Some Inequalities for the Curvature Image

Volume 2009, Article ID 320786,12pages doi:10.1155/2009/320786

Research Article

Some Inequalities for the

L

p

-Curvature Image

Wang Weidong,

Wei Daijun,

and Xiang Yu

1_{Department of Mathematics, China Three Gorges University, Daxue Road 8, Hubei Yichang 443002, China}

2_{Department of Mathematics, Hubei Institute for Nationalities, Hubei Enshi 445000, China}

Correspondence should be addressed to Wang Weidong,[email protected]

Received 16 May 2009; Revised 19 August 2009; Accepted 14 October 2009

Recommended by Peter Pang

Lutwak introduced the notion ofLp-curvature image and proved an inequality for the volumes of

convex body and itsLp-curvature image. In this paper, we first give an monotonic property ofLp

-curvature image. Further, we establish two inequalities for theLp-curvature image and its polar,

respectively. Finally, an inequality for the volumes ofLp-projection body andLp-curvature image

is obtained.

Copyrightq2009 Wang Weidong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetKn_{denote the set of convex bodiescompact, convex subsets with nonempty interiors}

in Euclidean space Rn_{, for the set of convex bodies containing the origin in their interiors}

and the set of origin-symmetric convex bodies inRn_{, we, respectively, writeK}n

oandKns. Let

Sn−1_{denote the unit sphere inR}n_{, and denote byVKthen-dimensional volume of bodyK,}

for the standard unit ballBinRn_{, and denoteω}

n VB. The groups of nonsingular linear

transformations and the group of special linear transformations are denoted byGLnand

SLn, respectively.

Suppose thatRis the set of real numbers. IfK ∈ Kn_{, then its support function,h} K

hK,·:Rn _{→ R, is defined bysee1, page 16}

hK, x maxx·y:y∈K, x∈Rn, 1.1

A convex bodyK ∈ Kn _{is said to have a curvature functionfK,· : S}n−1 _{→ R, if}

its surface area measureSK,·is absolutely continuous with respect to spherical Lebesgue measureS, and

dSK,·

dS fK,·. 1.2

ForK ∈ Kn

o, and realp ≥ 1, theLp-surface area measure,SpK,·, ofKis defined by

see2,3

dSpK,·

dSK,· hK,·

1−p_{. 1.3}

Equation 1.3 is also called Radon-Nikodym derivative, and the measure SpK,· is

absolutely continuous with respect to surface area measureSK,·.

A convex body K ∈ Kn

o is said to have aLp-curvature function see2fpK,· :

Sn−1 _{→ R, if its L}

p-surface area measure SpK,·is absolutely continuous with respect to

spherical Lebesgue measureS, and

dSpK,·

dS fpK,·. 1.4

IfKis a compact star shapedabout the origininRn_{, its radial function,ρ}

KρK,·:

Rn_{\ {0} → 0,∞, is defined bysee1, page 18}

ρK, x max{λ≥0 :λx∈K}, x∈Rn\ {0}. 1.5

IfρKis positive and continuous,Kwill be called a star bodyabout the origin. LetSnodenote

the set of star bodiesabout the origininRn_{. Two star bodiesKandLare said to be dilates}

of one anotherifρKu/ρLuis independent ofu∈Sn−1.

For the radial function, ifc >0, thensee1, page 18

ρKcx

1

cρKx. 1.6

From1.6, we have that, forμ >0,

ρμKx max

λ≥0 :λx∈μKmax

λ≥0 :λx μ ∈K

ρK

x μ

μρKx. 1.7

LetFn

o,Fsn denote the set of all bodies inKno,Kns, respectively, that have a positive

continuous curvature function.

Lutwak in2showed the notion ofLp-curvature image as follows. For eachK ∈ Fno

and realp≥1, defineΛpK∈ Sno, theLp-curvature image ofK, by

fpK,· ωn

VΛpK

Note that, forp1, this definition differs from the definition of classical curvature imagesee

2. For the study of classical curvature image1,4–7. Further, he proved that ifK∈ Fn

s andp≥1, then

VΛpK ≤ωn2p−n/pVKn−p/p 1.9

with equality if and only if Kis an ellipsoid centered at the origin.

In this paper, we continuously study theLp-curvature image for convex bodies. First,

we give a monotonic property ofLp-curvature image as follows.

Theorem 1.1. IfK, L∈ Fn

o,p≥1, andΛpK⊆ΛpL, then

VΛpK VKn−p/n≤V

ΛpL VLn−p/n 1.10

with equality fornp >1if and only ifKandLare dilates, for_{n /}p >1if and only ifK L, and for_{n /}p1if and only ifKandLare translation.

