Research Article Dual L p -Mixed Geominimal Surface Area and Related Inequalities

Transcription

1 Joural of Fuctio Spaces Volume 2016, Article ID , 10 pages Research Article Dual L p -Mixed Geomiimal Surface Area ad Related Iequalities Togyi Ma ad Yibi Feg College of Mathematics ad Statistics, Hexi Uiversity, Zhagye, Gasu , Chia Correspodece should be addressed to Togyi Ma; matogyi 123@163.com Received 11 April 2016; Accepted 9 Jue 2016 Academic Editor: Carlo Bardaro Copyright 2016 T. Ma ad Y. Feg. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The itegral formula of dual L p -geomiimal surface area is give ad the cocept of dual L p -geomiimal surface area is exteded to dual L p -mixed geomiimal surface area. Properties for the dual L p -mixed geomiimal surface areas are established. Some iequalities, such as aalogues of Alexadrov-Fechel iequalities, Blaschke-Sataló iequalities, ad affie isoperimetric iequalities for dual L p -mixed geomiimal surface areas, are also obtaied. 1. Itroductio The cocept of geomiimal surface area was itroduced by Petty [1] about 40 years ago, ad its L p -extesio was first itroduced by Lutwak [2, 3]. They have bee proved to be key igrediets i coectig affie differetial geometry, relative differetial geometry, ad Mikowski geometry. The basic theory cocerig geomiimal surface area is developed, ad a close coectio is established betwee this theory ad affie differetial geometry i [1]. The L p -geomiimal surface area is ow thought to be at the core of the rapidly developig L p -Bru-Mikowski theory. Hece, it receives a lot of attetio ad motivates extesios of some kow iequalities for geomiimal surface areas to L p -geomiimal surfaceareas.theseewiequalitiesofl p -type (p > 1) are stroger tha their classical couterparts. However, fidig a itegral expressio for the L p - geomiimal surface area seems to be itractable. This also leads to a big obstacle o extedig the L p -geomiimal surface area. Util more recetly, Zhu et al. [4] provided a itegral formula for L p -geomiimal surface area by p-petty body ad itroduced L p -mixed geomiimal surface areas which exteded the L p -geomiimal surface area. Thereout, they established some ew L p -affie isoperimetric iequalities. Recetly, Wag ad Qi [5] itroduced a cocept of dual L p -geomiimal surface area, which is a dual cocept for L p -geomiimal surface area ad belogs to the dual L p - Bru-Mikowski theory for star bodies also developed by Lutwak (see [6, 7]). The dual L p -Bru-Mikowski theory for star bodies ad a more extesive dual Orlicz-Bru- Mikowski theory for star bodies received cosiderable attetio (see, e.g., [8 21]), ad they have bee proved to be very powerful i solvig may geometric problems, for istace, the Busema-Petty problems (see, e.g., [6, 22 24]). I this paper, we show that the ifimum i the defiitio of dual L p -geomiimal surface area is a miimum ad provide a itegral formula for dual L p -geomiimal surface area by dual p-petty body. Moreover, we defie the dual L p - mixed geomiimal surface area ad establish some ew L p - affie isoperimetric iequalities for it. Our paper is orgaized as follows. I Sectio 2, we provide the ecessary backgroud, such as defiitios ad kow results which will be eeded. Sectio 3 icludes the basic theory of dual L p -geomiimal surface area, such as theorem of existece ad uiqueess for dual L p -geomiimal surface area, as well as the itegral defiitio of dual L p - geomiimal surface area. I Sectio 4, we itroduce the dual L p -mixed geomiimal surface area ad prove some importat properties, such as affie ivariat properties. We also obtai aalogues of Alexadrov-Fechel iequalities, Blaschke-Sataló iequalities, adaffieisoperimetric iequalities for dual L p -mixed geomiimal surface areas.

