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MAXIMUM AND MINIMUM OPERATORS OF CONVEX INTEGRANDS 7 1 0 HUHEHAN 2 n Abstract. For given convex integrands γi :Sn → R+ (where i = 1,2), the a functions γ and γ can be defined as natural way. In this paper, we J max min showthattheWulffshapeofγ (resp. theWulffshapeofγ )isexactly max min 1 theconvexhullof(Wγ ∪Wγ )(resp. Wγ ∩Wγ ). 1 2 1 2 3 ] G M 1. Introduction . Let n be a positive integer. Given a continuous function γ : Sn → R+, where th Sn ⊂Rn+1 is the unit sphereandR+ is the setconsistingofpositiverealnumbers, a theWulff shape associated with γ,denotedbyW ,isthe followingintersection(see γ m Figure 1), [ W = Γ . γ γ,θ 1 θ∈Sn \ v Here, Γ is the following half-space: γ,θ 6 5 Γγ,θ ={x∈Rn+1 |x·θ ≤γ(θ)}, 9 where the dot in the center stands for the dot product of two vectors x,θ ∈Rn+1. 8 This construction is well-known as Wulff’s construction of an equilibrium crystal 0 . 1 0 7 1 : v i X r a Figure 1. A Wulff shape W . γ introduced by G. Wulff in [13] (for details on Wulff shapes, see for instance [2, 7, 10, 11, 12]) For a continuous function γ :Sn →R+, set graph(γ)={(θ,γ(θ))∈Rn+1−{0}|θ ∈Sn}, 2010 Mathematics Subject Classification. 52A20,52A55,82D25. Key words and phrases. convexintegrand,maximumandminimum,Wulffshape. 1 2 HUHEHAN where(θ,γ(θ)) isthe polarplotexpressionforapointofRn+1−{0}. The mapping inv : Rn+1 −{0} → Rn+1 −{0}, defined as follows, is called the inversion with respect to the origin of Rn+1. 1 inv(θ,r)= −θ, . r (cid:18) (cid:19) Let Γ be the boundary of the convex hull of inv(graph(γ)). If the equality Γ = γ γ inv(graph(γ)) is satisfied, then γ is called a convex integrand (see Figure 2). The Figure 2. A convex integrand and its inversion. notion of convex integrand was first introduced by J. Taylor in [11] and it plays a keyroleforstudyingWulffshapes(fordetailsonconvexintegrands,seeforinstance [1, 4, 7]). For givenconvex integrands γ1, γ2, define γ and γ as natural way. max min γ :Sn →R+,γ (θ)=max{γ1(θ),γ2(θ)}. max max γ :Sn →R+,γ (θ)=min{γ1(θ),γ2(θ)}. min min Then the question naturally arises “What is the relationship between W ,W γ γ 1 2 and W (or W )? How are they related ?”. The main result of this paper is γ γ max min as follows (see Figure 3). Theorem 1. Let γ and γ beconvex integrands. Then the following two equalities 1 2 are holds. (1) W =convex hull of the (W ∪W ). γ γ γ max 1 2 (2) W =W ∩W . γ γ γ min 1 2 This paper organized as follows. In section 2, the preliminaries are given, and the proof of Theorem 1 and related topics are given in section 3 and section 4 respectively. 