Next, we establish an inequality for theLp-curvature image as follows.

Theorem 1.2. IfK∈ Fn

s, andp≥1, then

VΛpK ≤ωn/pn VK∗p−n/p 1.11

with equality if and only ifKis an ellipsoid.

Further, we get the following inequality for the polar of theLp-curvature image.

Theorem 1.3. IfK∈ Fn

o,ΛpK∈ Kno, andp≥1, then

VΛ∗pK

≤ωn/pn VKp−n/p 1.12

with equality forp >1if and only ifΛ∗_pKandKare dilates, and forp1if and only ifΛ∗_pKandK

are homothetic.

HereΛ∗_pKdenote the polar ofΛpK, rather thanΛpK∗. Compare with inequality1.9,

we see that inequality1.12may be regarded as a dual form of inequality1.9.

Finally, we obtain an interesting inequality for the Lp-curvature image and Lp

-projection bodyΠpKas follows.

Theorem 1.4. IfK∈ Fn

o,p≥1, then

VΠpK ≥V

ΛpK 1.13

2. Preliminaries

2.1. Polar of Convex Body

IfK∈ Kn

o, the polar body ofK,K∗, is defined bysee1, page 20

K∗x∈Rn:x·y≤1,∀y∈K. 2.1

From the definition 2.1, we know that if K ∈ Kn

o, then the support and radial

functions ofK∗, the polar body ofK, are defined, respectively, bysee1

hK∗

1

ρK, ρK

∗ 1

hK. 2.2

The Blaschke-Santal ´o inequality can be stated thatsee1or7: If K∈ Kn s, then

VKVK∗≤ωn2 2.3

with equality if and only if Kis an ellipsoid.

2.2.

L

-Mixed Volume

ForK, L∈ Kn

oandε >0, the FireyLp-combinationKpε·L∈ Knois defined bysee8

hKpε·L,· phK,·pεhL,·p 2.4

where “·” inε·Ldenotes the Firey scalar multiplication. IfK, L ∈ Kn

oinRn, then forp ≥ 1, theLp-mixed volume,VpK, L, of theKandLis

defined bysee9

n

pVpK, L εlim→0

VKpε·L −VK

ε . 2.5

Corresponding to eachK ∈ Kn

o, there is a positive Borel measure, SpK,·, onSn−1

such thatsee9

VpK, Q 1

n

Sn−1hQ, u

p

dSpK, u 2.6

for eachQ∈ Kno. The measureSpK,·is just theLp-surface area measure ofK.

From the formula2.6and definition1.3, we immediately get that

VpK, K

1

n

Sn−1

TheLp-Minkowski inequality states thatsee9ifK, L∈ Knoandp≥1,then

VpK, L≥VKn−p/nVLp/n 2.8

with equality forp >1 if and only ifKandLare dilates, and forp1 if and only ifKandL

are homothetic.

2.3.

L

-Dual Mixed Volume

ForK, L∈ Sn

o, andε >0, theLp-harmonic radial combinationK−pε·Lis the star body whose

radial function is defined bysee2

ρK−pε·L,· −pρK,·−pερL,·−p. 2.9

Note that here “ε·L” and the Firey scalar multiplication “ε·L” are different. IfK, L∈ Sn

o, forp≥1, theLp-dual mixed volume,V−pK, L, of theKandLis defined

bysee2

n

−pV−pK, L ε→lim0

VK−pε·L −VK

ε . 2.10

The definition above and the polar coordinate formula for volume give the following integral representation of theLp-dual mixed volumeV−pK, LofK, L∈ Sno:

V−pK, L 1

n

Sn−1ρK, u

np_{ρL, u}−p_{dSu, 2.11}

where the integration is with respect to spherical Lebesgue measureSonSn−1_.

From the formula2.11, it follows immediately that, for eachK∈ Sn

oandp≥1,

V−pK, K VK 1

n

Sn−1

ρK, undSu. 2.12

The Minkowski inequality for theLp-dual mixed volumeV−pis that ifK, L∈ Snoand

p≥1see2, then

V−pK, L≥VKnp/nVL−p/n 2.13

2.4.

L

-Affine Surface Area

Lutwak in2showed that for eachK ∈ Kn

oandp ≥1, theLp-affine surface area,ΩpK, of

Kcan be defined by

n−p/nΩpKnp/ninf

nVpK, Q∗VQp/n:Q∈Sno

. 2.14

Forp 1,ΩpKis just classical affine surface areaΩKby Leichtweiβsee4. Further,

Lutwak proved that ifK∈ Fn

oandp≥1, then theLp-affine surface area ofKhas the integral

representation

ΩpK

Sn−1fpK, u

n/np_{dSu. 2.15}

2.5.

