-Curvature Image

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

Research Article

Some Inequalities for the L

-Curvature Image

Wang Weidong,

Wei Daijun,

and Xiang Yu

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

2Department 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

LetKⁿdenote 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, writeKⁿ_oandKⁿ_s. Let Sⁿ⁻¹denote the unit sphere inRⁿ, and denote byVKthen-dimensional volume of bodyK, for the standard unit ballBinRⁿ, and denoteωn 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,hK hK,·:Rⁿ → R, is defined bysee1, page 16

hK, x max

x·y:y∈K

, x∈Rⁿ, 1.1

wherex·ydenotes the standard inner product ofxandy.

A convex bodyK ∈ Kⁿ is said to have a curvature functionfK,· : Sⁿ⁻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 Lp-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,ρKρK,·: 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. LetSⁿ_odenote 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∈μK max

λ≥0 :λx μ ∈K

ρK

x μ

μρKx. 1.7

LetFⁿ_o,F_sⁿ denote the set of all bodies inKⁿ_o,Kⁿ_s, respectively, that have a positive continuous curvature function.

Lutwak in2showed the notion ofLp-curvature image as follows. For eachK ∈ Fⁿ_o and realp≥1, defineΛpK∈ Sⁿ_o, theLp-curvature image ofK, by

fpK,· ωn

V

ΛpK ρ

ΛpK,· ^np. 1.8

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∈ F_sⁿandp≥1, then

V

ΛpK ≤ω^2p−n/pn VK^n−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

V

ΛpK VK^n−p/n≤V

ΛpL VL^n−p/n 1.10 with equality fornp >1 if and only ifKandLare dilates, forn /p >1 if and only ifK L, and forn /p1 if and only ifKandLare translation.

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

Theorem 1.2. IfK∈ Fⁿ_s, andp≥1, then

V

ΛpK ≤ω^n/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∈ Kⁿ_o, andp≥1, then

V Λ^∗_pK

≤ω^n/pn VK^p−n/p 1.12

with equality forp >1 if and only ifΛ^∗_pKandKare dilates, and forp1 if 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

V

ΠpK ≥V

ΛpK 1.13 with equality if and only ifKis an ellipsoid centered at the origin.

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^∗≤ω_n² 2.3

with equality if and only if Kis an ellipsoid.

2.2.Lp-Mixed Volume

ForK, L∈ Kⁿ_oandε >0, the FireyLp-combinationKpε·L∈ Kⁿ_ois defined bysee8 h

Kpε·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 lim

ε→0

V

Kpε·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⁻¹

hQ, u^pdSpK, u 2.6

for eachQ∈ Kⁿ_o. 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

hK, udSK, u VK. 2.7

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

VpK, L≥VK^n−p/nVL^p/n 2.8

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

2.3.Lp-Dual Mixed Volume

ForK, L∈ Sⁿ_o, andε >0, theLp-harmonic radial combinationK−pε·Lis the star body whose radial function is defined bysee2

ρ

K_−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 lim

ε→0

V

K−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∈ Sⁿ_o:

V−pK, L 1 n

Sⁿ⁻¹

ρK, u^npρL, u^−pdSu, 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∈ Sⁿ_oand p≥1see2, then

V_−pK, L≥VK^np/nVL^−p/n 2.13 with equality if and only ifKandLare dilates.

2.4.Lp-Affine Surface Area

Lutwak in2showed that for eachK ∈ Kⁿ_oandp ≥1, theLp-affine surface area,ΩpK, of Kcan be defined by

n^−p/nΩpK^np/ninf

nVpK, Q^∗VQ^p/n:Q∈Sⁿ_o

. 2.14

Forp 1,ΩpKis just classical affine surface areaΩKby Leichtweiβsee4. Further, Lutwak proved that ifK∈ F_oⁿandp≥1, then theLp-affine surface area ofKhas the integral representation

ΩpK

Sⁿ⁻¹

fpK, u^n/npdSu. 2.15

2.5.Lp-Projection Body

The notion of Lp-projection body is shown by Lutwak et al. see10. For K ∈ Kⁿ_o and p≥1, theLp-projection body,ΠpK, ofKis the origin-symmetric convex body whose support function is given by

h^p_Π

pKu 1

nωncn−2,p

Sⁿ⁻¹

|u·v|^pdSpK, v 2.16

for allu∈Sⁿ⁻¹. HereSpK,·is just theLp-surface area measure ofK, and

cn,p ω_np

ω2ωnω_p−1. 2.17

2.6.Lp-Centroid Body

Lutwak and Zhang in11introduced the notion ofLp-centroid body. For each compact star- shaped body about the originK⊂Rⁿand for real numberp≥1, the polar ofLp-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, wherecn,psatisfy2.17.

