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.
Keywords:polar dual; Lp-mixed volume; dual Lp-mixed volume; the Bourgain and Milman's inequality
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.
1.1 Lp-mixed volume and dual Lp-mixed volume
If , the Lp-mixed volume Vp(K, L) was defined by Lutwak (see ):
where Sp(K, ·) denotes a positive Borel measure on Sn-1.
The Lp 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 Lp-dual mixed volume was defined by Lutwak (see ):
where dS(u) signifies the surface area element on Sn-1 at u.
The following dual Lp-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,..., Kn-1] states that (see ): corresponding to the convex bodies in , there exists a convex body, unique up to translation, which we denote by[K1,..., Kn-1].
(1) V1([K1, ..., Kn-1], Kn) = V(K1, ..., Kn-1, Kn);
(2) [K1 + L1, K2, ..., Kn-1] = [K1, K2, ..., Kn-1] + [L1, K2, ..., Kn-1];
(3) [λ 1 K 1, ..., λn-1Kn-1] = λ1... λn-1 · [K1, ..., Kn-1];
The properties of mixed body play an important role in proving our main results.
1.3 Polar of convex body
It is easy to get that
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.
Proof From (1.1) and (1.2), it is easy
By definition of Lp-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
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 Lp-mixed volume of K and K*, Vp(K, K*) satisfies
Proof In view of the property (4) of the mixed body, we have
Form (1.4) and taking for K1 = K2 = ⋯ = Kn-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 .
where c is the constant of Bourgain and Milman's inequality.
Proof From (1.6) and (1.8), we have
The following theorem concerning Lp-dual mixed volumes will generalize Santaló inequality.
Proof From (1.6), we have
Multiply both sides of (2.10) and (2.11) and using Bourgain and Milman's inequality, we obtain
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.
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.