2. Preliminaries 2.1. Spherical convex body. For any point P ∈ Sn+1, let H(P) be the hemi- sphere centered at P, e e H(P)={Q∈Sn+1 |P ·Q≥0}. e e e e e MAXIMUM AND MINIMUM OPERATORS OF CONVEX INTEGRANDS 3 Figure 3. AnillustrationofTheorem1. Lefttop: a Wulff shape W , righttop: a Wulff shape W , left bottom: the Wulff shape γ γ 1 2 W , right bottom : the Wulff shape W . γ γ max min wherethedotinthecenterstandsforthescalarproductoftwovectorsP,Q∈Rn+2. Foranynon-emptysubsetW ⊂Sn+1,thespherical polar setof W,denotedbyW◦, is defined as follows: e e f W◦ = H(P). f f e f P\∈W f e Definition 1 ([9]). Let W be a subset of Sn+1. Suppose that there exists a point P ∈Sn+1 such that W ∩H(P)=∅. Then, W is said to be hemispherical. f Let P,Q be two points of Sn+1 such that (1−t)P +tQ is not the zero vector e f e f for any t∈[0,1]. Then, the following arc is denoted by PQ: e e e e (1−t)P +tQ PQ= ∈Sn+1 0≤ete≤1 . (||(1−t)P +tQ|| (cid:12) ) e e (cid:12) ee (cid:12) (cid:12) Definition 2 ([9]). Let W ⊂Sn+1ebe aehemispheri(cid:12)cal subset. (1) Suppose that PQ⊂W for any P,Q∈W. Then, W is said to be spherical f convex. ee f e e f f 4 HUHEHAN (2) SupposethatW isclosed,sphericalconvexandhasaninteriorpoint. Then, W is said to be a spherical convex body. f Definition 3 ([9]). Let W be a hemisphericalsubsetofSn+1. Then, the following f set, denoted by s-conv(W), is called the spherical convex hull of W. f k t P k s-conv(W)= f i=1 i i P ∈W, t =1, t ≥0f,k ∈N . k i i i (||Pi=1tiPei|| (cid:12)(cid:12) Xi=1 ) Lemma 2.1 ([f9]). For anPy hemisphe(cid:12)(cid:12)riceal sufbset W of Sn+1, the spherical convex e (cid:12) hull of W is the smallest spherical convex set containing W. f Lemmaf2.2 ([9]). Let W1,W2 ⊂ Sn+1. If W1 is a subfset of W2, then W2◦ is a subset of W◦. 1 f f f f f Lemma 2.3 ([9]). For any subset W of Sn+1, the inclusion W ⊂W◦◦ holds. f Especially, we have the following proposition. f f f Proposition 1 ([9]). For any non-empty closed hemispherical subset W ⊂ Sn+1, the equality s-conv(W)=(s-conv(W))◦◦ holds. f Lemma 2.4 (Maehara’slemma ([6, 9])). For any hemispherical finite subset W = f f {P1,...,Pk}⊂Sn+1, the following holds: ◦ f k k t P e (||ePiki==11tiiPeii|| (cid:12)(cid:12) Pi ∈X, Xi=1ti =1, ti ≥0) =H(P1)∩···∩H(Pk). (cid:12) e e e P (cid:12) e (cid:12) Maeara’slemmawasfirstgivenin[6],whichisausefulmethodtostudyspherical Wulff shapes. For English details, see for instance [9]. 2.2. An equivalentdefinition. In[9],anequivalentdefinitionofWulffshapewas given, which composed by following mappings. 1. Mapping Id:Rn+1 →Rn+1×{1} The mapping Id defined by Id(x)=(x,1). 