From definition2.18and equality2.2, ifK ∈ S_oⁿ, then theLp-centroid bodyΓpKof Kis the origin-symmetric convex body whose support function is given by

h^p_ΓpKu 1 cn,pVK

K

|u·x|^pdx 1

np cn,pVK

Sⁿ⁻¹

|u·v|^pρ_K^npvdSv

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∈F_oⁿ,p≥1 and the constantc >0, then

ΛpcKc^n−p/pΛpK. 3.1

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

fpcK,· c^n−pfpK,·, 3.2

this together with1.7and1.8, and notice thatVλQ λⁿVQforλ >0, we get that

ρ

ΛpcK,· ^np V

ΛpcK fpcK,·

ωn c^n−pfpK,·

ωn c^n−pρ

ΛpK,· ^np V

ΛpK ρ

c^n−p/pΛpK,·_np V

c^n−p/pΛpK , 3.3

that is,

ρ

ΛpcK,·

⎡

⎢⎣ V ΛpcK V

c^n−p/pΛpK

⎤

⎥⎦

1/np

ρ

c^n−p/pΛpK,·

, 3.4

and this together with formula2.12, we have that V

ΛpcK V

c^n−p/pΛpK

. 3.5

Hence, from3.4, then

ρ

ΛpcK,· ρ

c^n−p/pΛpK,·

, 3.6

and this yields3.1.

Ifφ∈SLn, Lutwaksee2proved that, forp≥1,

ΛpφKφ^−tΛpK, 3.7 whereφ^−tdenotes the inverse of the transpose ofφ.

Now we rewrite3.1as follows:

ΛpcKc^n−p/pΛpK c^n1/pc⁻¹ΛpK, 3.8

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

Proposition 3.2. IfK∈ F_oⁿ,p≥1, then forφ∈GLn,

ΛpφKdetφ^1/pφ^−tΛpK. 3.9 Lemma 3.3see2. IfK∈F_oⁿ,p≥1, then

VpK, Q^∗ ωn

V

ΛpK V_−p

ΛpK, Q 3.10

for allQ∈Sⁿ_o.

Lemma 3.4. IfK, L∈ S_oⁿ,p≥1, then for allQ∈ Sⁿ_o,

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−p

ΛpK, Q ≤V−p

ΛpL, Q 3.12 for all Q ∈ S_oⁿ, with equality in3.12if and only ifΛpK ΛpL by3.11. Using equality 3.10, then inequality3.12can be rewritten

V

ΛpK VpK, Q^∗≤V

ΛpL VpL, Q^∗, 3.13

for allQ∈ S_oⁿ. LetQ^∗L, together with2.7andLp-Minkowski inequality2.8, we have

V

ΛpL VL≥V

ΛpK VpK, L≥V

ΛpK VK^n−p/nVL^p/n. 3.14

Thus

V

ΛpK VK^n−p/n≤V

ΛpL VL^n−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 c^n−p/pΛpK.

Hencec 1 whenn /p, this means that ifn /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 forn /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

and this givesc 1, that is,L xK whenn > 1. Therefore, we see that equality holds in inequality1.10forn /p1 if and only ifKandLare translation.

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

VK ωn

V

ΛpK V−p

ΛpK, K^∗

≥ ωn

V

ΛpK V

ΛpK 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

V

ΛpK ^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

≥VK^n−p/nV

Λ^∗_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 Note that the proof of Lemma 3.5 can be found in 12. Here, for the sake of completeness, we present the proof as follows.

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

h^p_ΠpKu 1 nωncn−2,p

Sⁿ⁻¹

|u·v|^pdSpK, v 1

nωncn−2,p

Sⁿ⁻¹

|u·v|^pfpK, vdSv 1

ncn−2,pV ΛpK

Sⁿ⁻¹

|u·v|^pρ

ΛpK, v ^npdSv

3.25

for allu∈Sⁿ⁻¹. According to2.19, we also have that, for allu∈Sⁿ⁻¹,

h^p_ΓpΛpKu 1 np cn,pV

ΛpK

Sⁿ⁻¹

|u·v|^pρ

ΛpK, v ^npdSv. 3.26

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

h_ΠpKu h_ΓpΛpKu 3.27

for allu∈Sⁿ⁻¹. ThusΠpK ΓpΛpK.

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

ΓpK ≥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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