2 2 Joural of Fuctio Spaces Fially, we ivestigate the dual ith L p -mixed geomiimal surfaceareasadobtaiaaloguesofblaschke-satalóad affie isoperimetric iequalities i Sectio Prelimiaries ad Notatios Let K deote the set of covex bodies (compact, covex subsets with oempty iteriors) i Euclidea space R. For the set of covex bodies cotaiig the origi i their iteriors ad the set of covex bodies whose cetroids lie at the origi i R,wewriteK o ad K c,respectively.let V(K) deote the -dimesioal volume of a body K,adlet B deote the stadard Euclidea uit ball i R ad write ω = V(B) for its volume, ad let S 1 deote the uit sphere for B. For K K o, its support fuctio h K = h(k, ) : R \{o} [0, )is defied by x R \{o}, h(k, x) = max{ x, y : y K},where, is the stadard ier product o R. Associated with each K K o, oe ca uiquely defie its polar body K K o by K ={x R : x,y 1, y K}.Itiseasilyverifiedthat(K ) =Kif K K o. For K, L K ad α, β 0 (ot both zero), the Mikowski liear combiatio α K+β L K is defied by h(α K+β L, )=αh(k, ) +βh(l, ). (1) The classical Bru-Mikowski iequality (see [25]) states that for covex bodies K, L K ad real α, β 0 (ot both zero), the volume of the bodies ad the volume of their Mikowski liear combiatio α K+β L K are related by V (α K+β L) 1/ αv(k) 1/ +βv(l) 1/, (2) with equality if ad oly if K ad L are homothetic. For real p 1, K, L K o,adα, β 0 (ot both zero), the Firey liear combiatio, α K+ p β L, is defied by (see [26]) h(α K+ p β L, ) p =αh(k, ) p +βh(l, ) p. (3) FortheFireyliearcombiatioα K+ p β L,Firey [26] also established the L p -Bru-Mikowski iequality (a iequality that is also kow as the Bru-Mikowski-Firey iequality, see [14]). If p>1, α, β 0 (ot both zero), ad K, L K o,the V (α K+ p β L) p/ αv(k) p/ +βv(l) p/, (4) with equality if ad oly if K ad L are dilates. AsetK i R is star-shaped at o if o Kad for each x R \{o}, the itersectio K {cx:c 0}is a (possibly degeerate) compact lie segmet. If K R is star-shaped at the origi o,wedefieitsradialfuctioρ K for x R \{o} by ρ(k, x) = max{λ 0:λx K}.Ifρ K is positive ad cotiuous, the K is called a star body about the origi. S o deotes the set of star bodies (about the origi) i R.Two star bodies K ad L are dilates of oe aother if ρ K (u)/ρ L (u) is idepedet of u S 1.NotethatK S o cabeuiquely determied by its radial fuctio ρ K ( ) ad vice versa. If α> 0,wehave ρ K (αx) =α 1 ρ K (x), ρ αk (x) =αρ K (x). More geerally, from the defiitio of the radial fuctio, it follows immediately that for φ GL() the radial fuctioof the image φk = {φy : y K} of K S o is give by (see [27]) (5) ρ(φk,x)=ρ(k,φ 1 x), x R. (6) Obviously, for K, L S o, K L iff ρ K ρ L. (7) The radial Hausdorff metric betwee the star bodies K ad L is δ (K, L) = max u S 1 ρ K (u) ρ L (u). (8) Asequece{K i } of star bodies is said to be coverget to K if δ(k i,k) 0, as i. (9) Therefore, a sequece of star bodies K i coverges to K if ad oly if the sequece of radial fuctios ρ(k i, )coverges uiformly to ρ(k, ) (see [28, Theorem 7.9]). Accordig to the defiitios of the polar body for covex body, support fuctio, ad radial fuctio, it follows that for K K o h K (u) ρ K (u) =1, ρ K (u) h K (u) =1, u S 1. (10) Oeofthemostimportatiequalitiesicovexgeometry is the Blaschke-Sataló iequality about polar body(cf. [1, 27, 29]): If K K c,the V (K) V(K ) ω 2, (11) where the equality holds if ad oly if K is a ellipsoid. If K, L S o ad α, β 0 (ot both zero), the, for p 1, the radial harmoic L p -combiatio, α K + p β L S o, isdefiedby(see[3]) ρ(α K + p β L, ) p =αρ(k, ) p +βρ(l, ) p. (12) For p 1ad K, L S o,thedualharmoicl p-mixed volume, V p (K, L), is defied by p V V(K + p ε L) V(K) p (K, L) = lim. (13) ε 0 + ε

3 Joural of Fuctio Spaces 3 Let K, L S o ad p 1. The, the itegral represetatio of dual harmoic L p -mixed volume of K ad L, V p (K, L),is give (see [3]): V p (K, L) = 1 S 1 ρ +p K (u) ρ p L (u) ds (u). (14) With (5) ad (14) take together, we obtai for α>0 V p (αk, L) =α +p V p (K, L), V p (K, αl) =α p V (15) p (K, L). I [3], Lutwak proved the followig: For K, L S o ad α, β 0,ifp 1,the,forφ GL(), φ(α K + p β L)=α φk + p β φl. (16) Sice V(φK) = det (φ) V(K), forallk S o ad φ GL(), the followig follows from (6), (14), ad (15). Propositio 1. If p 1ad K, L S o,the,forφ GL(), V p (φk, φl) = det φ V p (K, L). (17) The case φ SL() of Propositio 1 reduces to the followig formula: V p (φk, φl) = V p (K, L). (18) This itegral represetatio of V p (, ), withhölder s iequality (see [30, p. 140]) together with the polar coordiate formula, immediately gives the followig: V p (K, L) V(K) +p V (L) p, (19) with equality if ad oly if K ad L are dilates. The followig result is a immediate cosequece