L

-Projection Body

The notion of Lp-projection body is shown by Lutwak et al. see10. For K ∈ Kno and

p≥1, theLp-projection body,ΠpK, ofKis the origin-symmetric convex body whose support

function is given by

hp_Π pKu

1

nωncn−2,p

Sn−1

|u·v|pdSpK, v 2.16

for allu∈Sn−1_{. HereS}

pK,·is just theLp-surface area measure ofK, and

cn,p

ωnp

ω2ωnωp−1.

2.17

2.6.

L

-Centroid Body

Lutwak and Zhang in11introduced the notion ofLp-centroid body. For each compact

star-shaped body about the originK⊂Rn_{and for real numberp≥1, the polar ofL}

p-centroid body,

Γ∗

pKrather thanΓpK∗, ofK is the origin-symmetric convex body, whose radial function

is defined by11

ρ−p_Γ∗ pKu

1

cn,pVK

K

|u·x|pdx 2.18

for allu∈Sn−1_{, wherec}

From definition2.18and equality2.2, ifK ∈ Sn

o, then theLp-centroid bodyΓpKof

Kis the origin-symmetric convex body whose support function is given by

hp_ΓpKu 1 cn,pVK

K

|u·x|pdx

1

np cn,pVK

Sn−1|u·v|

p

ρ_KnpvdSv

2.19

for allu∈Sn−1_.

3. The Proof of Theorems

In order to prove our theorems, the following lemmas are essential.

Lemma 3.1. IfK∈Fn

o,p≥1and the constantc >0, then

ΛpcKcn−p/pΛpK. 3.1

Proof. Forc >0, from1.3and1.4, then

fpcK,· cn−pfpK,·, 3.2

this together with1.7and1.8, and notice thatVλQ λn_{VQforλ >0, we get that}

ρΛpcK,· np

VΛpcK

fpcK,·

ωn c

n−pfpK,·

ωn c n−pρ

ΛpK,· np

VΛpK

ρ

cn−p/pΛpK,·

np

Vcn−p/p_Λ pK

,

3.3

that is,

ρΛpcK,·

⎡ ⎢

⎣ V

ΛpcK

Vcn−p/pΛpK

⎤ ⎥ ⎦

1/np

ρcn−p/pΛpK,·

, 3.4

and this together with formula2.12, we have that

VΛpcK V

cn−p/pΛpK

. 3.5

Hence, from3.4, then

ρΛpcK,· ρ

cn−p/pΛpK,·

, 3.6

Ifφ∈SLn, Lutwaksee2proved that, forp≥1,

ΛpφKφ−tΛpK, 3.7

whereφ−tdenotes the inverse of the transpose ofφ. Now we rewrite3.1as follows:

ΛpcKcn−p/pΛpK cn1/pc−1ΛpK, 3.8

this together with3.7and the factΛp−K −ΛpK, we easily get the following result.

Proposition 3.2. IfK∈ Fn

o,p≥1, then forφ∈GLn,

ΛpφKdetφ1/pφ−tΛpK. 3.9

Lemma 3.3see2. IfK∈Fon,p≥1, then

VpK, Q∗ ωn

VΛpK

V−pΛpK, Q 3.10

for allQ∈Sn o.

Lemma 3.4. IfK, L∈ Son,p≥1, then for allQ∈ Sno,

V−pK, Q V−pL, Q⇐⇒KL. 3.11

Proof. Taking Q K in 3.11, and using 2.12, we have that VK V−pL, K. Now

inequality2.13 givesVK ≥ VL, with equality if and only ifK and Lare dilates. Let

Q Lin3.11, and getVL ≥ VK. HenceVK VL, andK andLmust be dilates. ThusKL. In turn, whenKL,the result obviously is true.

Proof ofTheorem 1.1. SinceΛpK⊆ΛpL, then from formula2.11, we know

V−pΛpK, Q ≤V−p

ΛpL, Q 3.12

for all Q ∈ Sn

o, with equality in3.12if and only ifΛpK ΛpL by3.11. Using equality

3.10, then inequality3.12can be rewritten

VΛpK VpK, Q∗≤V

ΛpL VpL, Q∗, 3.13

for allQ∈ Sn

o. LetQ∗L, together with2.7andLp-Minkowski inequality2.8, we have

VΛpL VL≥V

ΛpK VpK, L≥V

Thus

VΛpK VKn−p/n≤V

ΛpL VLn−p/n 3.15

and this is just inequality1.10.

