Polar duals of convex and star bodies

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 n-dimensional Euclidean space ${ℝ}^n\left(n>2\right)$. Let ${\mathcal{K}}^n$ denotes the set of convex bodies (compact, convex subsets with non-empty interiors) in ${ℝ}^n$. 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 n-ball is denoted by ω_n.

We use V(K) for the n-dimensional volume of convex body K. $h\left(K,\cdot \right):S^{n-1}\to ℝ$, denotes the support function of $K\in {\mathcal{K}}^n$; i.e., for u ∈ S^n-l

$$h\left(K,u\right)=\mathsf{\text{Max}}\left\{u\cdot x:x\in K\right\},$$

(1.1)

where u · x denotes the usual inner product u and x in ${ℝ}^n$.

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 sup-norm on the space of continuous functions C(S^n-l).

Associated with a compact subset K of ${ℝ}^n$, which is star-shaped with respect to the origin, is its radial function $\rho \left(K,\cdot \right):S^{n-1}\to ℝ$, defined for u ∈ S^n-l, by

$$\rho \left(K,u\right)=\mathsf{\text{Max}}\left\{\lambda \ge \mathsf{\text{0:}}\lambda u\in K\right\}.$$

(1.2)

If ρ(K, ·) is positive and continuous, K will be called a star body. Let Sⁿ denotes the set of star bodies in ${ℝ}^n$. Let $\tilde{\delta }$ denotes the radial Hausdorff metric, as follows, if K, L∈ Sⁿ, then $\tilde{\delta }\left(K,L\right)={\left|{\rho }_K-{\rho }_L\right|}_{\infty }$ (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]):

$$V_p\left(K,L\right)=\frac{1}{n}{\int }_{S^{n-1}}h{\left(L,u\right)}^p\mathsf{\text{d}}{S}_p\left(K,u\right),$$

(1.3)

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

$$V_p\left(K,L\right)\ge V{\left(K\right)}^{\left(n-p\right)/n}V{\left(L\right)}^{p/n},$$

(1.4)

with equality if and only if K and L are homothetic.

If K, L ∈ Sⁿ, p ≥ 1, the L _p -dual mixed volume ${Ṽ}_{-p}\left(K,L\right)$ was defined by Lutwak (see [4]):

$${Ṽ}_{-p}\left(K,L\right)=\frac{1}{n}\underset{S^{n-1}}{\int }\rho {\left(K,u\right)}^{n+p}\rho {\left(L,u\right)}^{-p}\mathsf{\text{d}}S\left(u\right),$$

(1.5)

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

$${Ṽ}_{-p}{\left(K,L\right)}^n\ge V{\left(K\right)}^{n+p}V{\left(L\right)}^{-p},$$

(1.6)

with equality if and only if K and L are dilates.

1.2 Mixed bodies of convex bodies

If $K_1,\dots ,K_{n-1}\in {\mathcal{K}}^n$, the notation of mixed body [K₁,..., K _{n -1}] states that (see [5]): corresponding to the convex bodies $K_1,\dots ,K_{n-1}\in {\mathcal{K}}^n$ in ${ℝ}^n$, 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 $K_i\in {\mathcal{K}}^n,i=1,\dots ,n,L_1\in {\mathcal{K}}^n$ 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 -1}K _{n -1}] = λ₁... λ _{n -1} · [K₁, ..., K _{n -1}];

• (4) $\underset{n-1}{\underbrace{\left[K,\dots ,K\right]}}=K$.

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:

$$K^{*}=\left\{x\in {ℝ}^n:x\cdot y\le 1,y\in K\right\}.$$

It is easy to get that

$$\rho {\left(K,u\right)}^{-1}=h\left(K^{*},u\right).$$

(1.7)

Bourgain and Milman's inequality is stated as follows (see [6]).