of (19). Lemma 2. Suppose that p 1ad U S o such that K, L U.IfforallQ U V p (K, Q) = V p (L, Q) (20) or V p (Q, K) = V p (Q, L), the K=L. The cotiuity of the dual harmoic L p -mixed volume V p : S o S o (0, )is cotaied. Lemma 3. Supposethatsequeces{K i }, {L j } S o ad K i K S o, L j L S o.ifp 1,thelim i,j V p (K i,l j )= V p (K, L). Proof. Sice K i Kad L j Lare equivalet to ρ Ki ρ K ad ρ Lj ρ L,uiformlyoS 1,adρ K, ρ L are positively cotiuous o S 1,theρ Ki ad ρ Lj are uiformly bouded o S 1 (see [28, Theorem 7.9]). Hece, ρ +p K i ρ p L j ρ +p K, uiformly o S 1, ρ p L, uiformly o S 1. (21) Hece, ρ +p S 1 K i (u) ρ p L j (u) ds (u) ρ +p S 1 K (u) ρ p L (u) ds (u), if i, j. (22) Namely, lim i,j V p (K i,l j )= V p (K, L). The volume-ormalized dual coical measure d V K of K S o is defied by V(K)d V K = (1/)ρ K ds, wheres is Lebesgue measure o S 1.Weshallmakeuseofthefact that the volume-ormalized dual coical measure V K is a probability measure o S 1. The followig lemma will be eeded. Lemma 4 (see [3]). Let C deote the set of compact covex subsets of Euclidea -space R,adsupposeK i K o such that K i L C. If the sequece V(K i ) is bouded, the L K o. 3. The Dual L p -Geomiimal Surface Area Based o the otio of dual L p -mixed volumes, Wag ad Qi [5] defied the dual L p -geomiimal surface area as follows: For K S o,theduall p-geomiimal surface area, G p (K), of K is defied by ω p/ G p (K) (23) = if { V p (K, Q) V(Q ) p/ :Q K o }. For this otio of L p -dualgeomiimalsurfacearea,wag adqii[5]establishedthefollowigaffieisoperimetric iequality ad Blaschke-Satalótypeiequality:ForK S o ad p 1, G p (K) ω p/ V (K) (+p)/, (24) with equality if ad oly if K is a ellipsoid cetred at the origi. If K K c ad p 1,the G p (K) G p (K ) (ω ) 2, (25) with equality if ad oly if K is a ellipsoid. By the homogeeity of volume ad dual L p -mixed volume, the dual L p -geomiimal surface area could also be defied by G p (K) = if { V p (K, Q) :Q K o, V(Q )=ω }. (26) It will be show that the ifimum i the above defiitio is attaied.

4 4 Joural of Fuctio Spaces Theorem 5. If K S o ad p 1, the there exists a uique body K K o such that G p (K) = V p (K, K), V( K )=ω. (27) Proof. From the defiitio of G p (K), there exists a sequece {M i } K o such that V(M i ) = ω,with V p (K, B) V p (K, M i ),foralli,ad V p (K, M i ) G p (K).Toseethat the M i K o, i=1,2,...,areuiformlybouded,let R i =R(M i )=ρ(m i,u i ) = max {ρ (M i,u):u S 1 }, (28) where u i is ay of the poits i S 1 at which this maximum is attaied. Let r K = mi S 1ρ K.The,r K B K.From defiitio (14) of dual harmoic L p -mixed volume ad Jese s iequality, it follows that V p (K, B) V p (K, M i ) V (K) V (K) ρ ( K (u) S 1 ρ Mi (u) d V p K ) ( S 1 = ( ρ p K (u) S 1 ρ Mi (u) ) d V K ρ K (u) d V p K R ) i 1 =( ρ V (K) R K (u) +1 ds (u)) i S 1 ( r K V (K) R i S 1 ρ K (u) ds (u)) =( r p K ). R i p p (29) Namely, ω r p K (r K ) V R p (K, M i ) V p (K, B) < (30) i for a fixed K S o ;the,thesequece{m i} is uiformly bouded. Sice the sequece {M i } is uiformly bouded, the Blaschke selectio theorem guaratees the existece of a subsequece of M i, which will also be deoted by M i,ad acompactcovexl C,suchthatM i L.SiceV(M i )= ω, Lemma 4 gives L K o.now,m i L implies that M i L, ad sice V(M i )=ω,itfollowsthatv(l )=ω. Lemma 3 ca ow be used to coclude that L will serve as the desired body K. The uiqueess of the miimizig body is easily demostrated as follows. Suppose L 1,L 2 K o ad L 1 =L 2,such that V(L 1 )=ω =V(L 2 ),ad V p (K, L 1 )= V p (K, L 2 ). (31) Defie L K o by L= 1 2 L L 2. (32) Sice, obviously, L = 1 2 L L 2 (33) ad V(L 1 )=ω =V(L 2 ), it follows from Bru-Mikowski iequality (2) that V (L ) ω, (34) with equality if ad oly if L 1 =L 2. By formula (14) of dual L p -mixed volume, together with the covexity of φ(t) = t p (p 1),wehave V p (K, L) = 1 p ρ ( K (u) S 1 ρ 1/2 L /2 L 2 (u) ) ρ K (u) ds (u) = 1 S 1 ( ρ K (u) (1/2ρ L1 (u) +1/2ρ L2 (u)) 1 ) ρ K (u) ds (u) = 1 ( ρ K (u) S 1 2ρ L1 (u) + ρ p K (u) 2ρ L2 (u) ) ρ K (u) ds (u) 1 2 ( ρ p K (u) S 1 ρ L1 (u) ) ρ K (u) ds (u) S 1 ( ρ K (u) ρ L2 (u) ) p ρ K (u) ds (u) = 1 2 V p (K, L 1 )+ 1 2 V p (K, L 2 )= V p (K, L 1 ) = V p (K, L 2 ), p (35) with equality if ad oly if L 1 =L 2.Thus, V p (K, L) V(L ) p/ < V p (K, L 1 )V(L 1 ) p/ (36) = V p (K, L 2 )V(L 2 ) p/ is the cotradictio that would arise if it were the case that =L 2.Thiscompletestheproof. L 1 The uique body whose existece is guarateed by Theorem 5 will be deoted by T p K adwillbecalledthedual p-petty body of K.Thepolarbodyof T p K will be deoted by T p K rather tha ( T p K).Thus,forK S o,thebody T p K is defied by G p (K) = V p (K, T p K), (37) V( T p K) = ω.

