Open Access

Some Bonnesen-style Minkowski inequalities

Journal of Inequalities and Applications20142014:270

https://doi.org/10.1186/1029-242X-2014-270

Received: 29 April 2014

Accepted: 26 June 2014

Published: 23 July 2014

Abstract

In this paper, we obtain some Bonnesen-style Minkowski inequalities of mixed volumes of convex bodies K and L in the Euclidean space R n . Let L be the unit ball; we get some better Bonnesen-style isoperimetric inequalities than Dinghas’s result for  n 3 .

MSC:52A20, 52A40.

Keywords

Minkowski inequalityAleksandrov-Fenchel inequalityBonnesen-style inequalitymixed volumesisoperimetric inequality

1 Introduction

It is well known that the ball has the maximum volume among bodies of fixed surface area in the Euclidean space R n . That is, of all domains K with surface area S ( K ) and volume V ( K ) (cf. [1, 2]),
S ( K ) n n n ω n V ( K ) n 1 0 ,
(1)
with equality if and only if K is a ball. Here ω n denotes the volume of the unit ball,
ω n = 2 π n / 2 n Γ ( n / 2 ) ,

where Γ ( ) is the Gamma function.

The isoperimetric deficit
Δ n ( K ) = S ( K ) n n n ω n V ( K ) n 1
(2)
measures the deficit between the domain K and a ball of radius ( S ( K ) / n ω n ) 1 / ( n 1 ) . A Bonnesen-style isoperimetric inequality is of the form (cf. [24])
Δ n ( K ) = S ( K ) n n n ω n V ( K ) n 1 B K ,
(3)

where the quantity B K is a non-negative invariant of geometric significance of K and vanishes only when K is a ball.

Bonnesen himself proved several inequalities of the form (3) in the Euclidean plane (cf. [5, 6]), but he was not able to obtain direct generalizations of his two-dimensional results. This was done much later, first by Hadwiger [7] for n = 3 , and then by Dinghas [8] for arbitrary dimension. From then on, some Bonnesen-style inequalities in the higher dimensions and generalizations have been obtained by Osserman (cf. [1, 2]), Santaló (cf. [9]), Groemer and Schneider (cf. [10]), Zhang (cf. [11]), Zhou (cf. [4, 12]) and others. See references [1336] for more details. The following well-known Bonnesen-style inequality for a convex body K in the Euclidean space R n is due to Dinghas (cf. [8]):
S ( K ) n n n ω n V ( K ) n 1 ( S ( K ) 1 / ( n 1 ) ( n ω n ) 1 / ( n 1 ) r ) n ( n 1 ) ,
(4)

where r is the in-radius of K, and equality holds if and only if K is a ball.

In [11], some different forms of Bonnesen-style isoperimetric inequalities have been established associated with the mean width of K. Zhang obtained (cf. [11])
( M ( K ) 2 ) n / ( n 1 ) ( V ( K ) ω n ) 1 / ( n 1 ) ( V ( K ) ω n ) n / ( n 1 ) ( ( V ( K ) ω n ) 1 / n R 1 ) ,

where M ( K ) and R are the mean width and out-radius of K, respectively.

The Minkowski inequality of mixed volume is a natural generalization of the isoperimetric inequality (1) in the Euclidean space R n (cf. [27, 3739]). Let K, L be convex bodies in R n , then
V 1 ( K , L ) n V ( K ) n 1 V ( L ) ,
(5)

where V 1 ( K , L ) is the mixed volume of K and L and the equality holds if and only if K and L are homothetic.

Motivated by (2), we define the Minkowski homothetic deficit as
Δ n ( K , L ) = V 1 ( K , L ) n V ( K ) n 1 V ( L ) .
(6)
The Minkowski homothetic deficit Δ n ( K , L ) measures the homothety between K and L. Then a Bonnesen-style Minkowski inequality would be of the form
Δ n ( K , L ) = V 1 ( K , L ) n V ( K ) n 1 V ( L ) B K , L ,
(7)
where the quantity B K , L is an invariant of geometric significance about K and L with the following basic properties:
  1. 1.

    B K , L is non-negative;

     
  2. 2.

    B K , L vanishes only when K and L are homothetic.

     

Note that let L be the unit ball B and by S ( K ) = n V 1 ( K , B ) , the surface area of K, then the Minkowski homothetic deficit is just the isoperimetric deficit. Therefore, the Bonnesen-style Minkowski inequality (7) is more general than the Bonnesen-style isoperimetric inequality (3).

