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On the quermassintegrals of convex bodies

Chang Jian Zhao1* and Wing Sum Cheung2

Author Affiliations

1 Department of Mathematics, China Jiliang University, Hangzhou, 310018, P.R. China

2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China

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Journal of Inequalities and Applications 2013, 2013:264  doi:10.1186/1029-242X-2013-264

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/264


Received:18 March 2013
Accepted:9 May 2013
Published:27 May 2013

© 2013 Zhao and Cheung; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The well-known question for quermassintegrals is the following: For which values of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M1">View MathML</a> and every pair of convex bodies K and L, is it true that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M2">View MathML</a>

In 2003, the inequality was proved if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>. 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

1 Introduction

The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M5">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M6">View MathML</a>, for the ith elementary symmetric function of an n-tuple <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M7">View MathML</a> of positive real numbers. This is defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M8">View MathML</a> and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M9">View MathML</a>

In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M10">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M11">View MathML</a>.

The Marcus-Lopes inequality (see also [2], p.33]) states that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M12">View MathML</a>

(1.1)

for every pair of positive n-tuples x and y. This is a refinement of a further result concerning the symmetric functions, namely

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M13">View MathML</a>

(1.2)

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M14">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M15">View MathML</a> denote the submatrices obtained by deleting their ith row and column, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M16">View MathML</a>

(1.3)

The following generalization of (1.3) was established by Ky Fan [5]:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M17">View MathML</a>

(1.4)

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M18">View MathML</a> and if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M20">View MathML</a>

(1.5)

with equality if and only if K and L are homothetic, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M21">View MathML</a> 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M1">View MathML</a>, is it true that, for every pair of convex bodies K and L in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M18">View MathML</a>, one has

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M25">View MathML</a>

(1.6)

In 1991, the special case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M26">View MathML</a> 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M26">View MathML</a> and L is a ball.

In 2002, it was proved in [9] that (1.6) is true for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M28">View MathML</a> in the case where L is a ball.

Theorem AIfKis a convex body andBis a ball in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>, then for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M1">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M32">View MathML</a>

(1.7)

In 2003, it was proved in [8] that (1.6) holds true for every pair of convex bodies K and L in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M18">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a>.

Theorem BLet<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M37">View MathML</a>

(1.8)

is true for every pair of convex bodiesKandLin<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a>or<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>.

In this paper, following the above results, we prove the following interest results.

Theorem 1.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19">View MathML</a>and for every convex bodyKandLin<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>. Then the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M43">View MathML</a>

(1.9)

is a convex function on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M44">View MathML</a>if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a>or<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>.

Theorem 1.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M19">View MathML</a>and for every convex bodyKandLin<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M49">View MathML</a>

(1.10)

if and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a>or<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>.

2 Notations and preliminaries

The setting for this paper is an n-dimensional Euclidean space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M53">View MathML</a> denote the set of convex bodies (compact, convex subsets with non-empty interiors) in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>. 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M55">View MathML</a>. The volume of the unit n-ball is denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M56">View MathML</a>.

We use <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M57">View MathML</a> for the n-dimensional volume of a convex body K. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M58">View MathML</a> denote the support function of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M59">View MathML</a>; i.e., for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M60">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M61">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M62">View MathML</a> denotes the usual inner product u and x in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>.

Let δ denote the Hausdorff metric on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M53">View MathML</a>, i.e., for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M65">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M66">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M67">View MathML</a> denotes the sup-norm on the space of continuous functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M68">View MathML</a>.

Associated with a compact subset K of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>, which is star-shaped with respect to the origin, is its radial function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M70">View MathML</a>, defined for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M60">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M72">View MathML</a>

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M73">View MathML</a> is positive and continuous, K will be called a star body. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M74">View MathML</a> denote the set of star bodies in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M76">View MathML</a> denote the radial Hausdorff metric, as follows, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M77">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M78">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M79">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M80">View MathML</a>) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M81">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M80">View MathML</a>) are nonnegative real numbers, then of fundamental importance is the fact that the volume of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M83">View MathML</a> is a homogeneous polynomial in the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M81">View MathML</a> given by (see, e.g., [11] or [12])

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M85">View MathML</a>

(2.1)

where the sum is taken over all n-tuples (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M86">View MathML</a>) of positive integers not exceeding r. The coefficient <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M87">View MathML</a> depends only on the bodies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M88">View MathML</a> and is uniquely determined by (2.1). It is called the mixed volume of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M88">View MathML</a>, and is written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M90">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M91">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M92">View MathML</a>, then the mixed volume <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M93">View MathML</a> is written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M94">View MathML</a>. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M91">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M96">View MathML</a>, then the mixed volume <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M97">View MathML</a> is written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M21">View MathML</a> 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])

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M99">View MathML</a>

(2.2)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M100">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M101">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M102">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M103">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M104">View MathML</a>

known as formula ‘Steiner decomposition’.

On the other hand, for convex bodies K and L, (2.2) can show the following special case:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M105">View MathML</a>

(2.3)

3 Proof of main results

Proof of Theorem 1.1 If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M106">View MathML</a>, from (1.8), if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M109">View MathML</a>

(3.1)

Hence the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M110">View MathML</a> is a convex function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M111">View MathML</a> for every star body K and L if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>. □

Proof of Theorem 1.2 Let K be a convex body in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M29">View MathML</a>. For every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M115">View MathML</a>, we set

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M116">View MathML</a>

then from (2.3)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M117">View MathML</a>

Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M118">View MathML</a>

The derivative of the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M119">View MathML</a>

is thus given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M120">View MathML</a>

(3.2)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M121">View MathML</a> is a convex function if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>, hence by differentiating the both sides of (3.2), we obtain for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M124">View MathML</a>

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M125">View MathML</a>

if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>.

This can be equivalently written in the form

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M128">View MathML</a>

if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M3">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M4">View MathML</a>.

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/264/mathml/M131">View MathML</a>, we conclude Theorem 1.2. □

Competing interests

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.

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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