# Polar duals of convex and star bodies

- Chang-Jian Zhao
^{1}Email author, - Lian-Ying Chen
^{1}and - Wing-Sum Cheung
^{2}

**2012**:90

https://doi.org/10.1186/1029-242X-2012-90

© Zhao et al; licensee Springer. 2012

**Received: **17 December 2011

**Accepted: **17 April 2012

**Published: **17 April 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.

## Keywords

*L*

_{ p }-mixed volumedual

*L*

_{ p }-mixed volumethe Bourgain and Milman's inequality

## 1 Notations and preliminaries

The setting for this article is *n*-dimensional Euclidean space ${\mathbb{R}}^{n}\left(n>2\right)$. Let ${\mathcal{K}}^{n}$ denotes the set of convex bodies (compact, convex subsets with non-empty 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 *n*-ball is denoted by *ω*_{n}.

*V*(

*K*) for the

*n*-dimensional volume of convex body

*K*. $h\left(K,\cdot \right):{S}^{n-1}\to \mathbb{R}$, 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 sup-norm on the space of continuous functions *C*(*S*^{
n
}^{-l}).

*K*of ${\mathbb{R}}^{n}$, which is star-shaped with respect to the origin, is its radial function $\rho \left(K,\cdot \right):{S}^{n-1}\to \mathbb{R}$, 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

*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}.

*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.

*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*.

*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 ${K}_{1},\dots ,{K}_{n-1}\in {\mathcal{K}}^{n}$, the notation of mixed body [*K*_{1},..., *K*_{
n
}_{-1}] states that (see [5]): corresponding to the convex bodies ${K}_{1},\dots ,{K}_{n-1}\in {\mathcal{K}}^{n}$ in ${\mathbb{R}}^{n}$, there exists a convex body, unique up to translation, which we denote *by*[*K*_{1},..., *K*_{
n
}_{-1}].

_{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)
$\underset{n-1}{\underset{\u23df}{\left[K,\dots ,K\right]}}=K$.

The properties of mixed body play an important role in proving our main results.

### 1.3 Polar of convex body

*K*,

*K**is defined:

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

*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

*L*

_{ p }-mixed volume, we have

*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

This is just an inequality given by Ghandehari [9].

Let *L* = *B*, we have the following interesting result:

*K*be a convex body and

*K**its polar dual, then

*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

*K*

_{1}=

*K*

_{2}=

*⋯ = K*

_{ n }

_{-1}=

*K*in (2.1), we have

*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]).

**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

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).

## Declarations

### 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.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.