In this paper, we focus on Bonnesen-style Minkowski inequalities of type (7). Some B K , L are obtained. Let L be the unit ball; then we obtain stronger Bonnesen-style isoperimetric inequalities K than (4).

2 Preliminaries

A set of points K in the Euclidean space R n is convex if for all x , y K and 0 λ 1 , λ x + ( 1 λ ) y K . A domain is a set with nonempty interiors. A convex body is a compact convex domain. The set of convex bodies in R n is denoted by K n . Let K o n be the class of members of K n containing the origin in their interiors. Write V for an n-dimensional Lebesgue measure and H n 1 for an ( n 1 ) -dimensional Hausdorff measure. S n 1 denotes the surface of the unit ball in R n .

A convex body K R n is uniquely determined by its support function h K : R n R , where h K ( x ) = max { x y : y K } , for x R n . For the support function of the dilate c K = { c x : x K } of a convex body K we have
h c K = c h K , c > 0 .
(8)
Note that support functions are positively homogeneous of degree one and subadditive. It follows immediately from the definition of support functions that for convex bodies K and L
K L h K h L .
(9)
For a convex body K and each Borel set ω S n 1 , the reverse spherical image τ ( K , ω ) , of K at ω is the set of all boundary points of K which have an outer unit normal belonging to the set ω. Associated with each convex body K K o n there is a Borel measure S K on S n 1 called the Aleksandrov-Fenchel surface area measure of K, defined by
S K ( ω ) = H n 1 ( τ ( K , ω ) ) ,
for each Borel set ω S n 1 . Observe that for the surface area measure of the dilate cK of K we have
S c K = c n 1 S K , c > 0 .
The Minkowski sum of convex sets K 1 , , K m in R n is defined by
K 1 + + K m = { x 1 + + x m : x 1 K 1 , , x m K m } .
The mixed volume V ( K 1 , , K n ) of compact convex sets K 1 , , K n in R n is defined by
V ( K 1 , , K n ) = 1 n ! j = 1 n ( 1 ) n + j i 1 < < i k V ( K i 1 + + K i k ) .
The Aleksandrov-Fenchel inequality about the i th mixed volume is
V i ( K 1 , K 2 ) 2 V i + 1 ( K 1 , K 2 ) V i 1 ( K 1 , K 2 ) ,
(10)
where
V i ( K 1 , K 2 ) = V ( K 1 , , K 1 n i , K 2 , , K 2 i )

with K 1 appears n i times and K 2 appears i times and (10) holds as an equality if and only if K and L are homothetic.

Note that
V n ( K 1 , K 2 ) = V ( K 2 ) , V 0 ( K 1 , K 2 ) = V ( K 1 ) .
(11)
The following inequality for mixed volumes is the general Aleksandrov-Fenchel inequality: Let K 1 , , K n K and 1 m n . Then
V ( K 1 , , K n ) m i = 1 m V ( K i , , K i , K m + 1 , , K n ) .
Hence
V 1 ( K 1 , K 2 ) n 1 V ( K 1 ) n 2 V n 1 ( K 1 , K 2 ) .
(12)

Let K 2 = B , then V i ( K 1 , B ) = W i ( K 1 ) , the i th quermassintegral of the convex body K 1 .

The mixed volume has monotonicity: If K 1 K 1 , then
V ( K 1 , K 2 , , K n ) V ( K 1 , K 2 , , K n ) .
The mixed volume V 1 ( K , L ) of the convex bodies K , L K o n has the integral form
V 1 ( K , L ) = 1 n S n 1 h L d S K .
(13)
Since
V ( K ) = V 1 ( K , K ) ,
we have
V ( K ) = 1 n S n 1 h K d S K .
If B is the unit ball, then
n V 1 ( K , B ) = S ( K ) ,
the surface area of K. The mean width M ( K ) of K is
M ( K ) = 2 ω n V 1 ( B , K ) ,
that is,
M ( K ) = 2 n ω n S n 1 h K d S K .
The in-radius r ( K , L ) , out-radius R ( K , L ) of K with respect to L are, respectively, defined by
r ( K , L ) = sup { λ : x R n  and  x + λ L K } , R ( K , L ) = inf { λ : x R n  and  K x + λ L } .
Notice that always
r ( K , L ) R ( L , K ) = 1 .

When L is the unit ball, r ( K , L ) and R ( K , L ) are the radius of maximal inscribed and minimal circumscribed balls of K, respectively.