5 Joural of Fuctio Spaces 5 For K K, there exists a uique poit s(k) i the iterior of K, called the Sataló poit of K,suchthat(see[3]) V(( s (K) +K) ) (38) = mi {V (( x+k) ):x it K}, or, for the uique s(k) K,thisisequivaletto uh ( s (K) +K,u) (+1) ds (u) =0. (39) S 1 Let K s deote the set of covex bodies havig their SatalópoitattheorigiiR.Thus,wehave(see[3]) K K s iff K K c. (40) Let T ={ T K :s(k) =o, V( T )=ω }. (41) The ext result is a immediate cosequece of Theorem 5. Theorem 6. For each K S o, there exists a uique body T p K T with G p (K) = V p (K, T p K). The uique body T p K is called the dual p-petty body of K. By Theorem 6 ad the itegral represetatio (14) of dual harmoic L p -mixedvolume,wehavethefollowigitegral formula of G p (K). Propositio 7. For each K S o, there exists a uique covex body T = T p K T with G p (K) = ρ +p S 1 K (p) ρ p T 4. The Dual L p -Mixed Geomiimal Surface Area (u) ds (u). (42) Motivated by the defiitio of L p -mixed geomiimal surface area of Zhu et al. (see [4]), we ow defie the dual L p -mixed geomiimal surface area as follow: For each K i S o, i = 1,...,,adp 1, there exists a uique covex body (dual p-petty body of K i ) T i = T p K i T (i=1,...,)with G p (K 1,...,K )= ρ p T (u)] 1/ ds (u). [ρ +p S 1 K 1 (p) ρ p T 1 (u) ρ +p K (u) (43) G p (K 1,...,K ) will be called the dual L p -mixed geomiimal surface area of K 1,...,K S o. Let g p (K i,u) = ρ +p K i (p)ρ p (u). The, G T p (K 1,...,K ) i cabewritteasfollows: G p (K 1,...,K ) = [g p (K 1,u) g p (K,u)] 1/ (44) ds (u). S 1 ThefollowigpropositioswillprovidethatthedualL p - mixed geomiimal surface area is affie ivariat. Propositio 8. If K S o ad every φ GL(),the G p (φk) = det φ (+p)/ G p (K). (45) Proof. From defiitio (23) of dual L p -geomiimal surface area ad (18), for φ SL(),wehave G p (φk) = if { V p (φk, Q) : Q K o, V(Q ) =ω }=if { V p (K, φ 1 Q) : φ 1 Q K o, V((φ 1 Q) )=V(φ t Q )=ω }= G φ (K). (46) O the other had, for λ>0, it follows from (23) that G p (λk) =λ +p G p (K). (47) Therefore, for every φ GL(), wehave G p (φk) = det φ (+p)/ G p (K). (48) Propositio 9. If K S o,the,forφ GL(), det φ 1/ T p φk = φ T p K. (49) Proof. From the defiitio of T p ad Propositio 8, it follows that V p (K, T p K) = G p (K) = det φ (+p)/ G p (φk) = det φ (+p)/ V p (φk, T p φk). From the defiitio of T p, Propositio 9, ad (15), V p (K, T p K) = det φ (+p)/ V p (φk, T p φk) = det φ 1 V p (φk, det φ 1/ T p φk) = V p (K, φ 1 ( det φ 1/ T p φk)). Namely,fromLemma2,foreachφ GL(), (50) (51) det φ 1/ T p φk = φ T p K. (52) Propositio 10. If p 1ad K 1,...,K S o,the,forφ GL(), G p (φk 1,...,φK ) (53) = det φ (+p)/ G p (K 1,...,K ). I particular, if φ SL(), the G p (K 1,...,K ) is affie ivariat; that is, G p (φk 1,...,φK )= G p (K 1,...,K ). (54)

6 6 Joural of Fuctio Spaces Proof. Sice K S o,forφ GL() ad ay u S 1,wehave = det φ p/ φ 1 u ρk (V) +p ρ T p K (V) p g p (φk, u) = ρ (φk, u) +p ρ( T p φk, u) p = det φ p/ φ 1 u g p (K, V), (55) =ρ(φk,u) +p ρ( det φ 1/ φ T p K, u) p where V =(φ 1 u)/ φ 1 u S 1. Therefore, for φ GL(),we have G p (φk 1,...,φK )= det φ p/ φ 1 u [g p (K 1,u) g p (K,u)] 1/ ds(φ( S φ 1 u φ 1 u )) 1 φ 1 u = det φ (+p)/ G p (K 1,...,K ). (56) The dual mixed volume V(K 1,...,K ) of sets K 1,..., K S o is defied by V(K 1,...,K )= 1 S 1 ρ K1 (u) ρ K (u) ds (u). (57) The classical dual Alexadrov-Fechel iequalities for dual mixed volumes (cf. [27, 31, 32]) ca be writte as V(K 1,...,K ) m m 1 i=0 V(K 1,...,K m,k i,...,k i ), m (58) with equality if K m+1,...,k are dilates of each other. If m= 1,equalityholdstrivially. I particular, takig m=i the above iequality ad oticig that V(K) = V(K),wehave V(K 1,...,K ) V(K 1 ) V(K ), (59) with equality if ad oly if K 1,...,K are dilates. Take K 1 = =K i =K,K i+1 = = K =Bi V(K 1,...,K ),ad V (K,...,K,B,...,B ) fl W i (K), (60) i i where W i (K) is called the ith dual quermassitegral of K S o. The followig iequalities are the aalogous of dual Alexadrov-Fechel iequalities for dual L p -mixed geomiimal surface area. Theorem 11. If p 1ad K 1,...,K S o,the,for1 m, G p (K 1,...,K ) m with equality if K m+1,...,k are dilates of each other. If m= 1,equalityholdstrivially. I particular, if m=i the above iequality, the G p (K 1,...,K ) G p (K 1 ) G p (K ), (62) with equality if K i (1 i ) are dilates of each other. Proof. Let Y 0 (u) = [g p (K 1,u) g p (K m,u)] 1/ ad Y i+1 (u) = [g p (K i,u)] 1/ for i = 0,...,m 1.ByHölder s iequality (cf. [30]), we have G p (K 1,...,K ) = S 1 [g p (K 1,u) g p (K,u)] 1/ ds (u) = S 1 Y 0 (u) Y 1 (u) Y m (u) ds (u) = m 1 i=0 (Y 0 (u) Y i+1 (u) m ds (u)) 1/m m 1 G 1/m p i=0 (K 1,...,K m,k i,...,k i ). m (63) The equality i Hölder s iequality holds if ad oly if Y 0 (u)y m i+1 = λ m ij Y 0(u)Y m j+1 for some λ ij > 0 ad all 0 i = j m 1. This is equivalet to ρ +p K i (u)ρ p (u) = T i λ ij ρ+p K j (u)ρ p (u). From Propositio 9, T T p K = T p ck for j costat c>0.thus,theequalityholdsifk i ad K j are dilates of each other. A lemma of the followig type will be eeded. Lemma 12. If K K c ad p 1,the m 1 i=0 G p (K 1,...,K m,k i,...,k i ), m (61) G p (K) ω (2 p)/ V(K ) (+p)/, (64) with equality if ad oly if K is a ball cetred at the origi.

7 Joural of Fuctio Spaces 7 Proof. From defiitio (23) ad iequality (19), we have ω p/ V(K ) (+p)/ G p (K) = if { V p (K, Q) V(K ) (+p)/ V(Q ) p/ :Q K c } if { V p (K, Q) V p (K,Q ):Q K o }. (65) Sice K K o,takigq=k, it follows from iequalities (65) ad (11) that Namely, ω p/ V(K ) (+p)/ G p (K) (66) if {V (K) V(K ):K K o } ω2. G p (K) ω (2 p)/ V(K ) (+p)/. (67) By the equality coditio of (19) ad (65), we see that equality holds i (64) if ad oly if K is a ball cetred at the origi. Now, we prove the affie isoperimetric iequalities for dual L p -mixed geomiimal surface areas. Theorem 13. Let K 1,...,K K c ad p 1;the, G p (K 1,...,K ) G p (B,...,B) ( V(K 1,...,K ) (+p)/ ), (68) V (B,...,B) with equality if ad oly if K 1,...,K are balls cetred at the origi that are dilates of each other. Proof. From (77) i Sectio 5, it follows that T p B=B.The, g p (B, u) = 1, G p (B) = ω,ad G p (B,...,B) = ω.by iequalities (62), (64), ad (59), we have G p (K 1,...,K ) G p (B,...,B) ( V(K 1 ) V (B) V(K ) V (B) ) (+p)/ ( V(K 1,...,K ) (+p)/ ). V (B,...,B) (69) By the equality coditio of (62), (64), ad (59), we see that equality holds i (68) if ad oly if K is a ball cetred at the origi. Corollary 14. Let K 1,...,K K c ad p 1;the, Take K 1 = =K i =K, K i+1 = =K =Bi (70), ad we write G p (K,...,K,B,...,B ) fl. (71) i i Corollary 15. Let K K c ad p 1;the,fori=0,1,..., 1, ω (2+p)/ W i (K ) (+p)/, (72) with equality if ad oly if K is a ball cetred at the origi. 5. The Dual ith L p -Mixed Geomiimal Surface Area This sectio is maily dedicated to ivestigatig the dual ith L p -mixed geomiimal surface area. For K, L S o, p 1,adi R, we defie dual ith L pmixed geomiimal surface area, G p (K, L),ofK, L as ad write G p,i (K, L) = S 1 g p (K, u) ( i)/ g p (L, u) i/ ds (u) By Theorem 6, we have ad, obviously, (73) G p,i (K, B) =. (74) G p (B) = V p (B, T p B), (75) G p (B) =ω = V p (B, B). (76) Thus, the above two equatios ad the uiqueess part of Theorem 6 show that T p B=B. (77) Noticig that g p (B, u) = 1 for u S 1,the = g p (K, u) ( i)/ ds (u). (78) S 1 By (44), (73), ad (74), we have G p,0 (K) = G p (K), G p,i (K, K) = G p (K), (79) G p (K 1,...,K ) ω (2+p)/ V(K 1,...,K ) (+p)/, (70) G p,0 (K, L) = G p (K), G p, (K, L) = G p (L). (80) with equality if ad oly if K 1,...,K are balls cetred at the origi that are dilates of each other. The followig cyclic iequality for the dual ith L p -mixed geomiimal surface area will be established.

