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On the quermassintegrals of convex bodies

Journal of Inequalities and Applications20132013:264

https://doi.org/10.1186/1029-242X-2013-264

Received: 18 March 2013

Accepted: 9 May 2013

Published: 27 May 2013

Abstract

The well-known question for quermassintegrals is the following: For which values of i N and every pair of convex bodies K and L, is it true that

W i ( K + L ) W i + 1 ( K + L ) W i ( K ) W ˜ i + 1 ( K ) + W i ( L ) W i + 1 ( L ) ?

In 2003, the inequality was proved if and only if i = n 1 or i = n 2 . Following the problem, in the paper, we prove some interrelated results for the quermassintegrals of a convex body.

MSC:26D15, 52A30.

Keywords

symmetric functionconvex bodyquermassintegral

1 Introduction

The origin of this work is an interesting inequality of Marcus and Lopes [1]. We write E i ( x ) , 0 i n , for the i th elementary symmetric function of an n-tuple x = ( x 1 , , x n ) of positive real numbers. This is defined by E 0 ( x ) = 1 and
E i ( x ) = 1 j 1 < < j i n x j 1 x j 2 x j i , 1 i n .

In particular, E 1 ( x ) = x 1 + + x n , E 2 ( x ) = i j x i x j , , E n ( x ) = x 1 x 2 x n .

The Marcus-Lopes inequality (see also [2], p.33]) states that
E i ( x + y ) E i 1 ( x + y ) E i ( x ) E i 1 ( x ) + E i ( y ) E i 1 ( y )
(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
[ E i ( x + y ) ] 1 / i [ E i ( x ) ] 1 / i + [ E i ( y ) ] 1 / i .
(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 K i and L i denote the submatrices obtained by deleting their i th row and column, then
det ( K + L ) det ( K i + L i ) det ( K ) det ( K i ) + det ( L ) det ( L i ) .
(1.3)
The following generalization of (1.3) was established by Ky Fan [5]:
( det ( K + L ) det ( K i + L i ) ) 1 / k ( det ( K ) det ( K i ) ) 1 / k + ( det ( L ) det ( L i ) ) 1 / k .
(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 R n and if 0 i n 1 , then
W i ( K + L ) 1 / ( n i ) W i ( K ) 1 / ( n i ) + W i ( L ) 1 / ( n i ) ,
(1.5)

with equality if and only if K and L are homothetic, where W i ( K ) 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 0 i n 1 , i N , is it true that, for every pair of convex bodies K and L in R n , one has
W i ( K + L ) W i + 1 ( K + L ) W i ( K ) W i + 1 ( K ) + W i ( L ) W i + 1 ( L ) ?
(1.6)

In 1991, the special case i = 0 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 i = 0 and L is a ball.

In 2002, it was proved in [9] that (1.6) is true for all i = 1 , , n 1 in the case where L is a ball.

Theorem A If K is a convex body and B is a ball in R n , then for 0 i n 1 , i N ,
W i ( K + B ) W i + 1 ( K + B ) W i ( K ) W i + 1 ( K ) + W i ( B ) W i + 1 ( B ) .
(1.7)

In 2003, it was proved in [8] that (1.6) holds true for every pair of convex bodies K and L in R n if and only if i = n 2 or i = n 1 .

Theorem B Let 0 i n 1 , then
W i ( K + L ) W i + 1 ( K + L ) W i ( K ) W i + 1 ( K ) + W i ( L ) W i + 1 ( L )
(1.8)

is true for every pair of convex bodies K and L in R n if and only if i = n 1 or i = n 2 .

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

Theorem 1.1 Let 0 i n 1 and for every convex body K and L in R n . Then the function
g ( t ) = W i ( K + t L ) W i + 1 ( K + t L )
(1.9)

is a convex function on t [ 0 , + ) if and only if i = n 1 or i = n 2 .

Theorem 1.2 Let 0 i n 1 and for every convex body K and L in R n . Then
( n i ) W i + 2 ( K ) ( W i + 1 ( K ) 2 W i ( K ) W i + 2 ( K ) ) ( n i 2 ) W i ( K ) ( W i + 2 2 ( K ) W i + 1 ( K ) W i + 3 ( K ) )
(1.10)

if and only if i = n 1 or i = n 2 .

2 Notations and preliminaries

The setting for this paper is an n-dimensional Euclidean space R n . Let K n denote the set of convex bodies (compact, convex subsets with non-empty interiors) in R n . 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 S n 1 . The volume of the unit n-ball is denoted by ω n .

We use V ( K ) for the n-dimensional volume of a convex body K. Let h ( K , ) : S n 1 R denote the support function of K K n ; i.e., for u S n 1 ,
h ( K , u ) = Max { u x : x K } ,

where u x denotes the usual inner product u and x in R n .

Let δ denote the Hausdorff metric on K n , i.e., for K , L K n , δ ( K , L ) = | h K h L | , where | | denotes the sup-norm on the space of continuous functions C ( S n 1 ) .

Associated with a compact subset K of R n , which is star-shaped with respect to the origin, is its radial function ρ ( K , ) : S n 1 R , defined for u S n 1 by
ρ ( K , u ) = Max { λ 0 : λ u K } .