3 Bonnesen-style Minkowski inequalities associated with r ( K , L )

In this section, we derive some Bonnesen-style Minkowski inequalities associated with in-radius r ( K , L ) of K with respect to L. In [26], Diskant improved the Minkowski inequality of mixed volumes as follows.

Lemma 1 Let K, L be convex bodies in the Euclidean space R n , then
V 1 ( K , L ) n / ( n 1 ) V ( K ) V ( L ) 1 / ( n 1 ) ( V 1 ( K , L ) 1 / ( n 1 ) r ( K , L ) V ( L ) 1 / ( n 1 ) ) n ,
(14)

with equality if and only if K is homothetic to L.

Note that the right-hand side of (14) is non-negative for x + r ( K , L ) L K ( x R n ). By (13) we have
V 1 ( K , L ) = 1 n S n 1 h L d S K 1 n S n 1 h L d S r ( K , L ) L r ( K , L ) n 1 V ( L ) .

From Lemma 1 and using the inequality x n 1 y n 1 ( x y ) n 1 (for x y 0 ), a lower bound of the Minkowski deficit follows (cf. [26, 27]).

Proposition 1 Let K, L be convex bodies in the Euclidean space R n , then
Δ n ( K , L ) ( V 1 ( K , L ) 1 / ( n 1 ) r ( K , L ) V ( L ) 1 / ( n 1 ) ) n ( n 1 ) ,
(15)

where the inequality holds as an equality if and only if K and L are homothetic.

The following Bonnesen-style Minkowski inequality is stronger than (15) for n = 3 .

Theorem 1 Let K, L be convex bodies in the Euclidean space R 3 , then
Δ 3 ( K , L ) ( V 1 ( K , L ) 1 / 2 r ( K , L ) V ( L ) 1 / 2 ) 6 + 2 r ( K , L ) 3 V ( L ) 3 / 2 ( V 1 ( K , L ) 1 / 2 r ( K , L ) V ( L ) 1 / 2 ) 3 ,
(16)

with equality if and only if K is homothetic to L.

Proof Since V ( K ) r ( K , L ) 3 V ( L ) and by x 3 y 3 ( x y ) 3 (for x y 0 ), we have
V 1 ( K , L ) 3 / 2 + V ( K ) V ( L ) 1 / 2 V 1 ( K , L ) 3 / 2 + r ( K , L ) 3 V ( L ) 3 / 2 = V 1 ( K , L ) 3 / 2 ( r ( K , L ) V ( L ) 1 / 2 ) 3 + 2 r ( K , L ) 3 V ( L ) 3 / 2 ( V 1 ( K , L ) 1 / 2 r ( K , L ) V ( L ) 1 / 2 ) 3 + 2 r ( K , L ) 3 V ( L ) 3 / 2 .
Note that (14) can be rewritten as
V 1 ( K , L ) 3 / 2 V ( K ) V ( L ) 1 / 2 ( V 1 ( K , L ) 1 / 2 r ( K , L ) V ( L ) 1 / 2 ) 3 .
Multiplying by V 1 ( K , L ) 3 / 2 + V ( K ) V ( L ) 1 / 2 on both sides, we have
V 1 ( K , L ) 3 V ( K ) 2 V ( L ) ( V 1 ( K , L ) 1 / 2 r ( K , L ) V ( L ) 1 / 2 ) 3 ( V 1 ( K , L ) 3 / 2 + V ( K ) V ( L ) 1 / 2 ) .

By these inequalities, we complete the proof of the theorem. □

Let L be the unit ball and notice S ( K ) = 3 V 1 ( K , B ) in (16), we obtain the following Bonnesen-style isoperimetric inequality that strengthens Dinghas’s inequality (4) for n = 3 .

Corollary 1 Let K be a convex body in R 3 and r be the in-radius of K, then
Δ 3 ( K ) ( S ( K ) 1 / 2 ( 4 π ) 1 / 2 r ) 6 + 16 π 3 / 2 r 3 ( S ( K ) 1 / 2 ( 4 π ) 1 / 2 r ) 3 ,
(17)

with equality if and only if K is a ball.

For n 4 , we obtain a stronger Bonnesen-style Minkowski inequality as follows.

