IAENG Iteratioal Joural of Applied Mathematics, 46:1, IJAM_46_1_14 Some Iequalities for p-geomiimal Surface Area ad Related Results Togyi Ma, Yibi Feg Abstract The cocepts of p-affie ad p-geomiimal surface areas were itroduced by Lutwak I this paper, we establish some Bru-Mikowski type iequalities of p-geomiimal surface area combiig L p-polar curvature image with various combiatios of covex bodies Moreover, we discuss the equivalece of several iequalities, ad also obtai some results similar to p-geomiimal surface area for the p-affie surface area Idex Terms covex bodies, p-affie surface area, p- geomiimal surface area, Bru-Mikowski type iequality I INTRODUCTION LET K deote the set of covex bodies compact, covex subsets with oempty iteriors) i -dimesioal Euclidea space R For the set of covex bodies cotaiig the origi i their iteriors, the set of covex bodies whose cetroids lie at the origi ad the set of origi-symmetric covex bodies i R, we write Ko, Ke ad Kc, respectively So ad Sc respectively deote the set of star bodies about the origi) ad the set of origi-symmetric star bodies i R Let S 1 deote the uit sphere i R, ad let VK) deote the -dimesioal volume of a bodyk For the stadard uit ball B i R, we use ω = VB ) to deote its volume The study of affie surface area goes back to Blaschke [1] ad is about oe hudred years old It was geeralized to the p-affie surface area by Lutwak i [10] Sice the, cosiderable attetio has bee paid to the p-affie surface area, which is ow at the core of the rapidly developig L p - Bru-Mikowski theorysee articles [4], [5], [6], [8], [11], [1], [13], [14], [17], [19], [4], [8] or books [7], []) I particular, affie isoperimetric iequalities related to the p-affie surface area ca be foud i [10], [9] Aother fudametal cocept i covex geometry is geomiimal surface area, itroduced by Petty [19] more tha three decades ago As Petty explaied i [19], the geomiimal surface area coects the affie geometry, relative geometry ad Mikowski geometry Hece it receives a lot of attetio see [19], [0], [3]) The geomiimal surface area was exteded to p-geomiimal surface area by Lutwak i his semial paper [10] The p-geomiimal surface area shares may properties with the p-affie surface area For istace, both are affie ivariat ad have the same degree of homogeeity However, the p-geomiimal surface area is differet from the p-affie surface area For istace, ulike the p-affie surface area, p-geomiimal surface area has o Mauscript received Jue 6, 015; revised October 15, 015 This work was supported by the Natioal Natural Sciece Foudatio of Chia uder Grat 1156100 ad 113714, ad was supported by the Sciece ad Techology Pla of the Gasu Provice uder Grat 145RJZG7 T Y Ma is with the College of Mathematics ad Statistics, Hexi Uiversity, Zhagye, 734000, Chia e-mail: matogyi@16com Y B Feg is the College of Mathematics ad Statistics, Hexi Uiversity, Zhagye, 734000, Chia e-mail: fegyibi001@163com ice itegral expressio This leads to a big obstacle o extedig the p-geomiimal surface area There are may papers o p-affie ad p-geomiimal surface areas, see eg, [16], [18], [5], [6], [7], [8], [9], [30], [3] Based o the otio of L p -mixed volume, Lutwak itroduced the cocepts of p-affie ad p-geomiimal surface areas, respectively For p 1 ad K K o, the p-affie surface area, Ω pk), was defied i [10] by p Ωp K) +p = if{v p K,Q )VQ) p : Q S o } Here V p K,Q ) deotes the L p -mixed volume of K ad Q see Sectio II A) ad Q deotes the