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On the quermassintegrals of convex bodies

Abstract

The well-known question for quermassintegrals is the following: For which values of iN 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=n1 or i=n2. 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 E i (x), 0in, 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 ,1in.

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 0in1, 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 0in1, iN, 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,,n1 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 0in1, iN,

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=n2 or i=n1.

Theorem B Let 0in1, 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=n1 or i=n2.

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

Theorem 1.1 Let 0in1 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=n1 or i=n2.

Theorem 1.2 Let 0in1 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=n1 or i=n2.

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{ux:xK},

where ux 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{λ0:λuK}.

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+λB)= 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+λL)= 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=n1 or i=n2, 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=n1 or i=n2. □

Proof of Theorem 1.2 Let K be a convex body in R n . For every i0, we set

f i (t)= W i (K+tB),

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)=(ni) 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)=(ni)(ni1) 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=n1 or i=n2, hence by differentiating the both sides of (3.2), we obtain for t(0,+)

(ni) f i + 2 (t) f i + 1 2 (t)+(ni2) f i (t) f i + 1 (t) f i + 3 (t)2(ni1) f i (t) f i + 2 2 (t)0

if and only if i=n1 or i=n2.

This can be equivalently written in the form

(ni) f i + 2 (t) ( f i + 1 2 ( t ) f i ( t ) f i + 2 ( t ) ) (ni2) f i (t) ( f i + 2 2 ( t ) f i + 1 ( t ) f i + 3 ( t ) )

if and only if i=n1 or i=n2.

Letting t 0 + , we conclude Theorem 1.2. □

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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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Correspondence to Chang Jian Zhao.

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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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Keywords

  • symmetric function
  • convex body
  • quermassintegral