MSC: 26D15, 52A30.
Keywords:symmetric function; convex body; quermassintegral
The origin of this work is an interesting inequality of Marcus and Lopes . We write , , for the ith elementary symmetric function of an n-tuple of positive real numbers. This is defined by and
The Marcus-Lopes inequality (see also , 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 .
A matrix analogue of (1.1) is the following result of Bergstrom  (see also the article  and , 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
The following generalization of (1.3) was established by Ky Fan :
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.
In 1991, the special case was stated also in  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  that (1.6) is true for all in the case where L is a ball.
In 2003, it was proved in  that (1.6) holds true for every pair of convex bodies K and L in if and only if or .
In this paper, following the above results, we prove the following interest results.
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 .
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 , p.137])
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
then from (2.3)
The derivative of the function
is thus given by
This can be equivalently written in the form
The authors declare that they have no competing interests.
CJZ and WSC jointly contributed to the main results Theorems 1.1-1.2. All authors read and approved the final manuscript.
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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