Theorems of Helly’s Type for Partially-closed Half-spaces
1 Introduction
The well-known classical Helly’s Theorem(see[1]),which states that all members in a finite family of convex sets in the Euclidean space Rnhave a common point if arbitrary n+1 members in the family have a common point,is very profound and important in convex geometry and combinatorial geometry.Also it turns out that Helly’s Theorem has various elegant applications in many branches of mathematics(see[2]–[4]).For its importance,Helly’s theorem has gained much attention since it was found.Various kinds of extensions or generalizations of Helly’s Theorem were proved and the corresponding applications were established in the past years(see[5]–[8]).We refer readers to Danzer’s seminal paper[9]for general references.
There are examples showing that the finiteness of the family in Helly’s Theorem cannot be omitted in general.So,among the extensions or generalizations of Helly’s Theorem,one attempt is to find the conditions(for convex sets)under which Helly’s Theorem holds for in finite families.For instance,Helly’s Theorem holds for(not necessarily finite)families of compact convex sets(see[10]and[11]).
Theorems of Helly’s type for half-spaces with particular conditions appear often in many branches of mathematics.For instance,it was shown in[10]that Helly’s Theorem holds for families of closed half-spaces with the condition that the intersection of some finitely many members in the family is compact,from which the existence of center for Borel probabilities is con firmed,and also Theorem 3.1 in[12](see also[13])is in principle a theorem of Helly’s type in analytic form for families of closed half-spaces with another condition.In this paper,we present,under quite weaker conditions,theorems of Helly’s type for(not necessarily finite)families of partially closed half-spaces(see below for definition)in both analytic and geometric forms,along with examples showing that these conditions cannot be omitted in general.
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2 Notation and Definitions
We work with the Euclidean space Rnand do not distinguish points and vectors in Rn intentionally.Let Kndenote the family of all convex bodies(i.e.,compact convex sets with nonempty interior)in Rn.For any subset S⊂Rn,conv(S),cone(S),lin(S)denote the convex hull,the convex conical hull and the linear hull of S,respectively.A map T:Rn→ Rmis called affine if
![pagenumber_ebook=24,pagenumber_book=118](https://qkk.lw71.com/uploads/image/2018/2/ef37ebb99be84ecc24ab7b3c71b4c607.png)
In particular,an affine map f:Rn → R is called an affine function.It is known that f:Rn→R is affine if and only if
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![pagenumber_ebook=24,pagenumber_book=118](https://qkk.lw71.com/uploads/image/2018/2/d2bcefd8aee867e68ceb2d2e9d98edf9.png)
for some unique u∈ Rnand b∈ R,where〈·,·〉denotes the classical inner product.An affine function f(·)= 〈u,·〉+b is called non-trivial(or invertible)if uo,the origin or the zero vector.The set of affine functions on Rnis denoted by aff(Rn).In this paper,affine functions always mean the ones in aff(Rn)unless other mentioned.
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For ou ∈ Rnand b∈ R,the set Hu,b:={x ∈ Rn|〈u,x〉=b}is a hyperplane,and the set
and
are called closed(resp.open)half-spaces with outer and inner normal u respectively.It is trivial that
etc.,where the affine function
![pagenumber_ebook=24,pagenumber_book=118](https://qkk.lw71.com/uploads/image/2018/2/d8047c244b73cae43adb63e2a437edac.png)
Definition 2.1 For L⊂Hu,b,the set is called a partially-closed half-space with outer(resp.inner)normal u.We adopt
and
as generic symbols,to denote a partially-closed half-space(without specified L).
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(4)⇒ (1).Suppose that lin{ui0,···,uil}=cone{ui0,···,uil}with affinely independent ui0,···,uilfor some 1 ≤ l≤ n.Thenwhere αij≥ 0 with at least one αijpositive.Thus,setting
by Lemma 3.1 we have
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A familyof partially-closed half-spaces is called compactly-located if there are αi> 0,i∈ Λ,such that{αi(ui,bi)}i∈Λ ⊂ Rn+1 is compact(such a concept is suggested by the fact that
for any α >0),and well-located if int
where “int” denotes the interior.
3 Theorems of Helly’s Type for Half-spaces
In this section,we show theorems of Helly’s type for families of partially-closed half-spaces in both analytic and geometric forms.We call a family{fi|i∈Λ}of invertible affine functions compactly-located(resp.well-located)if the family{{fi≤0}|i∈ Λ}is compactly-located(resp.well-located).
where αik+λβik≥ 0(0≤ k ≤ l−1)and at least one of them is positive,a contradiction to the choice of l.