If K is a convex symmetric body in ${ℝ}^n$, then there exists a universal constant c> 0 such that

$$V\left(K\right)V\left(K^{*}\right)\ge c^n{\omega }_n^{\mathsf{\text{2}}}.$$

(1.8)

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, K₁, ..., K _{n- 1} are convex bodies in ${ℝ}^n$ 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

$$V_p\left(K,L\right)V_p\left(K^{*},L\right)\ge V_p{\left(B,L\right)}^2.$$

(2.1)

Proof From (1.1) and (1.2), it is easy

$$h\left(K,u\right)\ge \rho \left(K,u\right),K\in {\mathcal{K}}^n.$$

(2.2)

By definition of L _p -mixed volume, we have

$$V_p\left(K,L\right)=\frac{1}{n}\underset{S^{n-1}}{\int }h{\left(K,u\right)}^p\mathsf{\text{d}}{S}_p\left(L;u\right),$$

(2.3)

and

$$V_p\left(K^{*},L\right)=\frac{1}{n}\underset{S^{n-1}}{\int }h{\left(K^{*},u\right)}^p\mathsf{\text{d}}{S}_p\left(L,u\right).$$

(2.4)

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

$$\begin{array}{l}{n}^2{V}_p\left(K,L\right)V_p\left(K^{*},L\right)\\ =\left(\underset{S^{n-1}}{\int }h{\left(K,u\right)}^p\mathsf{\text{d}}{S}_p\left(K_1,\dots ,K_{n-1};u\right)\right)\left(\underset{S^{n-1}}{\int }\frac{1}{\rho {\left(K,u\right)}^p}\mathsf{\text{d}}{S}_p\left(K_1,\dots ,K_{n-1};u\right)\right)\\ \ge {\left(\underset{S^{n-1}}{\int }h{\left(K,u\right)}^{\frac{p}{2}}\cdot \frac{\mathsf{\text{1}}}{\rho {\left(K,u\right)}^{\frac{p}{2}}}\mathsf{\text{d}}{S}_p\left(K_1,\dots ,K_{n-1};u\right)\right)}^2\\ \ge {\left(\underset{S^{n-1}}{\int }\mathsf{\text{d}}{S}_p\left(K_1,\dots ,K_{n-1};u\right)\right)}^2\\ =n^2{V}_p^2\left(B,L\right).\end{array}$$

Taking p = n - 1 in (2.1) and in view of the property (1) of mixed body, we obtain the following result: If $K,K_1,\dots ,K_{n-1}\in {\mathcal{K}}^n$, then

$$V\left(K,K_1,\dots ,K_{n-1}\right)V\left(K^{*},K_1,\dots ,K_n\right)\ge V{\left(B,K_1,\dots ,K_{n-1}\right)}^2.$$

(2.5)

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

$$V_p\left(K,B\right)V_p\left(K^{*},B\right)\ge {\omega }_n^2.$$

(2.6)

Taking p = n-1 in (2.6), we have the following result which was given in [9]:

$$W_{n-1}\left(K\right)W_{n-1}\left(K^{*}\right)\ge {\omega }_n^2,$$

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

$$V_p{\left(K^{*},K\right)}^n\ge {\omega }_n^{2\left(n-p\right)}V{\left(K\right)}^{2p-n}.$$

(2.7)

Proof In view of the property (4) of the mixed body, we have

$$V_p\left(K,\left[K,\dots ,K\right]\right)=V_p\left(K,K\right)=V\left(K\right).$$

Form (1.4) and taking for K₁ = K₂ = ⋯ = K _{n -1} = K in (2.1), we have

$$\begin{array}{ll}\hfill V\left(K\right)V_p\left(K^{*},K\right)& \ge V_p^2\left(B,K\right)\\ \ge V{\left(B\right)}^{\frac{2\left(n-p\right)}{n}}V{\left(K\right)}^{\frac{2p}{n}}\\ ={\omega }_n^{\frac{2\left(n-p\right)}{n}}V{\left(K\right)}^{\frac{2p}{n}}.\end{array}$$