According to the conditions of equality that hold in inequalities3.12and2.8, we know that equality holds in inequality1.10forp >1 if and only ifKandLare dilates and

ΛpK ΛpL, and forp1 if and only ifKandLare homothetic andΛpK ΛpL.

For the casep >1 of equality that holds in1.10, we may supposeLcKc >0, and together withΛpK ΛpL, thenΛpK ΛpcK. Thus, from3.1, we haveΛpK cn−p/pΛpK.

Hencec 1 when_{n /}p, this means that if_{n /}p,then K L. Forn p > 1, we easily see thatKandLare dilates that impliyΛpK ΛpL. So we know that equality holds in inequality

1.10fornp >1 if and only ifKandLare dilates, and for_{n /}p >1 if and only ifKL. For the casep1 of equality that holds in1.10, we may takeLxcKc >0, x∈ Rn_{, then}

Λ1K Λ1L Λ1xcK. 3.16

ButS1xK,· SxK,· SK,·, thenf1xK,· fxK,· fK,·by1.2. By this together with2.15and3.10, we haveΩxK ΩKandVΛ1xK VΛ1K, respectively. Thus, from the definition1.8, we obtain that

ρΛ1xK,·n1 VΛ1xK

ωn fxK,·

VΛ1K

ωn fK,· ρΛ1K,· n1_,

3.17

hence

Λ1xK Λ1K. 3.18

From3.18and3.1, equality3.16can be rewritten as follows:

Λ1K Λ1xcK cn−1Λ1K, 3.19

Proof ofTheorem 1.2. LetQK∗in3.10, together with2.7and2.13, we have that

VK ωn VΛpK

V−pΛpK, K∗

≥ ωn

VΛpK

VΛpK np/nVK∗−p/n

ωnV

ΛpK p/nVK∗−p/n

3.20

with equality in inequality3.20if and only ifΛpKandK∗are dilates.

From this, and using the Blaschke-Santal ´o inequality2.3, then

VΛpK p/n≤

1

ωnVKVK ∗p/n

≤ωnVK∗p−n/n,

3.21

and equality holds in second inequality of3.21if and only ifKis an ellipsoid.

From3.21, we immediately obtain inequality1.11. According to the conditions of equality that hold in3.20and second inequality of3.21, we get equality in 1.11if and only ifKis an ellipsoid.

Proof ofTheorem 1.3. TakingQ ΛpKin3.10, and using2.12, then

Vp

K,Λ∗pK

ωn. 3.22

From3.22, and together with inequality2.8, we have

ωnVp

K,Λ∗pK

≥VKn−p/nVΛ∗pK

p/n

, 3.23

this inequality immediately gives1.12. According to equality conditions of inequality2.8, we get equality in1.12forp >1 if and only ifΛ∗_pKandKare dilates, and forp 1 if and only ifΛ∗_pKandKare homothetic.

The proof ofTheorem 1.4requires the following two lemmas.

Lemma 3.5. IfK∈ Fn

o,p≥1, then

ΠpK ΓpΛpK. 3.24

Proof. Using the definitions2.16,1.4, and1.8, we have

hp_ΠpKu 1 nωncn−2,p

Sn−1|u·v|

p

dSpK, v

1

nωncn−2,p

Sn−1|u

·v|pfpK, vdSv

1

ncn−2,pV

ΛpK

Sn−1

|u·v|pρΛpK, v npdSv

3.25

for allu∈Sn−1_{. According to2.19, we also have that, for allu∈S}n−1_,

hp_ΓpΛpKu 1 np cn,pV

ΛpK

Sn−1|u·v|

p_ρΛ

pK, v npdSv. 3.26

But2.17givesncn−2,p npcn,p; hence from3.25and3.26, we obtain

hΠpKu hΓpΛpKu 3.27

for allu∈Sn−1_{. ThusΠ}

pK ΓpΛpK.

Lemma 3.610 Lp-Busemann-Petty centroid inequality. IfK∈ Sno,p≥1, then

VΓpK ≥VK 3.28

with equality if and only ifKis an ellipsoid centered at the origin.

Proof ofTheorem 1.4. From3.28and3.24, we immediately get inequality1.13. According to the case of equality that holds in3.28, we see equality in 1.13if and only ifK is an ellipsoid centered at the origin.

Acknowledgment

This research is supported in part by the Natural Science Foundation of China Grant no. 10671117, Academic Mainstay Foundation of Hubei Province of China Grant no. D200729002, and Science Foundation of China Three Gorges University. The authors wish to thank the referees for their very helpful comments and suggestions on this paper.

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