Abstract
The wellknown 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; quermassintegral1 Introduction
The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write , , for the ith elementary symmetric function of an ntuple of positive real numbers. This is defined by and
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 and denote the submatrices obtained by deleting their ith row and column, then
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 and if , then
with equality if and only if K and L are homothetic, where is the ith 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 , , 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 [10] as an open question. In the same paper it was also mentioned that (1.6) follows directly from the AleksandrovFenchel inequality when and L is a ball.
In 2002, it was proved in [9] 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 [8] that (1.6) holds true for every pair of convex bodies K and L in if and only if or .
is true for every pair of convex bodiesKandLinif and only ifor.
In this paper, following the above results, we prove the following interest results.
Theorem 1.1Letand for every convex bodyKandLin. Then the function
is a convex function onif and only ifor.
Theorem 1.2Letand for every convex bodyKandLin. Then
2 Notations and preliminaries
The setting for this paper is an ndimensional Euclidean space . Let denote the set of convex bodies (compact, convex subsets with nonempty 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 nball is denoted by .
We use for the ndimensional 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 supnorm on the space of continuous functions .
Associated with a compact subset K of , which is starshaped 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 .
If () and () are nonnegative real numbers, then of fundamental importance is the fact that the volume of is a homogeneous polynomial in the given by (see, e.g., [11] or [12])
where the sum is taken over all ntuples () 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 [12], 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
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)
Therefore
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
This can be equivalently written in the form
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.
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