polar of body Q see Sectio II C) For p 1, Lutwak i [10] defied the p-geomiimal surface area, K), of K K o by ω p K) = if{v p K,Q)VQ ) p : Q K o } 1) Further, Lutwak obtaied the followig iequalities for the p-affie ad the p-geomiimal surface areas Lemma 11 Theorem 48 i [10]) Let K K e ad p 1 The Ω p K) +p +p ω p VK) p, ) with equality if ad oly if K is a ellipsoid Lemma 1 Theorem 31 i [10]) Let K K o adp 1 The K) ω p VK) p, 3) with equality if ad oly if K is a ellipsoid Lemma 13 [10] p 50) Let K Fo ad p 1 The Ω p K) +p ω ) p K), 4) with equality if ad oly if K is of p-elliptic type A covex body K K o is said to have a L p-curvature fuctio see [10]) f p K, ) : S 1 R, if its L p -surface area measure S p K, ) is absolutely cotiuous with respect to spherical Lebesgue measure S, ad ds p K, ) = f p K, ) ds Let Fo,F c deote the set of all bodies i K o,k c respectively, ad both of them have a positive cotiuous curvature fuctio If K Sc, ad p 1, the defie Λ p K F c, the L p- polar curvature image of K, by f p Λ pk, ) = ω VK) ρk, )+p 5) Whe p = 1, we write Λ 1K = ΛK, it is just the classical curvature image see [1], [14]); Whe p > 1, it was defied by Yua, Zhu, Lv ad Leg see [15], [30], [31]) Advace olie publicatio: 15 February 016)
IAENG Iteratioal Joural of Applied Mathematics, 46:1, IJAM_46_1_14 The followig theorems are our mai results: Combiig L p -polar curvature image with p-geomiimal surface area, we establish several Bru-Mikowski type iequalities of the p-geomiimal surface area Theorem 14 If p 1,K,L Kc, ad λ,µ 0 ot both zero), the Λ p λ K + p µ L) ) λ Λ p K)+µΛ pl), 6) with equality for p = 1 if ad oly if K ad L are homothetic, ad for p > 1 if ad oly if K ad L are dilates Here, λ K + p µ L deotes the L p -Firey combiatio of K ad L see 10)) Theorem 15 If 1 p,k,l K c, ad λ,µ 0 ot both zero), the Λ p λk + p µl) ) λ Λ pk)+µ Λ pl) 7) The reverse iequality holds whe p > Equality holds i every iequality whe p if ad oly if K is a dilate of L Here, λk + p µl deotes the L p -radial combiatio of K ad L see 13)) Theorem 16 If p 1,K,L K c, ad λ,µ 0 ot both zero), the Λ p λ K + p µ L) ) 1 λgp Λ pk) 1 +µ Λ pl) 1, 8) with equality if ad oly if K ad L are dilates Here, λ K + p µ L deotes the L p -harmoic radial combiatio of K ad L see 16)) Theorem 17 If p 1,K,L F c, ad λ,µ 0 ot both zero), the λk + p µl ) λ K)+µ L), 9) with equality for p = 1 if ad oly if K ad L are homothetic, for p > 1 if ad oly if K ad L are dilates Here, λk + p µl deotes the Blaschke L p -combiatio of K ad L see 3)) Please see the ext sectio for above iterrelated otatios, defiitios ad their backgroud materials The proofs of Theorems 14-17 will be give i Sectio III of this paper Moreover, we derive the equivalece of several iequalities i Sectio IV II PRELIMINARIES A L p -Firey Combiatio ad L p -mixed Volume If K K, the its support fuctio, h K = hk, ) : R, ), is defied by see [])hk,x) = max{x y : y K}, x R, where x y deotes the stadard ier product of x ad y For real p 1,K,L Ko, ad α,β 0 ot both zero), the L p -Firey combiatio, α K + p β L, is defied by see []) K o hα K + p β L, ) p = αhk, ) p +βhl, ) p 10) For p 1, the L p -mixed volume, V p K,L), of K,L, was defied i [9] by p V VK + p ε L) VL) pk,l) = lim It was show i [9] that correspodig to each K Ko there is a positive Borel measure S p K, ) o S 1 such that V p K,Q) = 1 hq,u) p ds p K,u) S 1 for all Q K o It turs out that the L p -surface area measure S p K, ) o S 1 is absolutely cotiuous with