2. Central projection α :Sn+1 →Rn+1×{1} N N,+ Denote the point (0,...,0,1)∈Rn+2 by N. The set Sn+1−H(−N) is denoted by Sn+1. Let α : Sn+1 → Rn+1 ×{1} be the central projection relative to N, N,+ N N,+ namely, αN is defined as follows for any (P1,...,Pn+1,Pn+2)∈SNn+,+1: P1 Pn+1 αN(P1,...,Pn+1,Pn+2)= ,..., ,1 . (cid:18)Pn+2 Pn+2 (cid:19) Through out remainder of this paper, let X = α−1 ◦ Id(X) for any non-empty subset X of Rn+1. For any Wulfff shape W ⊂ Rn+1, the spherical convex body W = α−1◦Id(W) called spherical Wulff shaepe of W. Then the following two are equivalent. f (1) W is a spherical Wulff shape. (2) W is a spherical convex body such that W ∩ H(−N) = ∅ and N is an ifnterior point of W. f f f MAXIMUM AND MINIMUM OPERATORS OF CONVEX INTEGRANDS 5 3. Spherical blow-up Ψ :Sn+1−{±N}→Sn+1 N N,+ Next, we consider the mapping Ψ :Sn+1−{±N}→Sn+1 defined by N N,+ 1 Ψ (P)= (N −(N ·P)P). N 1−(N ·P)2 ThemappingΨN wasfirsetintropducedin[8],hasthefolloweinegintriguingproperties: (1) For any P ∈Sn+1−{±N}, the equality P ·Ψ (P)=0 holds, N (2) for any P ∈Sn+1−{±N}, the property Ψ (P)∈RN +RP holds, N (3) for any Pe∈Sn+1−{±N}, the property Ne ·Ψ (Pe)>0 holds, N (4) the restreiction Ψ | : Sn+1−{N}→eSn+1−{N} ies a C∞ diffeo- N Sn+1−{N} N,+ N,+ N,+ morphisme. e 4. Spherical polar transform (cid:13):H◦(Sn+1)→H◦(Sn+1) Let H(Sn+1)be the set consistingofnon-empty compactset ofSn+1. Itis clear thatthesphericalpolarsetofSn+1istheemptyset. LetH◦(Sn+1)bethesubspace of H(Sn+1) defined as follows. H◦(Sn+1)={W ∈H(Sn+1)|W◦ 6=∅}. The spherical polar transform (cid:13) : H◦(Sn+1) → H◦(Sn+1), defined by (cid:13)(W) = f f W◦. Since W ⊂ W◦◦ for any W ∈ H◦(Sn+1), by Lemma 2.3, we know that W◦ ∈ H◦(Sn+1) for any W ∈ H◦(Sn+1). Thus, the spherical polar transform isfwell- dfefined. Itfis knfown that thefspherical polar transform is Lipchitz with refspect to the Pompeiu-Hafusdorff distance([3]). Moreover, the restriction of the spherical polar transform to the set consisting of spherical Wulff shape relative to P (see Definition 5) is an isometry with respect to the Pompeiu-Hausdorff distance ([3]). e Proposition 2 ([9]). Let γ : Sn → R+ be a continuous function. Then, Wγ is characterized as follows: W =Id−1◦α Ψ ◦α−1◦Id(graph(γ)) ◦ . γ N N N Proposition 3 ([9]). For any W(cid:16)u(cid:0)lff shape W , the following(cid:1) s(cid:17)et, too, is a Wulff γ shape: Id−1◦α α−1◦Id(W ) ◦ . N N γ Definition 4 ([9]). For any Wulff sh(cid:16)a(cid:0)pe W , the Wu(cid:1)lff(cid:17)shape givenin Proposition γ 3 is called the dual Wulff shape of W , denoted by DW . γ γ By Proposition 3, the following definition is reasonable. Definition 5 ([9]). Let P be a point of Sn+1. (1) A spherical convex body W such that W ∩H(−P) = ∅ and P ∈ int(W) e are satisfied is called a spherical Wulff shape relative to P. (2) Let W be a sphericalWulfffshape relativfeto P. Theen, the setWe◦ is calfled the spherical dual Wulff shape of the spherical Wulff shaepe W relative to P, dfenoted by DW. e f f Proposietion4 ([9]). Letfγ :Sn →R+ bea convex integrand. Let DWγ bethe dual Wulff shape of W . Then the boundary of the DW is exactly Γ =inv(graph(γ)). γ γ γ 6 HUHEHAN The Proposition2,givesa newpowerfulsphericalmethodto study Wulff shape. For example, the self-dual Wulff shape W = DW can be characterized by the γ γ induced spherical convex body of constant width π/2([5]), for related topics about spherical method see also [3, 4], for instance. 