# On the quermassintegrals of convex bodies

## 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

$\frac{{W}_{i}\left(K+L\right)}{{W}_{i+1}\left(K+L\right)}\ge \frac{{W}_{i}\left(K\right)}{{\stackrel{˜}{W}}_{i+1}\left(K\right)}+\frac{{W}_{i}\left(L\right)}{{W}_{i+1}\left(L\right)}?$

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 ${E}_{i}\left(x\right)$, $0\le i\le n$, for the i th elementary symmetric function of an n-tuple $x=\left({x}_{1},\dots ,{x}_{n}\right)$ of positive real numbers. This is defined by ${E}_{0}\left(x\right)=1$ and

${E}_{i}\left(x\right)=\sum _{1\le {j}_{1}<\cdots <{j}_{i}\le n}{x}_{{j}_{1}}{x}_{{j}_{2}}\cdots {x}_{{j}_{i}},\phantom{\rule{1em}{0ex}}1\le i\le n.$

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

$\frac{{E}_{i}\left(x+y\right)}{{E}_{i-1}\left(x+y\right)}\ge \frac{{E}_{i}\left(x\right)}{{E}_{i-1}\left(x\right)}+\frac{{E}_{i}\left(y\right)}{{E}_{i-1}\left(y\right)}$
(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

${\left[{E}_{i}\left(x+y\right)\right]}^{1/i}\ge {\left[{E}_{i}\left(x\right)\right]}^{1/i}+{\left[{E}_{i}\left(y\right)\right]}^{1/i}.$
(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 ${K}_{i}$ and ${L}_{i}$ denote the submatrices obtained by deleting their i th row and column, then

$\frac{det\left(K+L\right)}{det\left({K}_{i}+{L}_{i}\right)}\ge \frac{det\left(K\right)}{det\left({K}_{i}\right)}+\frac{det\left(L\right)}{det\left({L}_{i}\right)}.$
(1.3)

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

${\left(\frac{det\left(K+L\right)}{det\left({K}_{i}+{L}_{i}\right)}\right)}^{1/k}\ge {\left(\frac{det\left(K\right)}{det\left({K}_{i}\right)}\right)}^{1/k}+{\left(\frac{det\left(L\right)}{det\left({L}_{i}\right)}\right)}^{1/k}.$
(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 ${\mathbb{R}}^{n}$ and if $0\le i\le n-1$, then

${W}_{i}{\left(K+L\right)}^{1/\left(n-i\right)}\ge {W}_{i}{\left(K\right)}^{1/\left(n-i\right)}+{W}_{i}{\left(L\right)}^{1/\left(n-i\right)},$
(1.5)

with equality if and only if K and L are homothetic, where ${W}_{i}\left(K\right)$ 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

$\frac{{W}_{i}\left(K+L\right)}{{W}_{i+1}\left(K+L\right)}\ge \frac{{W}_{i}\left(K\right)}{{W}_{i+1}\left(K\right)}+\frac{{W}_{i}\left(L\right)}{{W}_{i+1}\left(L\right)}?$
(1.6)

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 $0\le i\le n-1$, $i\in \mathbb{N}$,

$\frac{{W}_{i}\left(K+B\right)}{{W}_{i+1}\left(K+B\right)}\ge \frac{{W}_{i}\left(K\right)}{{W}_{i+1}\left(K\right)}+\frac{{W}_{i}\left(B\right)}{{W}_{i+1}\left(B\right)}.$
(1.7)

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

$\frac{{W}_{i}\left(K+L\right)}{{W}_{i+1}\left(K+L\right)}\ge \frac{{W}_{i}\left(K\right)}{{W}_{i+1}\left(K\right)}+\frac{{W}_{i}\left(L\right)}{{W}_{i+1}\left(L\right)}$
(1.8)

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

$g\left(t\right)=\frac{{W}_{i}\left(K+tL\right)}{{W}_{i+1}\left(K+tL\right)}$
(1.9)

is a convex function on $t\in \left[0,+\mathrm{\infty }\right)$ 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

$\begin{array}{r}\left(n-i\right){W}_{i+2}\left(K\right)\left({W}_{i+1}{\left(K\right)}^{2}-{W}_{i}\left(K\right){W}_{i+2}\left(K\right)\right)\\ \phantom{\rule{1em}{0ex}}\ge \left(n-i-2\right){W}_{i}\left(K\right)\left({W}_{i+2}^{2}\left(K\right)-{W}_{i+1}\left(K\right){W}_{i+3}\left(K\right)\right)\end{array}$
(1.10)

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

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

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 $K,L\in {\mathcal{K}}^{n}$, $\delta \left(K,L\right)={|{h}_{K}-{h}_{L}|}_{\mathrm{\infty }}$, where ${|\cdot |}_{\mathrm{\infty }}$ denotes the sup-norm on the space of continuous functions $C\left({S}^{n-1}\right)$.

