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

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

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

The Marcus-Lopes inequality (see also [

2], p.33]) states that

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

(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

${[{E}_{i}(x+y)]}^{1/i}\ge {[{E}_{i}(x)]}^{1/i}+{[{E}_{i}(y)]}^{1/i}.$

(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

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

(1.3)

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

5]:

${\left(\frac{det(K+L)}{det({K}_{i}+{L}_{i})}\right)}^{1/k}\ge {\left(\frac{det(K)}{det({K}_{i})}\right)}^{1/k}+{\left(\frac{det(L)}{det({L}_{i})}\right)}^{1/k}.$

(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

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

(1.5)

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

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

(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* $0\le i\le n-1$,

$i\in \mathbb{N}$,

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

(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* $\frac{{W}_{i}(K+L)}{{W}_{i+1}(K+L)}\ge \frac{{W}_{i}(K)}{{W}_{i+1}(K)}+\frac{{W}_{i}(L)}{{W}_{i+1}(L)}$

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

(1.9)

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

(1.10)

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