Some Bonnesen style Minkowski inequalities

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R E S E A R C H

Open Access

Some Bonnesen-style Minkowski inequalities

Wenxue Xu

1,2

_{, Chunna Zeng}

3

_{and Jiazu Zhou}

1,4*

*_{Correspondence:}

[email protected]

1_{School of Mathematics and}

Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

4_{School of Mathematical Science,}

Kaili University, Kaili, Guizhou 556000, People’s Republic of China Full list of author information is available at the end of the article

Abstract

In this paper, we obtain some Bonnesen-style Minkowski inequalities of mixed volumes of convex bodiesKandLin the Euclidean spaceRn_{. Let}_L_{be the unit ball; we}

get some better Bonnesen-style isoperimetric inequalities than Dinghas’s result forn≥3.

MSC: 52A20; 52A40

Keywords: Minkowski inequality; Aleksandrov-Fenchel inequality; Bonnesen-style inequality; mixed volumes; isoperimetric inequality

1 Introduction

It is well known that the ball has the maximum volume among bodies of ﬁxed surface area in the Euclidean spaceRn. That is, of all domainsK with surface areaS(K) and volume

V(K) (cf.[, ]),

S(K)n–nnωnV(K)n–≥, ()

with equality if and only ifKis a ball. Hereωndenotes the volume of the unit ball,

ωn=

πn/ n(n/),

where(·) is the Gamma function. The isoperimetric deﬁcit

n(K) =S(K)n–nnωnV(K)n– ()

measures the deﬁcit between the domain K and a ball of radius (S(K)/nωn)/(n–).

A Bonnesen-style isoperimetric inequality is of the form (cf.[–])

n(K) =S(K)n–nnωnV(K)n–≥BK, ()

where the quantityBKis a non-negative invariant of geometric signiﬁcance ofKand

van-ishes only whenKis a ball.

Bonnesen himself proved several inequalities of the form () in the Euclidean plane (cf.[, ]), but he was not able to obtain direct generalizations of his two-dimensional results. This was done much later, ﬁrst by Hadwiger [] forn= , and then by Dinghas []

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for arbitrary dimension. From then on, some Bonnesen-style inequalities in the higher di-mensions and generalizations have been obtained by Osserman (cf.[, ]), Santaló (cf.[]), Groemer and Schneider (cf.[]), Zhang (cf.[]), Zhou (cf.[, ]) and others. See refer-ences [–] for more details. The following well-known Bonnesen-style inequality for a convex bodyKin the Euclidean spaceRn_{is due to Dinghas (}_cf._[]):

S(K)n–nnωnV(K)n–≥

S(K)/(n–)– (nωn)/(n–)r

n(n–)

, ()

whereris the in-radius ofK, and equality holds if and only ifKis a ball.

In [], some diﬀerent forms of Bonnesen-style isoperimetric inequalities have been es-tablished associated with the mean width ofK. Zhang obtained (cf.[])

M(K) 

n/(n–)

–

V(K) ωn

/(n–) ≥

V(K) ωn

n/(n–) V(K)

ωn

–/n

–R–

,

whereM(K) andRare the mean width and out-radius ofK, respectively.

The Minkowski inequality of mixed volume is a natural generalization of the isoperi-metric inequality () in the Euclidean spaceRn(cf.[, –]). LetK,Lbe convex bodies inRn_{, then}

V(K,L)n≥V(K)n–V(L), ()

whereV(K,L) is the mixed volume ofKandLand the equality holds if and only ifKand Lare homothetic.

Motivated by (), we deﬁne the Minkowski homothetic deﬁcit as

n(K,L) =V(K,L)n–V(K)n–V(L). ()

The Minkowski homothetic deﬁcitn(K,L) measures the homothety betweenKandL.

Then a Bonnesen-style Minkowski inequality would be of the form

n(K,L) =V(K,L)n–V(K)n–V(L)≥BK,L, ()

where the quantityBK,Lis an invariant of geometric signiﬁcance aboutK andLwith the

following basic properties:

. BK,Lis non-negative;

. BK,Lvanishes only whenKandLare homothetic.

Note that letLbe the unit ballBand byS(K) =nV(K,B), the surface area ofK, then the

Minkowski homothetic deﬁcit is just the isoperimetric deﬁcit. Therefore, the Bonnesen-style Minkowski inequality () is more general than the Bonnesen-Bonnesen-style isoperimetric in-equality ().