8 8 Joural of Fuctio Spaces Theorem 16. For K, L S o, p 1, i, j, k R,adi<j<k, we have G p,i (K, L) k j G p,k (K, L) j i G p,j (K, L) k i, (81) with equality if K ad L are dilates of each other. Proof. From defiitio (73) ad Hölder s iequality,it follows that, for p 1, G p,i (K, L) (k j)/(k i) G p,k (K, L) (j i)/(k i) That is, =[ g p (K, u) ( i)/ g p (L, u) i/ (k j)/(k i) ds (u)] S 1 [ g p (K, u) ( k)/ g p (L, u) k/ (j i)/(k i) ds (u)] S 1 ={ S 1 [g p (K, u) ( i)(k j)/(k i) g p (L, u) i(k j)/(k i) ] (k i)/(k j) (k j)/(k i) ds (u)} { S 1 [g p (K, u) ( k)(j i)/(k i) g p (L, u) k(j i)/(k i) ] (k i)/(j i) (j i)/(k i) ds (u)} g p (K, u) ( j)/(k i) g p (L, u) j/ ds (u) S 1 = G p,j (K, L). (82) G p,i (K, L) k j G p,k (K, L) j i G p,j (K, L) k i. (83) We obtai iequality (81). By the coditio of equality i Hölder s iequality, the equality holds i (81) if ad oly if, for ay u S 1, g p (K, u) ( i)/ g p (L, u) i/ g p (K, u) ( k)/ g p (L, u) k/ (84) is a costat; that is, g p (K, u)/g p (L, u) is a costat for ay u S 1. By the same argumet i the proof of Theorem 11, we coclude that equality holds if K ad L are dilates of each other. Takig L = B i Theorem 16 ad usig (74), we immediately obtai the followig. Corollary 17. For K S o, p 1, i, j, k R, adi<j<k, the k j G p,k (K) j i G p,j (K) k i, (85) with equality if K is a ball cetered at the origi. The, the followig Mikowski iequality for the dual ith L p -mixed geomiimal surface area will be obtaied. Theorem 18. For K, L S o, p 1,adi Rad the for i<0or i>, G p,i (K, L) G p (K) i G p (L) i, (86) ad for 0<i<, G p,i (K, L) G p (K) i G p (L) i. (87) Each iequality holds as a equality if K ad L are dilates of each other. For i=0or i=,(86)(or(87))isidetical. Proof. (i) For i<0,let(i, j, k) = (i, 0, ) i Theorem 16; we obtai G p,i (K, L) G p, (K, L) i G p,0 (K, L) i, (88) with equality if K ad L are dilates of each other. From (80), we ca get G p,i (K, L) G p (K) i G p (L) i, (89) with equality if K ad L are dilates of each other. (ii) For i >,let(i,j,k) = (0,,i) i Theorem 16; we obtai G p,0 (K, L) i G p,i (K, L) G p, (K, L) i, (90) with equality if K ad L are dilates of each other. From (80), we ca also get iequality (86). (iii) For 0<i<,let(i, j, k) = (0, i, ) i Theorem 16; we obtai G p,0 (K, L) i G p, (K, L) i G p,i (K, L), (91) with equality if K ad L are dilates of each other. From (80), we ca get iequality (87). (iv) For i=0(or i=), by (80), oe ca see (86) (or (87)) is idetical. Let L=Bi Theorem 18, G p (B) = ω,ad(74)will lead to the followig. Corollary 19. For K S o, p 1,adi Rad the for i<0or i>, ad for 0<i<, (ω ) i G p (K) i, (92) (ω ) i G p (K) i. (93) Each iequality holds as a equality if K is a ball cetered at the origi. For i=0or i=,(92)(or(93))isidetical. Now we will give a exteded form of iequality (24) as follows. Theorem 20. If K S o, p 1, i R,adi 0,the ω (( p)i p)/2 V (K) (+p)( i)/2, (94) with equality if ad oly if K is a ellipsoid cetred at the origi.

