Equivalence of Some Affine Isoperimetric Inequalities
© Wuyang Yu. 2009
Received: 24 May 2009
Accepted: 10 September 2009
Published: 28 September 2009
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.
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  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 .
Lutwak et al.  showed that the following results hold.
One purpose of this paper is to establish the equivalence of some affine isopermetric inequalities as follows.
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'.
Some notation and background material contained in Section 2.
2. Notations and Background Materials
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 .
3. Proof of the Results
From Corollary 3.2, we know that (1.9) is an affine isoperimetric inequality.
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).
Proof of Theorem 1.1.
Lemma 3.4 (see ).
Proof of Theorem 1.2.
Firstly, we prove that Theorem A implies Theorem A'.
Secondly, we prove that Theorem A' implies Theorem A.
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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