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# On the quermassintegrals of convex bodies

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 264 (2013)

## Abstract

The well-known question for quermassintegrals is the following: For which values of i\in \mathbb{N} and every pair of convex bodies *K* and *L*, is it true that

In 2003, the inequality was proved if and only if i=n-1 or i=n-2. Following the problem, in the paper, we prove some interrelated results for the quermassintegrals of a convex body.

**MSC:**26D15, 52A30.

## 1 Introduction

The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write {E}_{i}(x), 0\le i\le n, for the *i* th elementary symmetric function of an *n*-tuple x=({x}_{1},\dots ,{x}_{n}) of positive real numbers. This is defined by {E}_{0}(x)=1 and

In particular, {E}_{1}(x)={x}_{1}+\cdots +{x}_{n}, {E}_{2}(x)={\sum}_{i\ne j}{x}_{i}{x}_{j},\dots ,{E}_{n}(x)={x}_{1}{x}_{2}\cdots {x}_{n}.

The Marcus-Lopes inequality (see also [2], p.33]) states that

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

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 {K}_{i} and {L}_{i} denote the submatrices obtained by deleting their *i* th row and column, then

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

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 {\mathbb{R}}^{n} and if 0\le i\le n-1, then

with equality if and only if *K* and *L* are homothetic, where {W}_{i}(K) is the *i* th 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 0\le i\le n-1, i\in \mathbb{N}, is it true that, for every pair of convex bodies *K* and *L* in {\mathbb{R}}^{n}, one has

In 1991, the special case i=0 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 i=0 and *L* is a ball.

In 2002, it was proved in [9] that (1.6) is true for all i=1,\dots ,n-1 in the case where *L* is a ball.

**Theorem A** *If* *K* *is a convex body and* *B* *is a ball in* {\mathbb{R}}^{n}, *then for* 0\le i\le n-1, i\in \mathbb{N},

In 2003, it was proved in [8] that (1.6) holds true for every pair of convex bodies *K* and *L* in {\mathbb{R}}^{n} if and only if i=n-2 or i=n-1.

**Theorem B** *Let* 0\le i\le n-1, *then*

*is true for every pair of convex bodies* *K* *and* *L* *in* {\mathbb{R}}^{n} *if and only if* i=n-1 *or* i=n-2.

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

**Theorem 1.1** *Let* 0\le i\le n-1 *and for every convex body* *K* *and* *L* *in* {\mathbb{R}}^{n}. *Then the function*

*is a convex function on* t\in [0,+\mathrm{\infty}) *if and only if* i=n-1 *or* i=n-2.

**Theorem 1.2** *Let* 0\le i\le n-1 *and for every convex body* *K* *and* *L* *in* {\mathbb{R}}^{n}. *Then*

*if and only if* i=n-1 *or* i=n-2.

## 2 Notations and preliminaries

The setting for this paper is an *n*-dimensional Euclidean space {\mathbb{R}}^{n}. Let {\mathcal{K}}^{n} denote 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-1}. The volume of the unit *n*-ball is denoted by {\omega}_{n}.

We use V(K) for the *n*-dimensional volume of a convex body *K*. Let h(K,\cdot ):{S}^{n-1}\to \mathbb{R} denote the support function of K\in {\mathcal{K}}^{n}; *i.e.*, for u\in {S}^{n-1},

where u\cdot x denotes the usual inner product *u* and *x* in {\mathbb{R}}^{n}.

Let *δ* denote the Hausdorff metric on {\mathcal{K}}^{n}, *i.e.*, for K,L\in {\mathcal{K}}^{n}, \delta (K,L)={|{h}_{K}-{h}_{L}|}_{\mathrm{\infty}}, where {|\cdot |}_{\mathrm{\infty}} denotes the sup-norm on the space of continuous functions C({S}^{n-1}).

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

If \rho (K,\cdot ) is positive and continuous, *K* will be called a star body. Let {\mathcal{S}}^{n} denote the set of star bodies in {\mathbb{R}}^{n}. Let \tilde{\delta} denote the radial Hausdorff metric, as follows, if K,L\in {\mathcal{S}}^{n}, then \tilde{\delta}(K,L)={|{\rho}_{K}-{\rho}_{L}|}_{\mathrm{\infty}}.

If {K}_{i}\in {\mathcal{K}}^{n} (i=1,2,\dots ,r) and {\lambda}_{i} (i=1,2,\dots ,r) are nonnegative real numbers, then of fundamental importance is the fact that the volume of {\sum}_{i=1}^{r}{\lambda}_{i}{K}_{i} is a homogeneous polynomial in the {\lambda}_{i} given by (see, *e.g.*, [11] or [12])

where the sum is taken over all *n*-tuples ({i}_{1},\dots ,{i}_{n}) of positive integers not exceeding *r*. The coefficient {V}_{{i}_{1},\dots ,{i}_{n}} depends only on the bodies {K}_{{i}_{1}},\dots ,{K}_{{i}_{n}} and is uniquely determined by (2.1). It is called the mixed volume of {K}_{{i}_{1}},\dots ,{K}_{{i}_{n}}, and is written as V({K}_{{i}_{1}},\dots ,{K}_{{i}_{n}}). Let {K}_{1}=\cdots ={K}_{n-i}=K and {K}_{n-i+1}=\cdots ={K}_{n}=L, then the mixed volume V({K}_{1},\dots ,{K}_{n}) is written as {V}_{i}(K,L). If {K}_{1}=\cdots ={K}_{n-i}=K, {K}_{n-i+1}=\cdots ={K}_{n}=B, then the mixed volume {V}_{i}(K,B) is written as {W}_{i}(K) 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])

Let s=2, {\lambda}_{1}=1, {K}_{1}=K, {K}_{2}=B, 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:

## 3 Proof of main results

*Proof of Theorem 1.1* If s,t\in [0,\mathrm{\infty}), from (1.8), if and only if i=n-1 or i=n-2, we have

Hence the function g(t) is a convex function on [0,+\mathrm{\infty}) for every star body *K* and *L* if and only if i=n-1 or i=n-2. □

*Proof of Theorem 1.2* Let *K* be a convex body in {\mathbb{R}}^{n}. For every i\ge 0, we set

then from (2.3)

Therefore

The derivative of the function

is thus given by

Since {g}_{i}(x) is a convex function if and only if i=n-1 or i=n-2, hence by differentiating the both sides of (3.2), we obtain for t\in (0,+\mathrm{\infty})

if and only if i=n-1 or i=n-2.

This can be equivalently written in the form

if and only if i=n-1 or i=n-2.

Letting t\to {0}^{+}, we conclude Theorem 1.2. □

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## Acknowledgements

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

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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.

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Zhao, C.J., Cheung, W.S. On the quermassintegrals of convex bodies.
*J Inequal Appl* **2013**, 264 (2013). https://doi.org/10.1186/1029-242X-2013-264

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DOI: https://doi.org/10.1186/1029-242X-2013-264

### Keywords

- symmetric function
- convex body
- quermassintegral