# On the quermassintegrals of convex bodies

- Chang Jian Zhao
^{1}Email author and - Wing Sum Cheung
^{2}

**2013**:264

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

© Zhao and Cheung; licensee Springer 2013

**Received: **18 March 2013

**Accepted: **9 May 2013

**Published: **27 May 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.

## Keywords

## 1 Introduction

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

*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].

*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 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.

*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 ${\omega}_{n}$.

*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 $C({S}^{n-1})$.

*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 $\tilde{\delta}(K,L)={|{\rho}_{K}-{\rho}_{L}|}_{\mathrm{\infty}}$.

*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*.

known as formula ‘Steiner decomposition’.

*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

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

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

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

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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