In this paper, we focus on Bonnesen-style Minkowski inequalities of type (). SomeBK,L

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2 Preliminaries

A set of pointsK in the Euclidean space Rn_is_convex_{if for all}_x_,_y_∈_K _{and }_≤_λ_≤_,

λx+ ( –λ)y∈K. Adomainis a set with nonempty interiors. Aconvex bodyis a compact convex domain. The set of convex bodies in Rn _{is denoted by}_Kn_{. Let}_Kn

o be the class

of members ofKncontaining the origin in their interiors. WriteVfor ann-dimensional Lebesgue measure andHn–for an (n– )-dimensional Hausdorﬀ measure.Sn–denotes the surface of the unit ball inRn.

A convex bodyK⊂Rnis uniquely determined by its support functionhK :Rn→R,

wherehK(x) =max{x·y:y∈K}, forx∈Rn. For the support function of the dilatecK=

{cx:x∈K}of a convex bodyKwe have

hcK=chK, c> . ()

Note that support functions are positively homogeneous of degree one and subadditive. It follows immediately from the deﬁnition of support functions that for convex bodiesK

andL

K⊆L ⇐⇒ hK≤hL. ()

For a convex bodyKand each Borel setω⊂Sn–_{, the reverse spherical image}_τ₍_K_,_{ω), of} Katωis the set of all boundary points ofKwhich have an outer unit normal belonging to the setω. Associated with each convex bodyK∈Kn

o there is a Borel measureSK onSn–

called the Aleksandrov-Fenchel surface area measure ofK, deﬁned by

SK(ω) =Hn–

τ(K,ω),

for each Borel setω⊆Sn–. Observe that for the surface area measure of the dilatecKof

Kwe have

ScK=cn–SK, c> .

The Minkowski sum of convex setsK, . . . ,KminRnis deﬁned by

K+· · ·+Km={x+· · ·+xm:x∈K, . . . ,xm∈Km}.

The mixed volumeV(K, . . . ,Kn) of compact convex setsK, . . . ,KninRnis deﬁned by

V(K, . . . ,Kn) =



n!

n

j=

(–)n+j

i<···<ik

V(Ki+· · ·+Kik).

The Aleksandrov-Fenchel inequality about theith mixed volume is

Vi(K,K)≥Vi+(K,K)Vi–(K,K), ()

where

Vi(K,K) =V(K, . . . ,K

n–i

,K, . . . ,K

i

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withKappearsn–itimes andKappearsitimes and () holds as an equality if and only

ifKandLare homothetic. Note that

Vn(K,K) =V(K), V(K,K) =V(K). ()

The following inequality for mixed volumes is the general Aleksandrov-Fenchel inequal-ity: LetK, . . . ,Kn∈Kand ≤m≤n. Then

V(K, . . . ,Kn)m≥ m

i=

V(Ki, . . . ,Ki,Km+, . . . ,Kn).

Hence

V(K,K)n–≥V(K)n–Vn–(K,K). ()

LetK=B, thenVi(K,B) =Wi(K), theith quermassintegral of the convex bodyK.

The mixed volume has monotonicity: IfK⊂K, then

V(K,K, . . . ,Kn)≤V

K_,K, . . . ,Kn

.

The mixed volumeV(K,L) of the convex bodiesK,L∈Knohas the integral form

V(K,L) =



n

Sn–hLdSK. ()

Since

V(K) =V(K,K),

we have

V(K) = 

n

Sn–hKdSK.

IfBis the unit ball, then

nV(K,B) =S(K),

the surface area ofK. The mean widthM(K) ofKis

M(K) =  ωn

V(B,K),

that is,

M(K) = 

nωn

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The in-radiusr(K,L), out-radiusR(K,L) ofKwith respect toLare, respectively, deﬁned by

r(K,L) =supλ:x∈Rnandx+λL⊂K,

R(K,L) =infλ:x∈RnandK⊂x+λL. Notice that always

r(K,L)R(L,K) = .

WhenLis the unit ball,r(K,L) andR(K,L) are the radius of maximal inscribed and mini-mal circumscribed balls ofK, respectively.

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

In this section, we derive some Bonnesen-style Minkowski inequalities associated with in-radiusr(K,L) ofKwith respect toL. In [], Diskant improved the Minkowski inequality of mixed volumes as follows.