respect to SK, ), ad has the Rado-Nikodym derivative ds p K, ) dsk, ) = h1 p K, ) TheL p -Bru-Mikowski iequality was give by Lutwak i [9]: If K,L K o,λ,µ > 0, ad p 1, the Vλ K + p µ L) p/ λvk) p/ +µvl) p/, 11) with equality for p = 1 if ad oly if K ad L are homothetic, ad for p > 1 if ad oly if K ad L are dilates Takigλ = µ = 1 adl = K i 10), thel p-differece body, p K, of K was give by see [9]) p K = 1 K + p 1 K) 1) B L p -radial Combiatio ad L p -dual Mixed Volume If K is a compact star-shaped about the origi) set i R, the its radial fuctio, ρ K = ρk, ) : R \{0} [0, ), is defied by see []) ρk,u) = max{λ 0 : λu K}, u S 1 If ρ K is positive ad cotiuous, the K will be called a star body about the origi) Two star bodies K adlare said to be dilated of oe aother ifρ K u)/ρ L u) is idepedet of u S 1 If K,L So ad λ,µ 0 ot both zero), the for p > 0, the L p -radial combiatio, λk + p µ L So, is defied by see [3]) ρλk + p µl, ) p = λρk, ) p +µρl, ) p 13) For p 1, ad K,L S o, the L p -dual mixed volume, Ṽ p K,L), was defied i [3] by VK + p εl) VK) pṽpk,l) = lim The followig itegral represetatio for the L p -dual mixed volume was obtaied i [3]: If p 1, ad K,L So, the Ṽ p K,L) = 1 ρk,u) p ρl,u) p dsu), S 1 where S is the spherical Lebesgue measure o S 1 ie, the 1)-dimesioal Hausdorff measure) We shall eed the followig L p -dual Bru-Mikowski iequality see [3]): If K,L S o ad 0 < p, the VλK + p µl) p/ λvk) p/ +µvl) p/ 14) The reverse iequality holds whe p > Equality holds whe p if ad oly if K is a dilate of L Takig λ = µ = 1 ad L = K i 13), the L p-radial body, p K, of K is defied by p K = 1 K + 1 p K) 15) Advace olie publicatio: 15 February 016)
IAENG Iteratioal Joural of Applied Mathematics, 46:1, IJAM_46_1_14 C L p -harmoic Radial Combiatio adl p -harmoic Mixed Volume For K,L So, p 1 ad λ,µ 0 ot both zero), the L p -harmoic radial combiatio, λ Kˆ+ p µ L So, is defied bysee [10]) ρλ Kˆ+ p µ L, ) p = λρk, ) p +µρl, ) p 16) If K K o, the polar set, K, of K is defied by K = {x R : x y 1, for all y K} 17) From 17), we ca easily have K ) = K, ad h K = 1 ρ K, ρ K = 1 h K 18) for K K o By 10), 16) ad 18), it follows that if K,L K o ad λ,µ 0 ot both zero), the λ Kˆ+ p µ L = λ K + p µ L ) Defie the Sataló product of K K o by VK)VK ) The Blaschke-Sataló iequality see []) is oe of the fudametal affie isoperimetric iequalities It states that if K K c the VK)VK ) ω, with equality if ad oly if K is a ellipsoid For p 1 ad K,L S o, the L p-harmoic mixed volume, Ṽ pk,l), is defied by see [10]) VK + p ε L) VK) pṽ pk,l) = lim From the polar coordiate formula, the followig itegral represetatio was give i [10]: If p 1 ad K,L So, the Ṽ p K,L) = 1 ρk,u) +p ρl,u) p dsu) S 1 The Mikowski s iequality for the L p -harmoic mixed volume ca be stated that see [10]): Ifp 1 adk,l S o, the Ṽ p K,L) VK) +p VL) p, 19) with equality if ad oly if K ad L are dilates The Bru-Mikowski iequality for the L p -harmoic radial combiatio ca be stated that see [10]): Suppose K,L So,p 1 ad λ,µ > 0, the Vλ Kˆ+ p µ L) p/ λvk) p/ +µvl) p/, 0) with equality if ad oly if K ad L are dilates each other Takig λ = µ = 1 ad L = K i 16), the L p-harmoic radial body, p K, of K is defied by p K = 1 K + 1 p K) 1) D L p -affie Surface Area, L p -curvature Image ad Blaschke L p -combiatio I [10], Lutwak defied the L p -affie surface area as follows: For K Fo ad p 1, the L p-affie surface area, Ω p K), of K is defied by Ω p K) = f p K,u) +p dsu) S 1 Further, Lutwak [10] showed the otio of L p -curvature image as follows: For ayk Fo adp 1, defieλ pk So, the L p -curvature image of K, by ρλ p K, ) +p = VΛ pk) ω f p K, ) ) Note that for p = 1, this defiitio is differet from the classical curvature image see [14]) The defiitio of Blaschke L p -combiatio for covex bodies may be stated that see [9]) for K,L K c,λ,µ 0 ot both zero) ad p 1, the Blaschke L p - combiatio, λk + p µl K c, of K ad L is defied by ds p λk + p µl, ) = λds p K, )+µds p L, ) 3) Takig λ = µ = 1 ad L = K i 3), the Blaschke L p -body, p K Kc, of K is defied by see [9]) p K = 1 K + 1 p K) 4) From ) ad 3), Wag ad Leg [6] proved the followig L p -Bru-Mikowski iequality: If K,L Fc,λ,µ > 0 ad p 1, the VΛ p λk + p µl)) p/ λvλ p K) p/ +µvλ p L) p/, 5) with equality for p = 1 if ad oly if K ad L are homothetic, for p > 1 if ad oly if K ad L are dilates III PROOFS OF THEOREMS I this sectio, we prove Theorems 14-17 Takig L = Q i Propositio 34 of [31], we immediately give: Lemma 31 If p 1 ad K K c, the for ay Q K o, V p Λ pk,q) = ω Ṽ p K,Q )/VK) 6) Lemma 3 If p 1 ad K K c, the Proof Λ pk) = ω p VK) p 7) By 1), 6) ad 7), we have Λ pk) = ω p if{v p Λ pk,q)vq ) p : Q K o } = ω p if{ω Ṽ p K,Q )VQ ) p /VK) : Q K o } ω p : Q K o } = ω p VK) p if{vk) +p VQ ) p VQ ) p /VK) O the other had, from 1) ad 6), it follows that for ay Q K o Λ p K) ω p V p Λ p K,Q)VQ ) p = ω p Ṽ p K,Q )VQ ) p /VK) Advace olie publicatio: 15 February 016)
IAENG Iteratioal Joural of Applied Mathematics, 46:1, IJAM_46_1_14 Sice K K c, ad takig Q = K, we obtia Λ p pk) ω VK) p Above all, we yield equality 7) Proof of Theorem 14 From 7) ad 11), it follows that Λ p λ K + p µ L)) = ω p Vλ K + p µ L)) p λω p VK) p p +µω VL) p = λ Λ pk)+µ Λ pl) From the equality coditio of iequality 11), we kow that equality holds i 6) for p = 1 if ad oly if K ad L are homothetic, ad for p > 1 if ad oly if K ad L are dilates Accordig to 6) ad 1), we easily get that if K K c ad p 1, the Λ p pk)) = Λ p K) Proof of Theorem 15 It follows from 7) ad 14) that for 1 p, Λ pλk + p µl)) = ω p VλK + p µl)) p λω p VK) p p +µω VL) p = λ Λ p K)+µΛ p L) The reverse iequality holds whe p > From the equality coditio of iequality 14), we kow that equality holds i 7) whe p if ad oly if K is a dilate of L Together 7) with 15), we easily get that if K K c ad p, the Proof of Theorem 16 Λ p p K)) = Λ pk) By 7) ad 0), we have Λ pλ K + p µ L)) 1 ) 1Vλ K + p µ L)) p = ω p λ ω p ) 1VK) p p) +µ ω 1VL) p = λ Λ pk) 1 +µ Λ pl) 1 From the equality coditio of iequality 0), we kow that equality holds i 8) if ad oly if K ad L are dilates A immediate cosequece of Theorem 16 is: Corollary 33 With the same assumptios of Theorem I, if λ,µ > 0, the 4 Λ pλ K + p µ L)) 1 λ Λ pk)+ 1 µ Λ pl), 8) with equality if ad oly if K ad L are dilates each other Proof Usig Cauchy s iequality ad the arithmetic mea-harmoic mea iequality i 8), we have Λ p λ K + p µ L)) 1 λ Λ pk) 1 +µ Λ pl) 1 1 4λ Λ p K)+ 1 4λ Λ p L) This yields the desired iequality Combiig 8) with 1), we easily get that if K K c ad p 1, the Λ p p K)) = Λ p K) Lemma 34 For p 1, the mappig Λ p : Fc Sc is bijective Proof For the case p = 1, sice Λ = Λ 1 is the classical curvature image ad Λ : Sc F c is a bijectio see [14], p50), Λ 1 is a bijectio For p > 1, Λ p : S c F c was proved i Propositio 36 of [31] that it is also a bijectio Thus for p 1, Λ p : S c F c is bijective From the defiitio of the L p -polar curvature image Λ p, we kow that it is the iverse