学生于学习过程中所具有的良好科学态度、责任,即为高中学生可于明确了解事物科学本质后,学生自身应有的科学实验态度.因此,高中物理教师于物理实际教学活动中应将培养学生良好的科学态度作为重要工作内容,使学生具备责任感,促进高中学生核心素养发展.高中物理教师可于实际物理教学活动中为学生讲解同教学内容相关物理学家的生平,使学生于扩展自身视野的同时也可提升学生物理学习兴趣.如以高一人教版《物理》中《牛顿第一定律》一课为例,物理教师于讲述教学内容时,可向学生讲解牛顿挣脱传统观念束缚的过程,充分吸引学生课堂注意力,同时也可于潜移默化中培养学生养成良好的科学态度.
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(1)
(2)For each u ∈ Rn,〈ui,u〉≤ 0 for some i∈ Λ;
(3)o ∈ ri(conv{ui0,···,uil})for some affinely independent ui0,···,uil,1 ≤ l≤ n,where “ri” denotes the relative interior,i.e.,there are positive αiksuch that
(4)cone{ui0, ···,uil}=lin{ui0, ···,uil}for some affinely independent ui0, ···,uil,1≤l≤n.
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Remark 3.1 i)Proposition 3.1 is an extension of Theorem 3.1 in[12],where{fi|i∈ Λ}is assumed to be a finite family and fi∈ {f ∈ aff(Rn)|f(K)=[0,1]}for some convex body K.
ii)The family of closed half-spaces in Proposition 3.1 can be replaced by a family of partially-closed half-spaces(see the following Theorem 3.1).
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In order to prove Proposition 3.1,we need the following lemma.
Lemma 3.1 Let fi(·)= 〈ui,·〉+bi,i=0,1,···,m,be non-trivial affine functions.Then,for any(um+1,bm+1)∈ cone{(u0,b0),···,(um,bm)},we have
![pagenumber_ebook=25,pagenumber_book=119](https://qkk.lw71.com/uploads/image/2018/2/e9d5fa1d8e9033161280e13de359c746.png)
Proof. If then the equality is obvious;if
then this is just a reformulation of the well-known generalized Farkas lemma(see[14],p.60).
Proof of Proposition 3.1 Without loss of generality,we may clearly assume that{(ui,bi)}i∈Λ is compact.
Thus,sincewe have
with
,αik ≥ 0,which leads to
where αi0:=1+
> 0.We now let l be the smallest positive integer such that
(1)⇒(2).LetAssume that there is a u ∈ Rnsuch that〈ui,u〉> 0 for each i.Then,by the continuity of〈·,u〉and by the compactness of{ui}i∈Λ ⊂ Rnand{bi}i∈Λ ⊂ R(for{(ui,bi)}i∈Λ ⊂ Rn+1being compact),there exist constants δ> 0 and M > 0 such that〈ui,u〉≥ δ,|bi|≤ M for all i and in turn
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/a771708bb081c7a4036cbd2d2d188c97.png)
for all i and λ > 0.Thus,for any λ > δ/M,a contradiction to(1).
(2) ⇒ (3). Consider the linear space If V={o},then by the separation theorem for closed cones(cone{ui}i∈Λ is closed since{ui}i∈Λ is compact),there is a u ∈ Rnsuch that{ui}i∈Λ ⊂ {x| 〈u,x〉> 0},which contradicts(2).So dimV≥1.Now,it is easy to see by induction on the dimension of V that V=cone{ui0,···,uis}for some ui0,···,uis(1 ≤ s ≤ n).
根据目前的船舶进出口计划时间的安排原则,如果每艘船舶都准时离泊和靠泊,那么在离泊出口过程中就不会在港内与其他大型船舶产生相互影响。但随着洋山港货物吞吐量的不断增加,出口船舶因无法及时完货而延迟开航的情况逐渐增多,船舶一旦发生延迟开航,其离泊操纵可能会与驶经该泊位的其他大型船舶相互产生不利影响。
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/15098c2b25ff1ce3265b936c2eeb5663.png)
for some ui0,···,uilwith αik > 0 and claim that ui0,···,uilare affinely independent.Suppose that ui0,···,uilare not affinely independent.Then for some(not all zero) βi0,···,βilwith
Without loss of generality,let
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/74f61771855d9ca33b477b2eae92b63b.png)
Then
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/655f76aa4e723750cc52f86079163201.png)
The following proposition is in principle a theorem of Helly’s type in analytic form.