3. Proof of Theorem 1 Proof of assertion (1). By Proposition 2, Proposition 4 and Maehara’s lemma, for Wulff shape of γ the following is hold. max W =Id−1◦α Ψ ◦α−1◦Id(graph(γ )) ◦ γmax N N N max =Id−1◦α (cid:16)((cid:0)∂(DW ∩DW ))◦ (cid:1) (cid:17) N γ γ 1 2 (cid:16) (cid:17) (∗) =Id−1◦αN (DWγf∩DWγf)◦ 1 2 (cid:16) (cid:17) where ∂(DW ∩DW ) is the boundafry of (DWf ∩DW ). Thus it is sufficient γ γ γ γ 1 2 1 2 to prove the following. f f f f (DW ∩DW )◦ =s-conv(W ∪W ). γ γ γ γ 1 2 1 2 First, we show that f(DWγ ∩fDWγ )◦ is a sufbset off s-conv(Wγ ∪Wγ ). By 1 2 1 2 Maehara’s lemma, the following equality hold. f f ◦ f f (W ∪W )◦ = s-conv(W ∪W ) . γ γ γ γ 1 2 1 2 (cid:16) (cid:17) By Lemma 2.2, it is suffficienft to prove that (Wf ∪Wf )◦ is a subset of (DW ∩ γ γ γ 1 2 1 DW ). Let P be a point of (W ∪W )◦. Then it follows that, γ γ γ 2 1 2 f f f f e Wγ f⊂H(Pf) and Wγ ⊂H(P). 1 2 Thus P is a point of Wγ◦1f∩Wγ◦2 =DeWγ1 ∩fDWγ2. e Nexet, we show thaft s-confv(Wγ ∪Wfγ ) is afsubset of (DWγ ∩DWγ )◦. Since 1 2 1 2 DW ∩DW is a subset of DW , we have that γ γ γ 1 2 f 1 f f f f f (DfWγ1 ∩DWγ2)⊂Wγ◦1 By Lemma 2.2, f f f W =W◦◦ ⊂(DW ∩DW )◦. γ1 γ1 γ1 γ2 In the same way, it follows that f f f f W ⊂(DW ∩DW )◦. γ γ γ 2 1 2 Since s-conv(Wγ ∪Wγ ) isfthe smallefst convexfbody containing Wγ ∪Wγ and 1 2 1 2 (DW ∩DW )◦ is a convex body, it follows that γ γ 1 f2 f f f f f s-conv(Wγ ∪Wγ )⊂(DWγ ∩DWγ )◦. 1 2 1 2 f f f f MAXIMUM AND MINIMUM OPERATORS OF CONVEX INTEGRANDS 7 Proof of assertion (2). By Proposition 2, Proposition 4 and Maehara’s lemma, for Wulff shape of γ the following hold. min W =Id−1◦α (Ψ ◦α−1◦Id(graph(γ )))◦ γmin N N N min =Id−1◦α (cid:0)(∂(DW ∪DW ))◦ (cid:1) N γ γ 1 2 (cid:16) (cid:17) (∗∗) =Id−1◦αN (DWγf∪DWγf)◦ . 1 2 Thus it is sufficient to prove the fol(cid:16)lowing. (cid:17) f f (DW ∪DW )◦ =W ∩W . γ γ γ γ 1 2 1 2 First, we show that (DW ∪DW )◦ is a subset of W ∩W . Let P be a fγ1 fγ2 f f γ1 γ2 point of (DW ∪DW )◦. Then DW is a subset of H(P). It follows that γ γ γ 1 2 f f 1 f f e P ∈(DW )◦ =W f f f γ1 γ1 e Inthesameway,wehavethatP isapointofW . ThusP isapointofW ∩W . e f γf2 γ1 γ2 Next,weshowthatW ∩We isasubsetoff(DW ∪DeW )◦. LetPfbeapfoint γ γ γ γ 1 2 1 2 of W ∩W . Since W and W are spherical convex bodies, by Proposition 1, γ γ γ γ it follo1ws tha2t f1 f 2 f f e f f f f P ∈W = H(Q) and P ∈W = H(Q). γ γ 1 2 e f e f Q∈D\Wγ1 Q∈D\Wγ2 e f e e f e This implies DW ⊂H(P) and DW ⊂H(P). γ γ 1 2 Then it follows that DW ∪DW is a subset of H(P). By definition of spherical γ γ f1 2 e f e polar set, P is a point of (DW ∪DW )◦. (cid:3) γ γ f 1f 2 e Remark: Usually, for given convex integrands γ and γ , the inversion of graph(γ e) with respect tofthe origfin of Rn+1 is doe1s not eq2uivalent to Γ . min γmin Thus the function γ is not convex integrand in general case.On the other hand, min the inversion of graph(γ ) with respect to the origin of Rn+1 is exactly Γ , max γmax this implies γ is a convex integrand. max 4. More topics on maximum and minimum operators For given convex integrands γ1 and γ2, by (∗) and (∗∗) of proof of Theorem 1, we have the following relation between DW , DW and W , (resp. DW , γ γ γ γ 1 2 max 1 DW and W ). γ γ 2 min (1) W =D(DW ∩DW ). γ γ γ max 1 2 (2) W =D convex hull of (DW ∪DW ) . γ γ γ min 1 2 Since DW ∩DW is a subsetofthe convex hull of (DW ∪DW ), by Lemma γ γ(cid:0) (cid:1) γ γ 1 2 1 2 ◦ 2.2, it follows that s-conv (DW ∪DW ) is a subset of (DW ∩DW )◦. γ γ γ γ 1 2 1 2 This implies Wγ (cid:16)is a subset of dual of the(cid:17)Wγ . Moreover,as a corollary, we have the followinmginresult. f f max f f Corollary 1. Let W ,W be dual Wulff shapes. Then W is the dual Wulff γ γ γ 1 2 max shape of W . γ min 8 HUHEHAN ◦ Proof. Since (W ∪W )◦ and s-conv(W ∪W ) are same convex body, by γ γ γ γ 1 2 1 2 Theorem 1, it is sufficient to prov(cid:16)e that (cid:17) f f f f (W ∪W )◦ =W ∩W . γ γ γ γ 1 2 1 2 LetP be a pointof(W ∪W )◦. ThenW ∪W is a subsetofH(P). Itfollows γ1 fγ2 f γf1 γ2f that P is a point of W◦ ∩W◦ . Since W is the dual of W , we have that e fγ1 fγ2 fγ1 f γ2 e P ∈W◦ ∩W◦ =W ∩W . e f f γ1 fγ2 γ2 γ1 f On the other hand, since Wulff shapes W ,W are duals, we have that W ∪ e f f γf1 γ2f γ1 W is a subset of H(P), for any point P of W ∩W . This implies P is a point γ γ γ 2 1 2 f of (W ∪W )◦. (cid:3) γ γ f 1 2 e e f f e Acknowledgements f f The author wouldlike to express his sincere appreciationto Takashi Nishimura, for his kind advice. References [1] E. B. Batista, H. Han and T. Nishimura, Stability of C∞ convex integrands, to appear in KyushuJournalofMathematics. [2] Y.Giga,Surface Evolution Equations,MonographsofMathematics,99,Springer,2006. [3] H. Han and T. Nishimura, The spherical dual transform is an isometry for spherical Wulff shapes, toappearinStudiaMathematica. [4] H.HanandT.Nishimura,StrictlyconvexWulffshapesandC1 convexintegrands,toappear inProccedingsofAMS. [5] H. Han and T. 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Wulff, Zur frage der geschwindindigkeit des wachstrums und der auflo¨sung der krys- tallflachen,Z.KristallographineundMineralogie,34(1901),449–530. GraduateSchoolofEnvironmentandInformationSciences,YokohamaNationalUni- versity,Yokohama240-8501,Japan E-mail address: [email protected]