Taking p = n-1 in (2.7), we have the following result:

$$V{\left(K^{*},\underset{n-1}{\underbrace{K,\dots ,K}}\right)}^n\ge {\omega }_n^{2}V{\left(K\right)}^{n-2}.$$

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 $Ṽ\left(K^{*},\underset{n-1}{\underbrace{K,\dots ,K}}\right)$ was given by Ghandehari [9].

$$Ṽ{\left(K^{*},\underset{n-1}{\underbrace{K,\dots ,K}}\right)}^n\le {\omega }_n^{2}V{\left(K\right)}^{n-2}.$$

Theorem 2.3 Let K be a star body in ${ℝ}^n$, K* be the polar dual of K, then there exist a universal constant c> 0 such that

$$V{\left(K\right)}^{n+2p}{Ṽ}_{-p}{\left(K^{*},K\right)}^n\ge {\left(c^n{\omega }_n^2\right)}^{n+p},$$

(2.8)

where c is the constant of Bourgain and Milman's inequality.

Proof From (1.6) and (1.8), we have

$$\begin{array}{ll}\hfill {Ṽ}_{-p}\left(K^{*},K\right)& \ge V{\left(K^{*}\right)}^{\frac{n+p}{n}}V{\left(K\right)}^{-\frac{p}{n}}\\ ={\left(V\left(K^{*}\right)V\left(K\right)\right)}^{\frac{n+p}{n}}V{\left(K\right)}^{-\frac{n+2p}{n}}\\ \ge {\left(c^n{\omega }_n^{\mathsf{\text{2}}}\right)}^{\frac{n+p}{n}}V{\left(K\right)}^{-\frac{n+2p}{n}}.\end{array}$$

The following theorem concerning L _p -dual mixed volumes will generalize Santaló inequality.

Theorem 2.4 Let K₁ and K₂ be two star bodies, $K_1^{*}$ and $K_2^{*}$ be the polar dual of K₁ and K₂, then there exists a constant c, L _p -dual mixed volumes ${Ṽ}_{-p}\left(K_1,K_2\right)$ and ${Ṽ}_{-p}\left(K_1^{*},K_2^{*}\right)$satisfy

$${Ṽ}_{-p}\left(K_1,K_2\right){Ṽ}_{-p}\left(K_1^{*},K_2^{*}\right)\ge c^n{\omega }_n^2.$$

(2.9)

Proof From (1.6), we have

$${Ṽ}_{-p}\left(K_1,K_2\right)\ge {\left(K_1\right)}^{\frac{n+p}{n}}V{\left(K_2\right)}^{-\frac{p}{n}}.$$

(2.10)

For $K_1^{*}$ and $K_2^{*}$, we also have

$${Ṽ}_{-p}\left(K_1^{*},K_2^{*}\right)\ge V{\left(K_1^{*}\right)}^{\frac{n+p}{n}}V{\left(K_2^{*}\right)}^{-\frac{p}{n}}.$$

(2.11)

Multiply both sides of (2.10) and (2.11) and using Bourgain and Milman's inequality, we obtain

$$\begin{array}{ll}\hfill {Ṽ}_{-p}\left(K_1,K_2\right){Ṽ}_{-p}\left(K_1^{*},K_2^{*}\right)& \ge {\left(V\left(K_1\right)V\left(K_1^{*}\right)\right)}^{-\frac{p}{n}}{\left(V\left(K_2\right)V\left(K_2^{*}\right)\right)}^{-\frac{p}{n}}\\ \ge {\left(c^n{\omega }_n^{\mathsf{\text{2}}}\right)}^{\frac{n+p}{n}}{\left(c^n{\omega }_n^{\mathsf{\text{2}}}\right)}^{-\frac{p}{n}}\\ =c^n{\omega }_n^{\mathsf{\text{2}}}.\end{array}$$

Taking for K₁ = K₂ = K in (2.9) and in view of ${Ṽ}_{-p}\left(K_1,K_2\right)={Ṽ}_{-p}\left(K,K\right)=V\left(K\right)$, (2.9) changes to the Bourgain and Milman's inequality (1.8).