It follows from(3.1)that
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/608436856bec34350903ef5248e748ab.png)
l≤ n clearly since ui0,···,uilare affinely independent.
(3)⇒ (4).Supposethatui0,···,uil,l≤ n,areaffinely independentand o∈ri(conv{ui0,···,uil}),i.e.,with αik > 0.Then for each 0≤ k ≤ l,we have
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/20c780e84506c5efac97710aacef6896.png)
Thus
![pagenumber_ebook=26,pagenumber_book=120](https://qkk.lw71.com/uploads/image/2018/2/e77ba241dc81856e52a90ee85bf64ec7.png)
Clearly,when L=Ø(resp.L=Hu,b),we get back the open half-spaceor
(resp.the closed half-space
However,a partially-closed half-space may not even be convex since L may not be.
![pagenumber_ebook=27,pagenumber_book=121](https://qkk.lw71.com/uploads/image/2018/2/96fe2d4c1aef0d6f1d5e865214745141.png)
where we used the fact that{fi0≤ 0}{fil+1≤ 0}= Ø since{fil+1≤ 0}⊂ {fi0> 0}following from that writingand inf(resp.sup)f(K):=inf(resp.sup)f(x)for x∈Kx∈K f ∈ aff(Rn),we have intK
Ø,inf fi0(K)≥ 0,{fi0=supfi0(K)}={fil+1=inf fil+1(K)}and
![pagenumber_ebook=27,pagenumber_book=121](https://qkk.lw71.com/uploads/image/2018/2/d9e7a9cc0ca7d7b21d64d7c6091c766f.png)
Therefore,
![pagenumber_ebook=27,pagenumber_book=121](https://qkk.lw71.com/uploads/image/2018/2/3b49f273c3da728795c6dc4c66c3037d.png)
This completes the proof.
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Now,given L⊂{f=0},we denote{f≤L0}:={f<0}∪L.Notice that{f≤L0}={f< 0}when L= Ø,{f≤L0}={f≤ 0}when L={f=0}and{f≤L0}may not be convex since L may not be.Adopting such notation,we have the following extension of Proposition 3.1.
Theorem 3.1 Let{fi= 〈ui,·〉+bi}i∈Λ be compactly-located and well-located.Then the following statements are equivalent:
(1) for arbitrary Li⊂ {fi=0},i∈Λ;
(2)For each u ∈ Rn,〈ui,u〉≤ 0 for some i∈ Λ;
(3)The origin o ∈ ri(conv{ui0,···,uil})for some affinely independent ui0, ···,uil,1 ≤ l≤ n,i.e.,there are positive αiksuch that
(4)cone{ui0,···,uil}=lin{ui0,···,uil}for some affinely independent ui0,···,uil,1≤l≤n.
The proof of Theorem 3.1 follows from Proposition 3.1 and the following proposition.
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Proposition 3.2 Under the conditions in Proposition 3.1 or Theorem 3.1,Ø if and only if
and in turn if and only if
for arbitrary Li⊂ {fi=0},i∈Λ.
Proof.Sincefor arbitrary Li⊂{fi=0}, i∈Λ,and
implies
trivially,we need only to show that if
Now,suppose,i.e.,there is
We point out first that,for
there is a δ> 0 such that fi(x∗)≥ δ for i∈ Λ,which follows from the continuity of fi(x∗)on(ui,bi),the compactness of{(ui,bi)}i∈Λ and fi(x∗) > 0 for all i∈Λ(since x∗∈int
We then have,for λ<0,
![pagenumber_ebook=28,pagenumber_book=122](https://qkk.lw71.com/uploads/image/2018/2/be0a50c46a6d85ff9698596d06819fc7.png)
In this section,we discuss the conditions appearing in Proposition 3.1,Theorems 3.1 and 3.2.First,we show by example that the“compactly-located”and “well-located”conditions cannot be omitted in general in Proposition 3.1,Theorems 3.1 and 3.2.
