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On the quermassintegrals of convex bodies
Journal of Inequalities and Applications volume 2013, Article number: 264 (2013)
Abstract
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.
1 Introduction
The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write , , for the i th elementary symmetric function of an n-tuple of positive real numbers. This is defined by and
In particular, , .
The Marcus-Lopes inequality (see also [2], 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 [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 i th 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 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 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 , , 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 Aleksandrov-Fenchel 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 A If K is a convex body and B is 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 .
Theorem B Let , then
is 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.
Theorem 1.1 Let and for every convex body K and L in . Then the function
is a convex function on if and only if or .
Theorem 1.2 Let and for every convex body K and L in . 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 .
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 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 [12], 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)
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
if and only if or .
This can be equivalently written in the form
if and only if or .
Letting , we conclude Theorem 1.2. □
References
Marcus M, Lopes I: Inequalities for symmetric functions and Hermitian matrices. Can. J. Math. 1956, 8: 524–531. 10.4153/CJM-1956-059-0
Bechenbach EF, Bellman R: Inequalities. Springer, Berlin; 1961.
Bergstrom H: A triangle inequality for matrices. In Den Elfte Skandinaviski Matematiker-kongress. John Grundt Tanums Forlag, Oslo; 1952.
Bellman R: Notes on matrix theory - IV: an inequality due to Bergstrom. Am. Math. Mon. 1955, 62: 172–173. 10.2307/2306621
Fan K: Some inequalities concerning positive-definite Hermitian matrices. Proc. Camb. Philos. Soc. 1955, 51: 414–421. 10.1017/S0305004100030413
Fan K: Problem 4786. Am. Math. Mon. 1958, 65: 289. 10.2307/2310261
Mirsky L: Maximum principles in matrix theory. Proc. Glasg. Math. Assoc. 1958, 4: 34–37. 10.1017/S2040618500033827
Fradelizi M, Giannopoulos A, Meyer M: Some inequalities about mixed volumes. Isr. J. Math. 2003, 135: 157–179. 10.1007/BF02776055
Giannopoulos A, Hartzoulaki M, Paouris G: On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body. Proc. Am. Math. Soc. 2002, 130: 2403–2412. 10.1090/S0002-9939-02-06329-3
Dembo A, Cover TM, Thomas JA: Information theoretic inequalities. IEEE Trans. Inf. Theory 1991, 37: 1501–1518. 10.1109/18.104312
Schneider R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge; 1993.
Burago YD, Zalgaller VA: Geometric Inequalities. Springer, Berlin; 1988.
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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The authors declare that they have no competing interests.
Authors’ contributions
CJZ and WSC jointly contributed to the main results Theorems 1.1-1.2. All authors read and approved the final manuscript.
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Zhao, C.J., Cheung, W.S. On the quermassintegrals of convex bodies. J Inequal Appl 2013, 264 (2013). https://doi.org/10.1186/1029-242X-2013-264
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DOI: https://doi.org/10.1186/1029-242X-2013-264