- Open Access
Polar duals of convex and star bodies
© Zhao et al; licensee Springer. 2012
Received: 17 December 2011
Accepted: 17 April 2012
Published: 17 April 2012
In this article, some new inequalities about polar duals of convex and star bodies are established. The new inequalities in special case yield some of the recent results.
MR (2000) Subject Classification: 52A30.
1 Notations and preliminaries
The setting for this article is n-dimensional Euclidean space . Let denotes the set of convex bodies (compact, convex subsets with non-empty interiors) in . We reserve the letter u for unit vectors, and the letter B for the unit ball centered at the origin. The surface of B is S n -l. The volume of the unit n-ball is denoted by ωn.
where u · x denotes the usual inner product u and x in .
Let δ denotes the Hausdorff metric on , i.e., for , where | · | ∞ denotes the sup-norm on the space of continuous functions C(S n -l).
1.1 L p -mixed volume and dual L p -mixed volume
where S p (K, ·) denotes a positive Borel measure on S n -1.
with equality if and only if K and L are homothetic.
where dS(u) signifies the surface area element on S n -1 at u.
with equality if and only if K and L are dilates.
1.2 Mixed bodies of convex bodies
If , the notation of mixed body [K1,..., K n -1] states that (see ): corresponding to the convex bodies in , there exists a convex body, unique up to translation, which we denote by[K1,..., K n -1].
V1([K1, ..., K n -1], K n ) = V(K1, ..., K n -1, K n );
[K1 + L1, K2, ..., K n -1] = [K1, K2, ..., K n -1] + [L1, K2, ..., K n -1];
[λ 1 K 1, ..., λ n -1K n -1] = λ1... λ n -1 · [K1, ..., K n -1];
The properties of mixed body play an important role in proving our main results.
1.3 Polar of convex body
Bourgain and Milman's inequality is stated as follows (see ).
Different proofs were given by Pisier .
2 Main results
In this article, we establish some new inequalities on polar duals of convex and star bodies.
This is just an inequality given by Ghandehari .
Let L = B, we have the following interesting result:
with equality if and only if K is an n-ball.
where c is the constant of Bourgain and Milman's inequality.
The following theorem concerning L p -dual mixed volumes will generalize Santaló inequality.
Taking for K1 = K2 = K in (2.9) and in view of , (2.9) changes to the Bourgain and Milman's inequality (1.8).
C.-J. Zhao research was supported by National Natural Sciences Foundation of China (10971205). W.-S. Cheung research was partially supported by a HKU URG grant.
- Schneider R: Convex Boides: The Brunn-Minkowski Theory. Cambridge University Press Cambridge; 1993.View ArticleGoogle Scholar
- Gardner RJ: Geometric Tomography. Cambridge University Press New York; 1996.Google Scholar
- Lutwak E: The Brunn-Minkowski-Firey theory-I: mixed volumes and the Minkowski problem. J Diff Geom 1993, 38: 131–150.MathSciNetGoogle Scholar
- Lutwak E, Yang D, Zhang GY: L p affine isoperimetric inequalities. J Diff Geom 2000, 56: 111–132.MathSciNetGoogle Scholar
- Lutwak E: Volume of mixed bodies. Trans Am Math Soc 1986, 294: 487–500.MathSciNetView ArticleGoogle Scholar
- Bourgain J, Milman V: New volume ratio properties for convex symmetric bodies in [ineq]. Invent Math 1987, 88: 319–340.MathSciNetView ArticleGoogle Scholar
- Pisier G: The volume of convex bodies and Banach space geomery. Cambridge University Press Cambridge; 1989.View ArticleGoogle Scholar
- Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press Cambridge; 1934.Google Scholar
- Ghandehari M: Polar duals of convex bodies. Proc Am Math Soc 1991, 113(3):799–808.MathSciNetView ArticleGoogle Scholar
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