Maximum and minimum operators of convex integrands

0.17 MB·

by  Huhe Han

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.
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GraduateSchoolofEnvironmentandInformationSciences,YokohamaNationalUni-
versity,Yokohama240-8501,Japan
E-mail address: [e-mail]