9 Joural of Fuctio Spaces 9 Proof. By iequalities (92) ad (24), we ca immediately obtai iequality (94). As the extesio of iequality (25), we obtai a aalogue of Blaschke-Sataló iequalityforthedualith L p -mixed geomiimal surface area. Theorem 21. If K, L S o, p 1, i R,ad0 i,the G p,i (K, L) G p,i (K,L ) (ω ) 2, (95) ad equality holds for 0<i<if K ad L are dilated ellipsoids of each other cetered at the origi. The iequality holds as a equality for i=0(or i=)ifk (or L) isaellipsoidcetered at the origi. Proof. Give (87) together with (25), it follows that G p,i (K, L) G p,i (K,L ) ( G p (K) G p (K )) ( i)/ ( G p (L) G p (L )) i/ (ω ) 2( i)/ (ω ) 2i/ =(ω ) 2. (96) The equality holds for 0 < i < if K ad L are dilated ellipsoids of each other. The iequality holds as a equality for i=0(or i=)ifk (or L)isaellipsoid. Recall Ye s isoperimetric iequality (see [33]): If K S o, p (0, ) (, ),adtheduall p -surface area S p (K) = V p (K, B),the S p (K) S p (B) O the other had, if p (,0),the S p (K) S p (B) (V (K) V (B) ) (+p)/. (97) (V (K) V (B) ) (+p)/. (98) The equality i every iequality holds if ad oly if K is a origi-symmetric Euclidea ball. We ow establish geeralized isoperimetric iequalities for. Theorem 22. If K S o, p 1,adi R, thewehavethe followig. (i) If i 0, G p,i (B) 2 (+p)( i)/ (K) (V V (B) ), (99) with equality if K is a ball cetered at the origi. (ii) If i, G p,i (B) 2 (+p)( i)/ (K) (V V (B) ), (100) with equality if K is a ball cetered at the origi. Proof. (i) For i=0,by(79)ad(24),itfollowsthat G p (K) (+p)/ (K) (V G p (B) V (B) ). (101) ThisisWag siequality(24). For i =, by (74), (79), ad (80), the equality holds trivially i (100). For i<0,sice G p,i (B) = G p (B) = ω, by (24), (92), ad (94), we have ( G ( i)/ p (K) G p,i (B) G p (B) ). (102) Hece, for i < 0 ad p 1,theL p -affie isoperimetric iequalities (92) ad (24) imply that ( G ( i)/ p (K) G p,i (B) G p (B) ) 2 (+p)( i)/ V (K) ( V (B) ), (103) with equality if K is a ball cetered at the origi. (ii) For i=, by (74), (79), ad (80), the equality holds trivially i (100). We ow prove the case i>.iequality(93) ad the defiitio of dual L p -geomiimal surface area give the followig: ( G ( i)/ p (K) G p,i (B) G p (B) ) 2 (+p)( i)/ V (K) ( V (B) ), with equality if K is a ball cetered at the origi. The followig results are iterestig. (104) Theorem 23. Let K S o, p 1, i R,ad0 i ;the, ( S ( i)/ p (K) G p,i (B) S p (B) ), (105) with equality if K is a ball cetered at the origi. Proof. Iequality (93) ad the defiitio of dual L p - geomiimal surface area give the followig: i G ( G p,i (B) ) p (K) ( ) ω ( Ṽ i p (K, B) V p (B, B) ) i S p (K) =( S p (B) ), with equality if K is a ball cetered at the origi. (106)

10 10 Joural of Fuctio Spaces Competig Iterests The authors declare that they have o competig iterests. Ackowledgmets This work is supported by the Natioal Natural Sciece Foudatio of Chia (Grat o ), the Sciece ad Techology Pla of Gasu Provice (Grat o. 145RJZG227), the Youg Foudatio of Hexi Uiversity (Grat o. QN ), ad partly the Natioal Natural Sciece Foudatio of Chia (Grat o ). Refereces [1] C. M. Petty, Geomiimal surface area, Geometriae Dedicata, vol. 3, pp , [2] E. Lutwak, Exteded affie surface area, Advaces i Mathematics,vol.85,o.1,pp.39 68,1991. [3] E. Lutwak, The Bru-Mikowski-Firey theory II: affie ad geomiimal surface areas, Advaces i Mathematics, vol.118, o. 2, pp , [4] B.Zhu,J.Zhou,adW.Xu, L P mixed geomiimal surface area, Joural of Mathematical Aalysis ad Applicatios,vol.422,o. 2, pp , [5]W.D.WagadC.Qi, L p -dual geomiimal surface area, Joural of Iequalities ad Applicatios, vol. 2011, p. 6, [6] E. Lutwak, Itersectio bodies ad dual mixed volumes, Advaces i Mathematics, vol. 71, o. 2, pp , [7] E. Lutwak, Dual mixed volumes, Pacific Joural of