Associated with a compact subset 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 $u\in {S}^{n-1}$ by

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

If $\rho \left(K,\cdot \right)$ 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 $\stackrel{˜}{\delta }$ denote the radial Hausdorff metric, as follows, if $K,L\in {\mathcal{S}}^{n}$, then $\stackrel{˜}{\delta }\left(K,L\right)={|{\rho }_{K}-{\rho }_{L}|}_{\mathrm{\infty }}$.

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])

$V\left({\lambda }_{1}{K}_{1}+\cdots +{\lambda }_{n}{K}_{n}\right)=\sum _{{i}_{1},\dots ,{i}_{n}}{\lambda }_{{i}_{1}}\cdots {\lambda }_{{i}_{n}}{V}_{{i}_{1},\dots ,{i}_{n}},$
(2.1)

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 $V\left({K}_{{i}_{1}},\dots ,{K}_{{i}_{n}}\right)$. Let ${K}_{1}=\cdots ={K}_{n-i}=K$ and ${K}_{n-i+1}=\cdots ={K}_{n}=L$, then the mixed volume $V\left({K}_{1},\dots ,{K}_{n}\right)$ is written as ${V}_{i}\left(K,L\right)$. If ${K}_{1}=\cdots ={K}_{n-i}=K$, ${K}_{n-i+1}=\cdots ={K}_{n}=B$, then the mixed volume ${V}_{i}\left(K,B\right)$ is written as ${W}_{i}\left(K\right)$ 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])

$\begin{array}{r}V\left({\lambda }_{1}{K}_{1}+\cdots +{\lambda }_{s}{K}_{s}\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{{p}_{1}+\cdots +{p}_{r}=n}\sum _{1\le {i}_{1}<\cdots <{i}_{r}\le s}\frac{n!}{{p}_{1}!\cdots {p}_{r}!}{\lambda }_{{i}_{1}}^{{p}_{1}}\cdots {\lambda }_{{i}_{r}}^{{p}_{r}}V\left(\underset{{p}_{1}}{\underset{⏟}{{K}_{{i}_{1}},\dots ,{K}_{{i}_{1}}}},\dots ,\underset{{p}_{r}}{\underset{⏟}{{K}_{{i}_{r}},\dots ,{K}_{{i}_{r}}}}\right).\end{array}$
(2.2)

Let $s=2$, ${\lambda }_{1}=1$, ${K}_{1}=K$, ${K}_{2}=B$, we have

$V\left(K+\lambda B\right)=\sum _{i=0}^{n}{\left(}_{i}^{n}\right){\lambda }^{i}{W}_{i}\left(K\right),$

known as formula ‘Steiner decomposition’.

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

${W}_{i}\left(K+\lambda L\right)=\sum _{j=0}^{n-i}\left(\genfrac{}{}{0}{}{n-i}{j}\right){\lambda }^{j}V\left(\underset{n-i-j}{\underset{⏟}{K,\dots ,K}},\underset{i}{\underset{⏟}{B,\dots ,B}},\underset{j}{\underset{⏟}{L,\dots ,L}}\right).$
(2.3)

## 3 Proof of main results

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

$\begin{array}{rl}g\left(\frac{t+s}{2}\right)& =\frac{{W}_{i}\left(K+\frac{t+s}{2}L\right)}{{W}_{i+1}\left(K+\frac{t+s}{2}L\right)}\\ =\frac{{W}_{i}\left(\frac{K+tL}{2}+\frac{K+sL}{2}\right)}{{W}_{i+1}\left(\frac{K+tL}{2}+\frac{K+sL}{2}\right)}\\ \ge \frac{{W}_{i}\left(\frac{K+tL}{2}\right)}{{W}_{i+1}\left(\frac{K+tL}{2}\right)}+\frac{{W}_{i}\left(\frac{K+sL}{2}\right)}{{W}_{i+1}\left(\frac{K+sL}{2}\right)}\\ =\frac{1}{2}\frac{{W}_{i}\left(K+tL\right)}{{W}_{i+1}\left(K+tL\right)}+\frac{1}{2}\frac{{W}_{i}\left(K+sL\right)}{{W}_{i+1}\left(K+sL\right)}\\ =\frac{1}{2}\left(g\left(t\right)+g\left(s\right)\right).\end{array}$
(3.1)