Theorem 2 Let K, L be convex bodies in the Euclidean space R n ( n 4 ), then
Δ n ( K , L ) ( V 1 ( K , L ) 1 / ( n 1 ) r ( K , L ) V ( L ) 1 / ( n 1 ) ) n ( n 1 ) + 2 ( r ( K , L ) V ( L ) 1 / ( n 1 ) ) n ( n 2 ) ( V 1 ( K , L ) 1 / ( n 1 ) r ( K , L ) V ( L ) 1 / ( n 1 ) ) n + ( V 1 ( K , L ) V ( L ) ) n / ( n 1 ) r ( K , L ) n ( V 1 ( K , L ) 1 / ( n 1 ) r ( K , L ) V ( L ) 1 / ( n 1 ) ) n ( n 3 ) ,

with equality if and only if K is homothetic to L.

Proof Let p = V 1 ( K , L ) n / ( n 1 ) and q = V ( L ) n / ( n 1 ) r ( K , L ) n , then p q .
i = 2 n ( V 1 ( K , L ) n ( n i ) / ( n 1 ) ( V ( K ) n 1 V ( L ) ) ( i 2 ) / ( n 1 ) ) i = 2 n ( V 1 ( K , L ) n ( n i ) / ( n 1 ) ( V ( L ) 1 / ( n 1 ) r ( K , L ) ) n ( i 2 ) ) = ( p n 2 q n 2 ) + p q ( p n 4 + p n 5 q + + p n i q i 4 + + q n 4 ) + 2 q n 2 = ( p n 2 q n 2 ) + p q p n 3 q n 3 p q + 2 q n 2 ( p q ) n 2 + p q ( p q ) n 4 + 2 q n 2 ( p 1 / n q 1 / n ) n ( n 2 ) + p q ( p 1 / n q 1 / n ) n ( n 4 ) + 2 q n 2 .
That is,
i = 2 n ( V 1 ( K , L ) n ( n i ) / ( n 1 ) ( V ( K ) n 1 V ( L ) ) ( i 2 ) / ( n 1 ) ) ( V 1 ( K , L ) 1 / ( n 1 ) V ( L ) 1 / ( n 1 ) r ( K , L ) ) n ( n 2 ) + 2 ( V ( L ) 1 / ( n 1 ) r ( K , L ) ) n ( n 2 ) + ( V 1 ( K , L ) V ( L ) ) n / ( n 1 ) r ( K , L ) n ( V 1 ( K , L ) 1 / ( n 1 ) V ( L ) 1 / ( n 1 ) r ( K , L ) ) n ( n 4 ) .
Multiplying by i = 2 n ( V 1 ( K , L ) n ( n i ) / n 1 ( V ( K ) n 1 V ( L ) ) ( i 2 ) / ( n 1 ) ) both sides of (14) and via the formula
a n 1 b n 1 = ( a b ) ( a n 2 + a n 3 b + + a n i b i 2 + + a b n 3 + b n 2 ) ,
we obtain
V 1 ( K , L ) n V ( K ) n 1 V ( L ) ( V 1 ( K , L ) 1 / ( n 1 ) r ( K , L ) V ( L ) 1 / ( n 1 ) ) n × ( i = 2 n ( V 1 ( K , L ) n ( n i ) / ( n 1 ) ( V ( K ) n 1 V ( L ) ) ( i 2 ) / ( n 1 ) ) ) .

We complete the proof of Theorem 2. □

Let L be the unit ball and by S ( K ) = n V 1 ( K , B ) in Theorem 2; we obtain the following stronger Bonnesen-style isoperimetric inequality than Dinghas’s inequality (4) for n 4 .

Corollary 2 Let K be a convex body in R n ( n 4 ) and r be the in-radius of K, then
Δ n ( K ) ( S ( K ) 1 / ( n 1 ) ( n ω n ) 1 / ( n 1 ) r ) n ( n 1 ) + 2 ( ( n ω n ) 1 / ( n 1 ) r ) n ( n 2 ) ( S ( K ) 1 / ( n 1 ) ( n ω n ) 1 / ( n 1 ) r ) n + ( n ω n S ( K ) ) n / ( n 1 ) r n ( S ( K ) 1 / ( n 1 ) ( n ω n ) 1 / ( n 1 ) r ) n ( n 3 ) ,

with equality if and only if K is a ball.

4 Bonnesen-style Minkowski inequalities associated with the mean width

In this section, we derive some Bonnesen-style Minkowski inequalities associated with the mean width.

Lemma 2 Let K, L be convex bodies in R n , then
V 0 ( K , L ) V 1 ( K , L ) ( V 0 ( K , L ) V n ( K , L ) ) 1 / n V n 1 ( K , L ) V n ( K , L ) V 1 ( K , L ) n 1 V ( K ) n 2 V ( L ) ,
(18)

with equality if and only if K and L are homothetic.