Lemma  Let K,L be convex bodies in the Euclidean spaceRn,then V(K,L)n/(n–)–V(K)V(L)/(n–)≥

V(K,L)/(n–)–r(K,L)V(L)/(n–)

n

, ()

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

Note that the right-hand side of () is non-negative forx+r(K,L)L⊆K(x∈Rn). By () we have

V(K,L) =



n

Sn–hLdSK≥



n

Sn–hLdSr(K,L)L≥r(K,L)

n–_V₍_L_).

From Lemma  and using the inequalityxn–_–_yn–_≥₍_x_–_y₎n–_(for_x_≥_y_≥_{), a lower}

bound of the Minkowski deﬁcit follows (cf.[, ]).

Proposition  Let K,L be convex bodies in the Euclidean spaceRn,then

n(K,L)≥

V(K,L)/(n–)–r(K,L)V(L)/(n–)

n(n–)

, ()

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 () forn= .

Theorem  Let K,L be convex bodies in the Euclidean spaceR,then

(K,L)≥

V(K,L)/–r(K,L)V(L)/



+ r(K,L)V(L)/V(K,L)/–r(K,L)V(L)/



, ()

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Proof SinceV(K)≥r(K,L)_V₍_L_{) and by}_x_–_y_≥₍_x_–_y₎_(for_x_≥_y_≥_{), we have}

V(K,L)/+V(K)V(L)/≥V(K,L)/+r(K,L)V(L)/

=V(K,L)/–

r(K,L)V(L)/+ r(K,L)V(L)/

≥V(K,L)/–r(K,L)V(L)/



+ r(K,L)V(L)/.

Note that () can be rewritten as

V(K,L)/–V(K)V(L)/≥

V(K,L)/–r(K,L)V(L)/



.

Multiplying byV(K,L)/+V(K)V(L)/on both sides, we have

V(K,L)–V(K)V(L)≥

V(K,L)/–r(K,L)V(L)/



V(K,L)/+V(K)V(L)/

.

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

Let L be the unit ball and notice S(K) = V(K,B) in (), we obtain the following

Bonnesen-style isoperimetric inequality that strengthens Dinghas’s inequality () for

n= .

Corollary  Let K be a convex body inRand r be the in-radius of K,then

(K)≥

S(K)/– (π)/r+ π/rS(K)/– (π)/r, ()

with equality if and only if K is a ball.

Forn≥, we obtain a stronger Bonnesen-style Minkowski inequality as follows.

Theorem  Let K,L be convex bodies in the Euclidean spaceRn₍_n_≥_),_then

n(K,L)≥

V(K,L)/(n–)–r(K,L)V(L)/(n–)

n(n–)

+ r(K,L)V(L)/(n–)n(n–)V(K,L)/(n–)–r(K,L)V(L)/(n–)

n

+V(K,L)V(L)

n/(n–)

r(K,L)nV(K,L)/(n–)–r(K,L)V(L)/(n–)

n(n–)

,

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

Proof Letp=V(K,L)n/(n–)andq=V(L)n/(n–)r(K,L)n, thenp≥q.

n

i=

V(K,L)n(n–i)/(n–)

V(K)n–V(L)(i–)/(n–)

≥

n

i=

V(K,L)n(n–i)/(n–)

V(L)/(n–)r(K,L)n(i–)

=pn––qn–+pqpn–+pn–q+· · ·+pn–iqi–+· · ·+qn–+ qn–

=pn––qn–+pq·p

n–_–_qn– p–q + q

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≥(p–q)n–+pq(p–q)n–+ qn–

≥p/n_–_q/nn(n–)₊_pq_p/n_–_q/nn(n–)_{+ }_qn–_.

That is,

n

i=

V(K,L)n(n–i)/(n–)

V(K)n–V(L)(i–)/(n–)

≥V(K,L)/(n–)–V(L)/(n–)r(K,L)

n(n–)

+ V(L)/(n–)r(K,L)n(n–) +V(K,L)V(L)

n/(n–)

r(K,L)nV(K,L)/(n–)–V(L)/(n–)r(K,L)

n(n–)

.

