The well-known question for quermassintegrals is the following: For which values of and every pair of convex bodies K and L, is it true that
In 2003, the inequality was proved if and only if or . Following the problem, in the paper, we prove some interrelated results for the quermassintegrals of a convex body.
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
In particular, , .
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.
If K and L are convex bodies in and if , then
with equality if and only if K and L are homothetic, where is the ith quermassintegral of K (see Section 2).
Question For which values of , , is it true that, for every pair of convex bodies K and L in , one has
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.
Theorem AIfKis a convex body andBis a ball in , then for , ,
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 .
Theorem BLet , then
is true for every pair of convex bodiesKandLin if and only if or .
In this paper, following the above results, we prove the following interest results.
Theorem 1.1Let and for every convex bodyKandLin . Then the function
is a convex function on if and only if or .
Theorem 1.2Let and for every convex bodyKandLin . Then
if and only if or .
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 .
We use for the n-dimensional volume of a convex body K. Let denote the support function of ; i.e., for ,
where denotes the usual inner product u and x in .
Let δ denote the Hausdorff metric on , i.e., for , , where denotes the sup-norm on the space of continuous functions .
Associated with a compact subset K of , which is star-shaped with respect to the origin, is its radial function , defined for by
If is positive and continuous, K will be called a star body. Let denote the set of star bodies in . Let denote the radial Hausdorff metric, as follows, if , then .
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])
Let , , , , we have
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 , from (1.8), if and only if or , we have
Hence the function is a convex function on for every star body K and L if and only if or . □
Proof of Theorem 1.2 Let K be a convex body in . For every , we set
then from (2.3)
The derivative of the function
is thus given by
Since is a convex function if and only if or , hence by differentiating the both sides of (3.2), we obtain for
if and only if or .
This can be equivalently written in the form
if and only if or .
Letting , we conclude Theorem 1.2. □
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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