Proof By inequality (12), we have
V n 1 ( K , L ) V ( L ) V 1 ( K , L ) n 1 V ( K ) n 2 V ( L ) .
By the Aleksandrov-Fenchel inequality (10) we have
V 0 ( K , L ) V 1 ( K , L ) V 1 ( K , L ) V 2 ( K , L ) V i ( K , L ) V i + 1 ( K , L ) V n 1 ( K , L ) V n ( K , L ) .
Therefore
V 0 ( K , L ) V 1 ( K , L ) ( V 0 ( K , L ) V n ( K , L ) ) 1 / n V n 1 ( K , L ) V n ( K , L ) .

 □

Theorem 3 Let K, L be convex bodies in R n , then
Δ n ( K , L ) V ( K ) n 2 V ( L ) V 1 ( K , L ) ( V n 1 ( K , L ) V n ( K , L ) ( V 0 ( K , L ) V n ( K , L ) ) 1 / n ) ,
(19)

with equality if and only if K and L are homothetic.

Proof Via (18), we have
V 1 ( K , L ) n 1 V ( K ) n 2 V ( L ) V 0 ( K , L ) V 1 ( K , L ) V n 1 ( K , L ) V n ( K , L ) ( V 0 ( K , L ) V n ( K , L ) ) 1 / n .
That is
V 1 ( K , L ) n V ( K ) n 1 V ( L ) V ( K ) n 2 V ( L ) V 1 ( K , L ) ( V n 1 ( K , L ) V n ( K , L ) ( V 0 ( K , L ) V n ( K , L ) ) 1 / n ) .

 □

The following Bonnesen-style inequality is a direct consequence of Theorem 3.

Theorem 4 Let K be a convex body in R n , then
Δ n ( K ) n n 1 ω n S ( K ) V ( K ) n 2 ( M ( K ) 2 ( V ( K ) ω n ) 1 / n ) ,

with equality if and only if K is a ball.

Lemma 3 Let K, L be convex bodies in R n , then
V 1 ( K , L ) n 1 V 1 ( K , L ) n 2 ( V 1 ( K , L ) n V ( K ) n 1 V ( L ) ) V ( K ) n 2 V ( L ) V 0 ( K , L ) V 1 ( K , L ) V n 1 ( K , L ) V n ( K , L ) V 1 ( K , L ) n 1 V ( K ) n 2 V ( L ) .
(20)
Proof The Minkowski inequality (5) gives
V 1 ( K , L ) n 1 V 1 ( K , L ) n 2 ( V 1 ( K , L ) n V ( K ) n 1 V ( L ) ) V ( K ) n 2 V ( L ) V 0 ( K , L ) V 1 ( K , L ) .

The above inequality together with (18) leads to Lemma 3. □

We are now in a position to prove the following Bonnesen-style Minkowski inequality.

Theorem 5 Let K, L be convex bodies in R n , then
Δ n ( K , L ) V ( K ) 2 n 4 V ( L ) 2 V 1 ( K , L ) n 2 ( V n 1 ( K , L ) V n ( K , L ) V 0 ( K , L ) V 1 ( K , L ) ) 2 ,
(21)

with equality if and only if K and L are homothetic.

Proof From (20) we have
V 1 ( K , L ) n 2 ( V 1 ( K , L ) n V ( K ) n 1 V ( L ) ) V ( K ) n 2 V ( L ) V n 1 ( K , L ) V n ( K , L ) V 0 ( K , L ) V 1 ( K , L ) .

 □

The following Bonnesen-style inequality is a direct consequence of Theorem 5 when L is the unit ball.

Theorem 6 Let K be a convex body in R n , then
S ( K ) n n n ω n V ( K ) n 1 n 2 n 2 ω n 2 V 2 n 4 S n 2 ( M ( K ) 2 n V ( K ) S ( K ) ) 2 ,

with equality if and only if K is a ball.

Declarations

Acknowledgements

The authors would like to thank two anonymous referees for many helpful comments and suggestions that directly lead to the improvement of the original manuscript. The authors are supported in part by NSFC (No. 11271302 and No. 11326073), Fundamental Research Funds for the Central Universities (No. XDJK2014C164), the Ph.D. Program of Higher Education Research Funds (No. 2012182110020).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Southwest University
(2)
Faculty of Education, Southwest University
(3)
College of Mathematics Science, Chongqing Normal University
(4)
School of Mathematical Science, Kaili University

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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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.