Research

# On the quermassintegrals of convex bodies

Chang Jian Zhao1* and Wing Sum Cheung2

Author Affiliations

1 Department of Mathematics, China Jiliang University, Hangzhou, 310018, P.R. China

2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China

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Journal of Inequalities and Applications 2013, 2013:264  doi:10.1186/1029-242X-2013-264

 Received: 18 March 2013 Accepted: 9 May 2013 Published: 27 May 2013

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.

### Abstract

The well-known question for quermassintegrals is the following: For which values of and every pair of convex bodies K and L, is it true that

In 2003, the inequality was proved if and only if or . Following the problem, in the paper, we prove some interrelated results for the quermassintegrals of a convex body.

MSC: 26D15, 52A30.

##### Keywords:
symmetric function; convex body; quermassintegral

### 1 Introduction

The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write , , for the ith elementary symmetric function of an n-tuple of positive real numbers. This is defined by and

In particular, , .

(1.1)

for every pair of positive n-tuples x and y. This is a refinement of a further result concerning the symmetric functions, namely

(1.2)

A discussion of the cases of equality is contained in the reference [1].

A matrix analogue of (1.1) is the following result of Bergstrom [3] (see also the article [4] and [5], p.67] for an interesting proof): If K and L are positive definite matrices, and if and denote the submatrices obtained by deleting their ith row and column, then

(1.3)

The following generalization of (1.3) was established by Ky Fan [5]:

(1.4)

The proof is based on a minimum principle; see also Ky Fan [6] and Mirsky [7].

There is a remarkable similarity between inequalities about symmetric functions (or determinants of symmetric matrices) and inequalities about the mixed volumes of convex bodies. For example, the analogue of (1.2) in the Brunn-Minkowski theory is as follows.

If K and L are convex bodies in and if , then

(1.5)

with equality if and only if K and L are homothetic, where is the ith quermassintegral of K (see Section 2).

In view of this analogue, Milman asked if there exists a version of (1.1) or (1.3) in the theory of mixed volumes (see [8,9]).

Question For which values of , , is it true that, for every pair of convex bodies K and L in , one has

(1.6)

In 1991, the special case was stated also in [10] as an open question. In the same paper it was also mentioned that (1.6) follows directly from the Aleksandrov-Fenchel inequality when and L is a ball.

In 2002, it was proved in [9] that (1.6) is true for all in the case where L is a ball.

Theorem AIfKis a convex body andBis a ball in, then for, ,

(1.7)

In 2003, it was proved in [8] that (1.6) holds true for every pair of convex bodies K and L in if and only if or .

Theorem BLet, then

(1.8)

is true for every pair of convex bodiesKandLinif and only ifor.

In this paper, following the above results, we prove the following interest results.

Theorem 1.1Letand for every convex bodyKandLin. Then the function

(1.9)

is a convex function onif and only ifor.

Theorem 1.2Letand for every convex bodyKandLin. Then

(1.10)

if and only ifor.

### 2 Notations and preliminaries

The setting for this paper is an n-dimensional Euclidean space . Let denote 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 . The volume of the unit n-ball is denoted by .

We use for the n-dimensional volume of a convex body K. Let denote the support function of ; i.e., for ,

where denotes the usual inner product u and x in .

Let δ denote the Hausdorff metric on , i.e., for , , where denotes the sup-norm on the space of continuous functions .

Associated with a compact subset K of , which is star-shaped with respect to the origin, is its radial function , defined for by

If is positive and continuous, K will be called a star body. Let denote the set of star bodies in . Let denote the radial Hausdorff metric, as follows, if , then .

If () and () are nonnegative real numbers, then of fundamental importance is the fact that the volume of is a homogeneous polynomial in the given by (see, e.g., [11] or [12])

(2.1)

where the sum is taken over all n-tuples () of positive integers not exceeding r. The coefficient depends only on the bodies and is uniquely determined by (2.1). It is called the mixed volume of , and is written as . Let and , then the mixed volume is written as . If , , then the mixed volume is written as and is called the quermassintegral of a convex body K.

It is convenient to write relation (2.1) in the form (see [12], p.137])

(2.2)

Let , , , , we have

known as formula ‘Steiner decomposition’.

On the other hand, for convex bodies K and L, (2.2) can show the following special case:

(2.3)

### 3 Proof of main results

Proof of Theorem 1.1 If , from (1.8), if and only if or , we have

(3.1)

Hence the function is a convex function on for every star body K and L if and only if or . □

Proof of Theorem 1.2 Let K be a convex body in . For every , we set

then from (2.3)

Therefore

The derivative of the function

is thus given by

(3.2)

Since is a convex function if and only if or , hence by differentiating the both sides of (3.2), we obtain for

if and only if or .

This can be equivalently written in the form

if and only if or .

Letting , we conclude Theorem 1.2. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

CJZ and WSC jointly contributed to the main results Theorems 1.1-1.2. All authors read and approved the final manuscript.

### Acknowledgements

First author is supported by the National Natural Science Foundation of China (10971205). Second author is partially supported by a HKU URG grant.

### References

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