of the L p -curvature image Λ p This implies that Λ p is a bijectio o the class of origi-symmetric bodies for p 1 Proof of Theorem 17 It follows from 5) thatλ p = Λ 1 p is the iverse image of Λ p By Lemma 34, equatio 7) ad iequality 5), we have λk + p µl) = Λ pλ p λk + p µl) = ω p VΛ p λk + p µl)) p λω p = λ K)+µ L) VΛ p K) p p +µω VΛ p L) p From the equality coditio of 5), we kow that equality holds i 9) for p = 1 if ad oly if K ad L are homothetic, ad for p > 1 if ad oly if K ad L are dilates By 9) ad 4), we easily get that if K F c ad p 1, the p K) = K) IV THE EQUIVALENCE OF SEVERAL INEQUALITIES Defie M p = {K F o : there exists a Q K o with f p K, ) = hq, ) +p) }, ad call it the p-elliptic type if K M p see [10]) The followig lemma is a direct cosequece of Lemma 13 Lemma 41 Suppose K M p ad p 1, the Ω p K) +p = ω ) p K) 9) Let Fe deote the set of all bodies i K e which has a positive cotiuous curvature fuctio Combiig iequality ) with iequality 3), it follows from Lemma 41 that Theorem 4 Suppose K Fe ad p 1 If K M p, the iequality 3) is equivalet to iequality ) Lutwak [10] proved the followig Blaschke-Sataló type iequality for p-affie surface area Theorem 410 i [10]): If p 1 ad K Ke, the Ω p K)Ω p K ) ω ), 30) with equality if ad oly if K is a ellipsoid From 9) ad 30), we get the followig Blaschke-Sataló type iequality for p-geomiimal surface area Advace olie publicatio: 15 February 016)
IAENG Iteratioal Joural of Applied Mathematics, 46:1, IJAM_46_1_14 Theorem 43 For p 1 ad K K e, if K M p, the K) K ) ω ), 31) with equality if ad oly if K is a ellipsoid If p 1 ad K Ko, the there exists a uique body T p K Ko such thatsee see Propositio 33 i [10]) K) = V p K,T p K) ad VT p K) = ω A body i Ko will be called p-selfmiimal if T p K ad K are dilates of each other For K Ko, Lutwak [10] defied the p-geomiimal area ratio of K by Gp K) ) 1/p VK) p, ad proved that the p-geomiimal area ratios are mootoe o-decreasig i p see Theorem 63 i [10]): If K Ko, ad 1 p q, the Gp K) ) 1/p Gq K) ) 1/q VK) p VK) q, 3) with equality if ad oly if K is p-selfmiimal For K Ko, Lutwak [10] defied the p-affie area ratio of K by Ωp K) +p ) 1/p +p VK) p, ad also obtaied that the p-affie area ratios are mootoe o-decreasig i p see Propositio 513 i [10]): If K Fo, ad 1 p q, the Ωp K) +p ) 1/p Ωq K) +q ) 1/q +p VK) p +q VK) q, 33) with equality if ad oly if K ad Λ p K are dilates The equatio 9) implies that if K M p ad p 1, the Ωp K) +p ) 1/p Gp K) ) 1/p +p VK) p = ω VK) p 34) It is clear from 34) that for K M p iequality 3) ad iequality 33) are equivalet Lutwak proved the followig iequalities 35) ad 36) for the p-affie area ratio of K ad the p-geomiimal area ratio of K Obviously, they are also equivalet for K M p If K Fo, ad p 1, the see Propositio 47 i [10]) Ωp K) +p ) 1/p VK)VK ), 35) +p VK) p with equality if ad oly if K ad Λ p K are dilates If K Ko, ad p 1, the see Propositio 6 i [10]) Gp K) ) 1/p VK)VK )/ω, 36) VK) p with equality if ad oly if K is p-selfmiimal We ote that due to equality 9), Theorems 14-17 have obvious aalogs for the p-affie surface area ACKNOWLEDGMENT The referee of this paper proposed may very valuable commets ad suggestios to improve the accuracy ad readability of the origial mauscript We would like to express our most sicere thaks to the aoymous referee REFERENCES [1] W Blaschke, Vorlesuge über Differetialgeometrie II, Affie Differetialgeometrie, 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