Abstract
The wellknown question for quermassintegrals is the following: For which values of
In 2003, the inequality was proved if and only if
MSC: 26D15, 52A30.
Keywords:
symmetric function; convex body; quermassintegral1 Introduction
The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write
In particular,
The MarcusLopes inequality (see also [2], p.33]) states that
for every pair of positive ntuples 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
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 BrunnMinkowski theory is as follows.
If K and L are convex bodies in
with equality if and only if K and L are homothetic, where
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
In 1991, the special case
In 2002, it was proved in [9] that (1.6) is true for all
Theorem AIfKis a convex body andBis a ball in
In 2003, it was proved in [8] that (1.6) holds true for every pair of convex bodies K and L in
Theorem BLet
is true for every pair of convex bodiesKandLin
In this paper, following the above results, we prove the following interest results.
Theorem 1.1Let
is a convex function on
Theorem 1.2Let
if and only if
2 Notations and preliminaries
The setting for this paper is an ndimensional Euclidean space
We use
where
Let δ denote the Hausdorff metric on
Associated with a compact subset K of
If
If
where the sum is taken over all ntuples (
It is convenient to write relation (2.1) in the form (see [12], p.137])
Let
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
Hence the function
Proof of Theorem 1.2 Let K be a convex body in
then from (2.3)
Therefore
The derivative of the function
is thus given by
Since
if and only if
This can be equivalently written in the form
if and only if
Letting
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
CJZ and WSC jointly contributed to the main results Theorems 1.11.2. All authors read and approved the final manuscript.
Acknowledgements
First author is supported by the National Natural Science Foundation of China (10971205). Second author is partially supported by a HKU URG grant.
References

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