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Polar duals of convex and star bodies
Journal of Inequalities and Applications volume 2012, Article number: 90 (2012)
Abstract
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 ndimensional Euclidean space {\mathbb{R}}^{n}\left(n>2\right). Let {\mathcal{K}}^{n} denotes the set of convex bodies (compact, convex subsets with nonempty interiors) in {\mathbb{R}}^{n}. 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 nball is denoted by ω_{n}.
We use V(K) for the ndimensional volume of convex body K. h\left(K,\cdot \right):{S}^{n1}\to \mathbb{R}, denotes the support function of K\in {\mathcal{K}}^{n}; i.e., for u ∈ S^{n}^{l}
where u · x denotes the usual inner product u and x in {\mathbb{R}}^{n}.
Let δ denotes the Hausdorff metric on {\mathcal{K}}^{n}, i.e., for K,L\in {\mathcal{K}}^{n},\delta \left(K,L\right)={\left{h}_{K}{h}_{L}\right}_{\infty}, where  · _{ ∞ } denotes the supnorm on the space of continuous functions C(S^{n}^{l}).
Associated with a compact subset K of {\mathbb{R}}^{n}, which is starshaped with respect to the origin, is its radial function \rho \left(K,\cdot \right):{S}^{n1}\to \mathbb{R}, defined for u ∈ S^{n}^{l}, by
If ρ(K, ·) is positive and continuous, K will be called a star body. Let S^{n} denotes the set of star bodies in {\mathbb{R}}^{n}. Let \stackrel{\u0303}{\delta} denotes the radial Hausdorff metric, as follows, if K, L∈ S^{n}, then \stackrel{\u0303}{\delta}\left(K,L\right)={\left{\rho}_{K}{\rho}_{L}\right}_{\infty} (See [1, 2]).
1.1 L_{ p }mixed volume and dual L_{ p }mixed volume
If K,L\in {\mathcal{K}}^{n}, the L_{ p }mixed volume V_{ p }(K, L) was defined by Lutwak (see [3]):
where S_{ p }(K, ·) denotes a positive Borel measure on S^{n}^{1}.
The L_{ p } analog of the classical Minkowski inequality (see [3]) states that: If K and L are convex bodies, then
with equality if and only if K and L are homothetic.
If K, L ∈ S^{n}, p ≥ 1, the L_{ p }dual mixed volume {\u1e7c}_{p}\left(K,L\right) was defined by Lutwak (see [4]):
where dS(u) signifies the surface area element on S^{n}^{1} at u.
The following dual L_{ p }Minkowski inequality was obtained in [2]: 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 {K}_{1},\dots ,{K}_{n1}\in {\mathcal{K}}^{n}, the notation of mixed body [K_{1},..., K_{ n }_{1}] states that (see [5]): corresponding to the convex bodies {K}_{1},\dots ,{K}_{n1}\in {\mathcal{K}}^{n} in {\mathbb{R}}^{n}, there exists a convex body, unique up to translation, which we denote by[K_{1},..., 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 {K}_{i}\in {\mathcal{K}}^{n},i=1,\dots ,n,{L}_{1}\in {\mathcal{K}}^{n} and λ_{i}> 0,

(1)
V_{1}([K_{1}, ..., K_{ n }_{1}], K_{ n }) = V(K_{1}, ..., K_{ n }_{1}, K_{ n });

(2)
[K_{1} + L_{1}, K_{2}, ..., K_{ n }_{1}] = [K_{1}, K_{2}, ..., K_{ n }_{1}] + [L_{1}, K_{2}, ..., K_{ n }_{1}];

(3)
[λ _{1} K _{1}, ..., λ_{ n }_{1}K_{ n }_{1}] = λ_{1}... λ_{ n }_{1} · [K_{1}, ..., K_{ n }_{1}];

(4)
\underset{n1}{\underset{\u23df}{\left[K,\dots ,K\right]}}=K.
The properties of mixed body play an important role in proving our main results.
1.3 Polar of convex body
For K\in {\mathcal{K}}^{n}, the polar body of K, K* is defined:
It is easy to get that
Bourgain and Milman's inequality is stated as follows (see [6]).
If K is a convex symmetric body in {\mathbb{R}}^{n}, then there exists a universal constant c> 0 such that
Different proofs were given by Pisier [7].
2 Main results
In this article, we establish some new inequalities on polar duals of convex and star bodies.
Theorem 2.1 If K, K_{1}, ..., K_{ n }_{1} are convex bodies in {\mathbb{R}}^{n} and let L = [K_{1}, ..., 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
and
Multiply both sides of (2.3) and (2.4), in view of (1.7) and (2.2) and using the CauchySchwarz inequality (see [8]), 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 K,{K}_{1},\dots ,{K}_{n1}\in {\mathcal{K}}^{n}, then
This is just an inequality given by Ghandehari [9].
Let L = B, we have the following interesting result:
Let K be a convex body and K* its polar dual, then
Taking p = n1 in (2.6), we have the following result which was given in [9]:
with equality if and only if K is an nball.
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 K_{1} = K_{2} = ⋯ = K_{ n }_{1} = K in (2.1), we have
Taking p = n1 in (2.7), we have the following result:
This is just an inequality given by Ghandehari [9]. The cases p = 1 and n = 2 give Steinhardt's and Firey's result (see [7]).
A reverse inequality about \u1e7c\left({K}^{*},\underset{n1}{\underset{\u23df}{K,\dots ,K}}\right) was given by Ghandehari [9].
Theorem 2.3 Let K be a star body in {\mathbb{R}}^{n}, 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 K_{1} and K_{2} be two star bodies, {K}_{1}^{*} and {K}_{2}^{*} be the polar dual of K_{1} and K_{2}, then there exists a constant c, L_{ p }dual mixed volumes {\u1e7c}_{p}\left({K}_{1},{K}_{2}\right) and {\u1e7c}_{p}\left({K}_{1}^{*},{K}_{2}^{*}\right)satisfy
Proof From (1.6), we have
For {K}_{1}^{*} and {K}_{2}^{*}, we also have
Multiply both sides of (2.10) and (2.11) and using Bourgain and Milman's inequality, we obtain
Taking for K_{1} = K_{2} = K in (2.9) and in view of {\u1e7c}_{p}\left({K}_{1},{K}_{2}\right)={\u1e7c}_{p}\left(K,K\right)=V\left(K\right), (2.9) changes to the Bourgain and Milman's inequality (1.8).
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Acknowledgements
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.
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The authors declare that they have no competing interests.
Authors' contributions
CJZ, LYC and WSC 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/10.1186/1029242X201290
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DOI: https://doi.org/10.1186/1029242X201290