Hence the function $g\left(t\right)$ is a convex function on $\left[0,+\mathrm{\infty }\right)$ 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

${f}_{i}\left(t\right)={W}_{i}\left(K+tB\right),$

then from (2.3)

$\begin{array}{rl}{f}_{i}\left(t+\epsilon \right)& ={W}_{i}\left(\left(K+tB\right)+\epsilon B\right)\\ =\sum _{j=0}^{n-i}\left(\genfrac{}{}{0}{}{n-i}{j}\right){\epsilon }^{j}{W}_{i+j}\left(K+tB\right)\\ ={f}_{i}\left(t\right)+\epsilon \left(n-i\right){f}_{i+1}\left(t\right)+O\left({\epsilon }^{2}\right).\end{array}$

Therefore

${f}_{i}^{\prime }\left(t\right)=\left(n-i\right){f}_{i+1}\left(t\right).$

The derivative of the function

${g}_{i}\left(t\right)=\frac{{f}_{i}\left(t\right)}{{f}_{i+1}\left(t\right)}=\frac{{W}_{i}\left(K+tB\right)}{{W}_{i+1}\left(K+tB\right)}$

is thus given by

${g}_{i}^{\prime }\left(t\right)=\left(n-i\right)-\left(n-i-1\right)\frac{{f}_{i}\left(t\right){f}_{i+2}\left(t\right)}{{f}_{i+1}^{2}\left(t\right)}.$
(3.2)

Since ${g}_{i}\left(x\right)$ 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 $t\in \left(0,+\mathrm{\infty }\right)$

$\left(n-i\right){f}_{i+2}\left(t\right){f}_{i+1}^{2}\left(t\right)+\left(n-i-2\right){f}_{i}\left(t\right){f}_{i+1}\left(t\right){f}_{i+3}\left(t\right)-2\left(n-i-1\right){f}_{i}\left(t\right){f}_{i+2}^{2}\left(t\right)\ge 0$

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

This can be equivalently written in the form

$\left(n-i\right){f}_{i+2}\left(t\right)\left({f}_{i+1}^{2}\left(t\right)-{f}_{i}\left(t\right){f}_{i+2}\left(t\right)\right)\ge \left(n-i-2\right){f}_{i}\left(t\right)\left({f}_{i+2}^{2}\left(t\right)-{f}_{i+1}\left(t\right){f}_{i+3}\left(t\right)\right)$

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

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

## References

1. Marcus M, Lopes I: Inequalities for symmetric functions and Hermitian matrices. Can. J. Math. 1956, 8: 524–531. 10.4153/CJM-1956-059-0

2. Bechenbach EF, Bellman R: Inequalities. Springer, Berlin; 1961.

3. Bergstrom H: A triangle inequality for matrices. In Den Elfte Skandinaviski Matematiker-kongress. John Grundt Tanums Forlag, Oslo; 1952.

4. Bellman R: Notes on matrix theory - IV: an inequality due to Bergstrom. Am. Math. Mon. 1955, 62: 172–173. 10.2307/2306621

5. Fan K: Some inequalities concerning positive-definite Hermitian matrices. Proc. Camb. Philos. Soc. 1955, 51: 414–421. 10.1017/S0305004100030413

6. Fan K: Problem 4786. Am. Math. Mon. 1958, 65: 289. 10.2307/2310261

7. Mirsky L: Maximum principles in matrix theory. Proc. Glasg. Math. Assoc. 1958, 4: 34–37. 10.1017/S2040618500033827

8. Fradelizi M, Giannopoulos A, Meyer M: Some inequalities about mixed volumes. Isr. J. Math. 2003, 135: 157–179. 10.1007/BF02776055

9. Giannopoulos A, Hartzoulaki M, Paouris G: On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body. Proc. Am. Math. Soc. 2002, 130: 2403–2412. 10.1090/S0002-9939-02-06329-3

10. Dembo A, Cover TM, Thomas JA: Information theoretic inequalities. IEEE Trans. Inf. Theory 1991, 37: 1501–1518. 10.1109/18.104312

11. Schneider R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge; 1993.

12. Burago YD, Zalgaller VA: Geometric Inequalities. Springer, Berlin; 1988.

## 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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Correspondence to Chang Jian Zhao.

### 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.

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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