Multiplying byn_i_=(V(K,L)n(n–i)/n–(V(K)n–V(L))(i–)/(n–)) both sides of () and via

the formula

an––bn–= (a–b)an–+an–b+· · ·+an–ibi–+· · ·+abn–+bn–,

we obtain

V(K,L)n–V(K)n–V(L)≥

V(K,L)/(n–)–r(K,L)V(L)/(n–)

n

×

_n

i=

V(K,L)n(n–i)/(n–)

V(K)n–V(L)(i–)/(n–)

.

We complete the proof of Theorem .

LetLbe the unit ball and byS(K) =nV(K,B) in Theorem ; we obtain the following

stronger Bonnesen-style isoperimetric inequality than Dinghas’s inequality () forn≥.

Corollary  Let K be a convex body inRn(n≥)and r be the in-radius of K,then

n(K)≥

S(K)/(n–)– (nωn)/(n–)r

n(n–)

+ (nωn)/(n–)r

n(n–)

S(K)/(n–)_{– (}_n_ω

n)/(n–)r

n

+nωnS(K)

n/(n–)

rnS(K)/(n–)– (nωn)/(n–)r

n(n–)

,

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  Let K,L be convex bodies inRn_,_then

V(K,L) V(K,L)≤

V(K,L) Vn(K,L)

/n

≤Vn–(K,L) Vn(K,L) ≤

V(K,L)n–

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Proof By inequality (), we have

Vn–(K,L) V(L) ≤

V(K,L)n– V(K)n–_V₍_L₎.

By the Aleksandrov-Fenchel inequality () we have

V(K,L) V(K,L)

≤ V(K,L) V(K,L)

≤ · · · ≤ Vi(K,L)

Vi+(K,L)

≤ · · · ≤Vn–(K,L) Vn(K,L)

.

Therefore

V(K,L) V(K,L)

≤

V(K,L) Vn(K,L)

/n

≤Vn–(K,L) Vn(K,L)

.

Theorem  Let K,L be convex bodies inRn,then

n(K,L)≥V(K)n–V(L)V(K,L)

Vn–(K,L) Vn(K,L)

–

V(K,L) Vn(K,L)

/n

, ()

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

Proof Via (), we have

V(K,L)n– V(K)n–_V₍_L₎–

V(K,L) V(K,L) ≥

–

V(K,L) Vn(K,L)

/n

.

That is

V(K,L)n–V(K)n–V(L)≥V(K)n–V(L)V(K,L)

–

V(K,L) Vn(K,L)

/n

.

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

Theorem  Let K be a convex body inRn_,_then

n(K)≥nn–ωnS(K)V(K)n–

M(K)

 –

V(K) ωn

/n

,

Lemma  Let K,L be convex bodies inRn,then V(K,L)n––

V(K,L)n–(V(K,L)n–V(K)n–V(L)) V(K)n–_V₍_L₎

≤V(K,L) V(K,L)≤

Vn–(K,L) Vn(K,L) ≤

V(K,L)n–

V(K)n–_V₍_L₎. () Proof The Minkowski inequality () gives

V(K,L)n––

V(K,L)n–(V(K,L)n–V(K)n–V(L)) V(K)n–_V₍_L₎ ≤

V(K,L) V(K,L)

.

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

Theorem  Let K,L be convex bodies inRn_,_then

n(K,L)≥

V(K)n–V(L)

V(K,L)n–

–V(K,L)

V(K,L)



, ()

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

Proof From () we have

V(K,L)n–(V(K,L)n–V(K)n–V(L)) V(K)n–_V₍_L₎ ≥

–V(K,L)

V(K,L)

.

The following Bonnesen-style inequality is a direct consequence of Theorem  whenL

is the unit ball.

Theorem  Let K be a convex body inRn_,_then

S(K)n–nnωnV(K)n–≥

nn–_ω

nVn–

Sn–

M(K)

 –

nV(K)

S(K) 

,

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China.}2_{Faculty of}

Education, Southwest University, Chongqing, 400715, People’s Republic of China. 3_{College of Mathematics Science,}

Chongqing Normal University, Chongqing, 401331, People’s Republic of China.4_{School of Mathematical Science, Kaili}

University, Kaili, Guizhou 556000, People’s Republic of China.

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).

Received: 29 April 2014 Accepted: 26 June 2014 Published:23 Jul 2014 References

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10.1186/1029-242X-2014-270

Cite this article as:Xu et al.:Some Bonnesen-style Minkowski inequalities.Journal of Inequalities and Applications