ellipsoid of the unit cube is _Bn , that is, _Ep Q=_Bn , so we have V(K)=V(P) by the fact V(Q)=²ⁿ . Following Lemma 3.4, _Ep Q=_Bn , _Ep K=[varphi]..._Bn , (3.27) and the left inequality of Theorem A, we have [figure omitted; refer to PDF] where the inclusion with an equality if and only if K 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 _Ep E⊇_Ep K and _Ep E=E by Lemma 3.4, we have [figure omitted; refer to PDF] with an equality holds if and only if K is a centered ellipsoid. On the other hand, suppose that P=[varphi]Q for some [varphi]∈GL(n) , then V(K)=V(P)=|det [varphi]|V(Q)=|det [varphi]|²ⁿ , so |det [varphi]|=V(K)/²ⁿ . Following Theorem A' and Lemma 3.4, we have [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] with an equality if and only if K is a parallelotope. By (3.29) and (3.31), we know that Theorem A' implies Theorem A.

Acknowledgments

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

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