- Research Article
- Open Access

# Equivalence of Some Affine Isoperimetric Inequalities

- Wuyang Yu
^{1}Email author

**2009**:981258

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

© Wuyang Yu. 2009

**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.

## Keywords

- Convex Body
- Radial Function
- Mixed Volume
- Minkowski Inequality
- Projection Body

## 1. Introduction

*if*

*is a convex body in*

*, then for*

*,*

*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
.

*if*

*is a star body (about the origin) in*

*, then for*

*,*

*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].

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

Theorem A.

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.

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.

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'

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

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

*polar body*of , that is,

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
.

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

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

with an equality if and only if and are dilates.

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.

Proof.

Corollary 3.2.

Proof.

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

Lemma 3.3.

Proof.

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.

Lemma 3.4 (see [8]).

Proof of Theorem 1.2.

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

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

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.

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

## References

- Campi S, Gronchi P:
**The -Busemann-Petty centroid inequality.***Advances in Mathematics*2002,**167**(1):128–141. 10.1006/aima.2001.2036MathSciNetView ArticleMATHGoogle Scholar - Campi S, Gronchi P:
**On the reverse -Busemann-Petty centroid inequality.***Mathematika*2002,**49**(1–2):1–11. 10.1112/S0025579300016004MathSciNetView ArticleMATHGoogle Scholar - Lutwak E, Yang D, Zhang G:
**A new ellipsoid associated with convex bodies.***Duke Mathematical Journal*2000,**104**(3):375–390. 10.1215/S0012-7094-00-10432-2MathSciNetView ArticleMATHGoogle Scholar - Lutwak E, Yang D, Zhang G:
**affine isoperimetric inequalities.***Journal of Differential Geometry*2000,**56**(1):111–132.MathSciNetMATHGoogle Scholar - Lutwak E, Yang D, Zhang G:
**The Cramer-Rao inequality for star bodies.***Duke Mathematical Journal*2002,**112**(1):59–81. 10.1215/S0012-9074-02-11212-5MathSciNetView ArticleMATHGoogle Scholar - Lutwak E, Zhang G:
**Blaschke-Santaló inequalities.***Journal of Differential Geometry*1997,**47**(1):1–16.MathSciNetMATHGoogle Scholar - Werner E:
**The -affine surface area and geometric interpretations.***Rendiconti del Circolo Matematico di Palermo*2002,**70:**367–382.MathSciNetMATHGoogle Scholar - Lutwak E, Yang D, Zhang G:
**John ellipsoids.***Proceedings of the London Mathematical Society*2005,**90**(2):497–520. 10.1112/S0024611504014996MathSciNetView ArticleMATHGoogle Scholar - Yu W, Leng G, Wu D:
**Dual John ellipsoids.***Proceedings of the Edinburgh Mathematical Society*2007,**50**(3):737–753.MathSciNetView ArticleMATHGoogle Scholar - Lu F, Leng G:
**Volume inequalities for -John ellipsoids and their duals.***Glasgow Mathematical Journal*2007,**49**(3):469–477.MathSciNetMATHGoogle Scholar - Lutwak E:
**The Brunn-Minkowski-Firey theory—I: mixed volumes and the Minkowski problem.***Journal of Differential Geometry*1993,**38**(1):131–150.MathSciNetMATHGoogle Scholar - Gardner RJ:
*Geometric Tomography, Encyclopedia of Mathematics and Its Applications*.*Volume 58*. Cambridge University Press, Cambridge, UK; 1995:xvi+424.MATHGoogle Scholar - Schneider R:
*Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications*.*Volume 44*. Cambridge University Press, Cambridge, UK; 1993:xiv+490.Google Scholar

## Copyright

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.