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Polar duals of convex and star bodies
Journal of Inequalities and Applications volume 2012, Article number: 90 (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 Sn-l. The volume of the unit n-ball is denoted by ωn.
We use V(K) for the n-dimensional volume of convex body K. , denotes the support function of ; i.e., for u ∈ Sn-l
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(Sn-l).
Associated with a compact subset K of , which is star-shaped with respect to the origin, is its radial function , defined for u ∈ Sn-l, by
1.1 L p -mixed volume and dual L p -mixed volume
If , the L p -mixed volume V p (K, L) was defined by Lutwak (see ):
where S p (K, ·) denotes a positive Borel measure on Sn-1.
The L p analog of the classical Minkowski inequality (see ) states that: If K and L are convex bodies, then
with equality if and only if K and L are homothetic.
If K, L ∈ Sn, p ≥ 1, the L p -dual mixed volume was defined by Lutwak (see ):
where dS(u) signifies the surface area element on Sn-1 at u.
The following dual L p -Minkowski inequality was obtained in : If K and L are star bodies, then
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].
The following is a list of the properties of mixed body: It is symmetric, linear with respect to Minkowski linear combinations, positively homogeneous, and for and λi> 0,
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
For , the polar body of K, K* is defined:
It is easy to get that
Bourgain and Milman's inequality is stated as follows (see ).
If K is a convex symmetric body in , then there exists a universal constant c> 0 such that
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.
Theorem 2.1 If K, K1, ..., K n- 1 are convex bodies in and let L = [K1, ..., K n- 1], then the L p -mixed volumes V p (K, L), V p (K*, L), V p (B, L) satisfy
Proof From (1.1) and (1.2), it is easy
By definition of L p -mixed volume, we have
Multiply both sides of (2.3) and (2.4), in view of (1.7) and (2.2) and using the Cauchy-Schwarz inequality (see ), we obtain
Taking p = n - 1 in (2.1) and in view of the property (1) of mixed body, we obtain the following result: If , then
This is just an inequality given by Ghandehari .
Let L = B, we have the following interesting result:
Let K be a convex body and K* its polar dual, then
Taking p = n-1 in (2.6), we have the following result which was given in :
with equality if and only if K is an n-ball.
Corollary 2.2 The L p -mixed volume of K and K*, V p (K, K*) satisfies
Proof In view of the property (4) of the mixed body, we have
Form (1.4) and taking for K1 = K2 = ⋯ = K n -1 = K in (2.1), we have
Taking p = n-1 in (2.7), we have the following result:
A reverse inequality about was given by Ghandehari .
Theorem 2.3 Let K be a star body in , K* be the polar dual of K, then there exist a universal constant c> 0 such that
where c is the constant of Bourgain and Milman's inequality.
Proof From (1.6) and (1.8), we have
The following theorem concerning L p -dual mixed volumes will generalize Santaló inequality.
Theorem 2.4 Let K1 and K2 be two star bodies, and be the polar dual of K1 and K2, then there exists a constant c, L p -dual mixed volumes and satisfy
Proof From (1.6), we have
For and , we also have
Multiply both sides of (2.10) and (2.11) and using Bourgain and Milman's inequality, we obtain
Taking for K1 = K2 = K in (2.9) and in view of , (2.9) changes to the Bourgain and Milman's inequality (1.8).
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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.
The authors declare that they have no competing interests.
C-JZ, L-YC and W-SC jointly contributed to the main results Theorems 2.1, 2.3, and 2.4. All authors read and approved the final manuscript.
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Zhao, CJ., Chen, LY. & Cheung, WS. Polar duals of convex and star bodies. J Inequal Appl 2012, 90 (2012). https://doi.org/10.1186/1029-242X-2012-90
- polar dual
- L p -mixed volume
- dual L p -mixed volume
- the Bourgain and Milman's inequality