If ρ ( K , ) is positive and continuous, K will be called a star body. Let S n denote the set of star bodies in R n . Let δ ˜ denote the radial Hausdorff metric, as follows, if K , L S n , then δ ˜ ( K , L ) = | ρ K ρ L | .

If K i K n ( i = 1 , 2 , , r ) and λ i ( i = 1 , 2 , , r ) are nonnegative real numbers, then of fundamental importance is the fact that the volume of i = 1 r λ i K i is a homogeneous polynomial in the λ i given by (see, e.g., [11] or [12])
V ( λ 1 K 1 + + λ n K n ) = i 1 , , i n λ i 1 λ i n V i 1 , , i n ,
(2.1)

where the sum is taken over all n-tuples ( i 1 , , i n ) of positive integers not exceeding r. The coefficient V i 1 , , i n depends only on the bodies K i 1 , , K i n and is uniquely determined by (2.1). It is called the mixed volume of K i 1 , , K i n , and is written as V ( K i 1 , , K i n ) . Let K 1 = = K n i = K and K n i + 1 = = K n = L , then the mixed volume V ( K 1 , , K n ) is written as V i ( K , L ) . If K 1 = = K n i = K , K n i + 1 = = K n = B , then the mixed volume V i ( K , B ) is written as W i ( K ) 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])
V ( λ 1 K 1 + + λ s K s ) = p 1 + + p r = n 1 i 1 < < i r s n ! p 1 ! p r ! λ i 1 p 1 λ i r p r V ( K i 1 , , K i 1 p 1 , , K i r , , K i r p r ) .
(2.2)
Let s = 2 , λ 1 = 1 , K 1 = K , K 2 = B , we have
V ( K + λ B ) = i = 0 n ( i n ) λ i W i ( K ) ,

known as formula ‘Steiner decomposition’.

On the other hand, for convex bodies K and L, (2.2) can show the following special case:
W i ( K + λ L ) = j = 0 n i ( n i j ) λ j V ( K , , K n i j , B , , B i , L , , L j ) .
(2.3)

3 Proof of main results

Proof of Theorem 1.1 If s , t [ 0 , ) , from (1.8), if and only if i = n 1 or i = n 2 , we have
g ( t + s 2 ) = W i ( K + t + s 2 L ) W i + 1 ( K + t + s 2 L ) = W i ( K + t L 2 + K + s L 2 ) W i + 1 ( K + t L 2 + K + s L 2 ) W i ( K + t L 2 ) W i + 1 ( K + t L 2 ) + W i ( K + s L 2 ) W i + 1 ( K + s L 2 ) = 1 2 W i ( K + t L ) W i + 1 ( K + t L ) + 1 2 W i ( K + s L ) W i + 1 ( K + s L ) = 1 2 ( g ( t ) + g ( s ) ) .
(3.1)

Hence the function g ( t ) is a convex function on [ 0 , + ) for every star body K and L if and only if i = n 1 or i = n 2 . □

Proof of Theorem 1.2 Let K be a convex body in R n . For every i 0 , we set
f i ( t ) = W i ( K + t B ) ,
then from (2.3)
f i ( t + ε ) = W i ( ( K + t B ) + ε B ) = j = 0 n i ( n i j ) ε j W i + j ( K + t B ) = f i ( t ) + ε ( n i ) f i + 1 ( t ) + O ( ε 2 ) .
Therefore
f i ( t ) = ( n i ) f i + 1 ( t ) .
The derivative of the function
g i ( t ) = f i ( t ) f i + 1 ( t ) = W i ( K + t B ) W i + 1 ( K + t B )
is thus given by
g i ( t ) = ( n i ) ( n i 1 ) f i ( t ) f i + 2 ( t ) f i + 1 2 ( t ) .
(3.2)
Since g i ( x ) is a convex function if and only if i = n 1 or i = n 2 , hence by differentiating the both sides of (3.2), we obtain for t ( 0 , + )
( n i ) f i + 2 ( t ) f i + 1 2 ( t ) + ( n i 2 ) f i ( t ) f i + 1 ( t ) f i + 3 ( t ) 2 ( n i 1 ) f i ( t ) f i + 2 2 ( t ) 0

if and only if i = n 1 or i = n 2 .

This can be equivalently written in the form
( n i ) f i + 2 ( t ) ( f i + 1 2 ( t ) f i ( t ) f i + 2 ( t ) ) ( n i 2 ) f i ( t ) ( f i + 2 2 ( t ) f i + 1 ( t ) f i + 3 ( t ) )

if and only if i = n 1 or i = n 2 .

Letting t 0 + , we conclude Theorem 1.2. □

Declarations

Acknowledgements

First author is supported by the National Natural Science Foundation of China (10971205). Second author is partially supported by a HKU URG grant.

Authors’ Affiliations

(1)
Department of Mathematics, China Jiliang University, Hangzhou, P.R. China
(2)
Department of Mathematics, The University of Hong Kong, Hong Kong, P.R. China

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Copyright

© Zhao and Cheung; licensee Springer 2013

This article is published under license to BioMed Central Ltd. 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.

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