Now,as a consequence of Theorem 3.1,we have the following theorem of Helly’s type for partially-closed half-spaces.For simplicity,we write for
Theorem 3.2 Let familybe compactly-located and well-located,Li ⊂ Hiand
for arbitrary i1,i2,···,in+1 ∈ Λ,then we have
![pagenumber_ebook=28,pagenumber_book=122](https://qkk.lw71.com/uploads/image/2018/2/939178c09bf1be5a0d9d8d2116dce8b2.png)
Proof.SupposeThen by the proof of(1)⇒ (3)in Theorem 3.1,there are affinely independent ui0,ui1,···,uiland positive αi0,αi1,···,αil,1 ≤ l≤ n,such that
Thus,by the proof of(3)⇒ (1)in Theorem 3.1 again,we have
![pagenumber_ebook=28,pagenumber_book=122](https://qkk.lw71.com/uploads/image/2018/2/ce60b5f8cc65f7e3efeaa558905960c2.png)
a contradiction since l≤n.The proof is completed.
Proposition 3.1 Let{fi= 〈ui, ·〉+bi}i∈Λ be compactly-located and well-located.Then the following statements are equivalent:
4 Further Discussions
which leads to fi((1−λ)x0+λx∗)→ −∞ uniformly for i∈ Λ.Thus,there is λ0< 0 such that fi((1− λ0)x0+ λ0x∗)< 0 for all i∈ Λ,i.e.,and so
The proof is completed.
Example 4.1 In R2,let uθ=(−cosθ,−sinθ),bθ=1,θ∈ Λ :=(0,π).Then the familyis well-located but not compactly-located.Now it is easy to see that
for arbitrary θ0, θ1, ···, θnbut
This example shows that the “compactlylocated”condition cannot be omitted in general in Proposition 3.1,Theorems 3.1 and 3.2.
Example 4.2 In R2,let u1=(1,1),u2=(1,−1),u3=(0,−1),b1=b2=b3=0 and fi(·):= 〈ui,·〉+bi,i=1,2,3(observing that the family{f1,f2,f3}is not welllocated since intThen(0,0)∈int(cone{u1,u2,u3})(also cone{u1,u2,u3}=lin{u1,u2,u3}),however,
This example shows that,in Theorem 3.1,the“well-located”condition cannot even be replaced by
Next example shows that the“well-located”condition cannot be replaced by in Theorem 3.2 either.
Example 4.3 In R2,let u0+=(1,0),u0− =(−1,0),b0+=b0− =0,bi+=bi− =
,i=1,2,···,and
bi−,i=0,1,2,···.Then it is easy to see that the family
is compactly-located.Also it is easy to check that
![pagenumber_ebook=29,pagenumber_book=123](https://qkk.lw71.com/uploads/image/2018/2/245519e79f9d417e252ecffe38b90bbf.png)
and so inti.e.,the family F is not well-located.Now,for arbitrary
∈F(where±is specified),
![pagenumber_ebook=29,pagenumber_book=123](https://qkk.lw71.com/uploads/image/2018/2/08f57b3f91584664a507c1dfc592b99e.png)
where j=max{j1,j2,j3}.However,since for any(a,b)∈R2,0}for each j with j>b,we have
![pagenumber_ebook=29,pagenumber_book=123](https://qkk.lw71.com/uploads/image/2018/2/379df51797315861d24d15432ba8125a.png)
At the end,we point out that without the “well-located” condition,the following conclusion still holds.
Theorem 4.1 Let{fi= 〈ui,·〉+bi}i∈Λ be a compactly-located family and the statements
(1),(2),(3),(4)be the same as in Theorem 3.1.Then,(1)⇒(2)⇔(3)⇔(4).
Proof. The arguments for(1)⇒(2)⇒(3)⇒(4)are exactly the same as those for Theorem 3.1.So we need only to show(4) ⇒ (2).Suppose that cone{ui0,···,uil}=lin{ui0,···,uil}for some affinely independent ui0,···,uil,1 ≤ l≤ n.If there exists u ∈ Rn such that〈ui,u〉> 0 for all i∈ Λ,in particular,〈uik,u〉> 0 for 0≤ k≤ l,then,observingfor some α ≥ 0 with at least one α > 0 since lin{u,···,u}=ij ij0 i0il cone{ui0,···,uil},we have 〈−ui0,u〉= −〈ui0,u〉< 0 and
a contradiction.The proof is completed.
Remark 4.1 As expected,since a convex body in Rncan be expressed as the intersection of some(closed)half-spaces,the theorems obtained in this paper will play roles in the study for convex bodies.However,we leave this topic to other papers.
泥巴又飞奔去了远方,买回一个书包。左小龙把零件们都放到了书包里,拉上拉链,把手在地上搓了搓,推着摩托车,泥巴也在旁边跟着扶着,两人艰难的花了几个小时把摩托车推到了修车铺。在修车铺的门口,左小龙说:“我走不动了,我们就在这里靠一靠。”
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