Mathematics,vol.58,o.2,pp ,1975. [8] A. Berig, The isoperimetrix i the dual Bru-Mikowski theory, Advaces i Mathematics,vol.254,pp.1 14,2014. [9]P.Dulio,R.J.Garder,adC.Peri, Characterizigthedual mixed volume via additive fuctioals, Idiaa Uiversity Mathematics Joural,vol.65,o.1,pp.69 91,2016. [10] R. J. Garder, The dual Bru-Mikowski theory for bouded BORel sets: dual affie quermassitegrals ad iequalities, Advaces i Mathematics,vol.216,o.1,pp ,2007. [11] R. J. Garder ad S. Vassallo, Iequalities for dual isoperimetric deficits, Mathematika, vol. 45, o. 2, pp , [12] R. J. Garder ad S. Vassallo, Stability of iequalities i the dual Bru-Mikowski theory, Joural of Mathematical Aalysis ad Applicatios,vol.231,o.2,pp ,1999. [13] R. J. Garder ad S. Vassallo, The Bru-MINkowski iequality, MINkowski s first iequality, ad their duals, Joural of Mathematical Aalysis ad Applicatios,vol.245,o.2,pp , [14] E. Lutwak, Cetroid bodies ad dual mixed volumes, Proceedigs of the Lodo Mathematical Society,vol.60,o.2,pp , [15] Y. Li ad W. Wag, The L p -dual mixed geomiimal surface area for multiple star bodies, Joural of Iequalities ad Applicatios,vol.2014,o.1,article456,pp.1 10,2014. [16] E. Milma, Dual mixed volumes ad the slicig problem, Advaces i Mathematics,vol.207,o.2,pp ,2006. [17] T. Ma ad W. Wag, Dual Orlicz geomiimal surface area, Joural of Iequalities ad Applicatios, vol. 2016, article 56, [18] T. Ma, The miimal dual orlicz surface area, Taiwaese Joural of Mathematics,vol.20,o.2,pp ,2016. [19] G. Y. Zhag, Cetered bodies ad dual mixed volumes, Trasactios of the America Mathematical Society,vol.345,o. 2, pp , [20] B. Zhu, J. Zhou, ad W. Xu, Dual Orlicz-Bru-Mikowski theory, Advaces i Mathematics,vol.264,pp ,2014. [21] D. Zou ad G. Xiog, Orlicz legedre ellipsoids, Joural of Geometric Aalysis,vol.26,o.3,pp ,2016. [22] R. J. Garder, A positive aswer to the Busema-Petty problem i three dimesios, Aals of Mathematics,vol.140, o. 2, pp , [23] R. J. Garder, A. Koldobsky, ad T. Schlumprecht, A aalytic solutio to the Busema-Petty problem o sectios of covex bodies, Aals of Mathematics,vol.149,o.2,pp ,1999. [24] G. Zhag, A positive solutio to the Busema-Petty problem i R 4, Aals of Mathematics,vol.149,o.2,pp ,1999. [25] R. J. Garder, The Bru-Mikowski iequality, Bulleti of the America Mathematical Society,vol.39,pp ,2002. [26] W. J. Firey, p-meas of covex bodies, Mathematica Scadiavica,vol.10,pp.17 24,1962. [27] R. Scheider, Covex Bodies: The Bru-Mikowski Theory, Cambridge Uiversity Press, Cambridge, UK, 2d editio, [28] W. Rudi, Priciples of Mathematical Aalysis, McGraw-Hill, New York, NY, USA, [29] E.LutwakadG.Zhag, Blaschke-Sataló iequalities, Joural of Differetial Geometry,vol.47,o.1,pp.1 16,1997. [30] G. H. Hardy, J. E. Littlewood, ad G. Pólya, Iequalities, Cambridge Uiversity Press, Cambridge, UK, [31] R. J. Garder, Geometric Tomography, Cambridge Uiversity Press, New York, NY, USA, 2d editio, [32] M. Ludwig, C. Schütt, ad E. Werer, Approximatio of the Euclidea ball by polytopes, Studia Mathematica, vol.173,o. 1, pp. 1 18, [33] D. Ye, Dual Orlicz-Bru-Mikowski theory: Orlicz φ-radial additio, Orlicz L φ -dual mixed volume ad related iequalities,

11 Advaces i Operatios Research Advaces i Decisio Scieces Joural of Applied Mathematics Algebra Joural of Probability ad Statistics The Scietific World Joural Iteratioal Joural of Differetial Equatios Submit your mauscripts at Iteratioal Joural of Advaces i Combiatorics Mathematical Physics Joural of Complex Aalysis Iteratioal Joural of Mathematics ad Mathematical Scieces Mathematical Problems i Egieerig Joural of Mathematics Discrete Mathematics Joural of Discrete Dyamics i Nature ad Society Joural of Fuctio Spaces Abstract ad Applied Aalysis Iteratioal Joural of Joural of Stochastic Aalysis Optimizatio