Polar duals of convex and star bodies

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.

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 ω_n.

We use V(K) for the n-dimensional volume of convex body K. , denotes the support function of ; i.e., for u ∈ S^n-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(S^n-l).

Associated with a compact subset K of , which is star-shaped with respect to the origin, is its radial function , defined for u ∈ S^n-l, by

If ρ(K, ·) is positive and continuous, K will be called a star body. Let Sⁿ denotes the set of star bodies in . Let denotes the radial Hausdorff metric, as follows, if K, L∈ Sⁿ, then (See [1,2]).

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 [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 ∈ Sⁿ, p ≥ 1, the L_p-dual mixed volume 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 , the notation of mixed body [K₁,..., K_n-1] states that (see [5]): 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 λ_i> 0,

(1) V₁([K₁, ..., K_n-1], K_n) = V(K₁, ..., K_n-1, K_n);

(2) [K₁ + L₁, K₂, ..., K_n-1] = [K₁, K₂, ..., K_n-1] + [L₁, K₂, ..., K_n-1];

(3) [λ ₁ K ₁, ..., λ_n-1K_n-1] = λ₁... λ_n-1 · [K₁, ..., K_n-1];

(4) .

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 [6]).

If K is a convex symmetric body in , then there exists a universal constant c> 0 such that

Different proofs were given by Pisier [7].

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

and

Multiply both sides of (2.3) and (2.4), in view of (1.7) and (2.2) and using the Cauchy-Schwarz 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 , 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 = n-1 in (2.6), we have the following result which was given in [9]:

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:

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 was given by Ghandehari [9].

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).

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.

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