Review on spherically convex analysis I

GUO Qi

（School of Mathematical Sciences，SUST，Suzhou 215009，China）

Abstract： This is the first part of a brief survey of recent progress in spherically convex analysis. A very brief introduction is given on the history and developments in various aspects of the spherical convexity. Then it focuses on the basic definitions and properties of spherically convex sets including the binary operations and the classical transformations，the combinatorial and structural properties on the Euclidean unit sphere Sn-1. Several basic questions are also proposed.

Key words： spherical convexity；spherical binary operation；combinatorial property；structural property

1 Introduction and preliminaries

This article is the first part of a brief survey of recent advances in the so-called spherical convex analysis.The concept of spherically convex sets in the Euclidean spheres （called spherical spaces） appears as early as the one of convex sets in the Euclidean spaces （see，for example，Refs. [1-6] etc） while the concept of spherically convex functions emerges quite later （see Refs. [7-11] etc）. Against the situation of Euclidean convexity，the spherical convexity theory，even the part of spherically convex sets，develops very slowly due to the lack of analytic methods in the last century.

In the last decade，the spherical convexity regains a lot of attentions for the pure mathematical interests or more importantly for the requirements in applications，such as in spherical optimizations （Refs. [12-14]） or the like （Refs. [15-16]）. With the development，it becomes gradually an unavoidable topic to establish the systematic and fundamental theory of spherical convexity as basic tools in the study. Meanwhile，the investigations on other topics become active as well，such as the spherical convex geometric analysis （Refs. [17-21]） and the classical spherical convex geometry （Refs. [22-26]） and spherical covering/packing problems （Refs. [27-29]）.Also，it is on the way to establish the basic theory of spherical convex analysis after Refs. [10，30-31] where the authors adopt an analytic approach to investigate spherical convexity.

Here，we present mainly the recent progress in one part of the spherical convex analysis，focusing on the topics related to spherically convex sets. The readers may read Ref. [32] and the references therein for more details and information. Topics related to spherically convex functions and spherical convex geometry are left to the second part of this survey.

Now，we recall some necessary notations and definitions.

Rnand Sn-1denote the Euclidean n-space and the unit sphere in Rnrespectively. “〈·，·〉” and ||·|| denote the standard inner product and the norm induced by “〈·，·〉” on Rn，respectively. Often we also view Rnas an——————————affine space，so we will not distinguish vectors and points intentionally. The origin （or zero vector） of Rnis always denoted by the lettero. R，R+and R*+stand for the set of reals，nonnegative and positive reals respectively.

For non-zerou∈Rnandα∈R，Hu，αdenotes the hyperplane {x∈Rn|〈u，x〉=α}. The sets of the formare called respectively the open and the closed half space deter-mined byu，α（andare defined in a similar way）. Whenα=0，we write simplyHu，Hu+，instead of

Hu，0，，respectively. A set of the formV∩Sn-1is called ak-sphere （0≤k≤n-1），whereVis a （k+1）-

dimensional subspace of Rn. Observe that a 0-sphere is of the form {x，-x} for somex∈Sn-1. A set of the form:=∩Sn-1（resp. Su+:=Hu+∩Sn-1） is called a closed （resp. open） hemisphere （and Su-are defined in a similar way）. A set of the form∩withv≠±uis called a lune.

Nonzerox，y∈Rnare called （a pair of） antipodes （or antipodal） ify=-x. For other notations and terms refer to Ref. [33].

There were various （equivalent or inequivalent） definitions in history，which may lead to different families of spherically convex sets （Refs. [1-2，7]）. Here we will work with the one popularly used in the past two or three decades. A subsetC⊂Sn-1is called spherically convex （s-convex for brevity） ifRC:={tu|u∈C，t∈R+} is convex in Rn；if furtherCcontains no antipodes，then it is called a propers-convex set. This definition can be restated in geometric language as：C⊂Sn-1iss-convex if the short arc connectinguandvis contained inCwhenever nonantipodalu，v∈C. We point out that，in fact，almost all definitions for so-called spherically convex sets are given in geometric language before Refs. [10，30] （we will explain the reason for such a phenomena later）.

For thes-convexity we pick up to work with，we adopt the analytic definition given in Refs. [30] （just for propers-convex sets），[10] and [32] （for general cases）.

Forx，y∈Rn，definex+sy:=ρ（x+y），whereρ:Rn→Sn-1∪{o} is the radial projection defined as

which has the following properties：

（i）ρ◦ρ=ρ，where ◦denotes the composition of operators；

（ii）ρ（tx）=ρ（x），ρ（-x） =-ρ（x） forx∈Rnandt＞0；

（iii）ρ（x）=xif and only ifx∈Sn-1orx＝o.

The composition “+s” is called the spherical addition on Rn. In terms of spherical addition，the spherically convex combination composition was introduced in Refs. [10，30]：

Fork≥1，x1，x2，…，xk∈Rnand nonnegativeλ1，λ2，…，λkwith，define

called a spherically convex combination （ans-convex combination for brevity） ofx1，x2，…，xk. Instead，（s）（λx+（1-λ）y） is often written asλx+s（1-λ）y.

It is easy to see （Ref. [10] for a precise proof） that for non-antipodalu，v∈Rn（u=vis allowed），[u，v]s:={λu+s（1-λ）v|0≤λ≤1} is just the short arc connectingu，v. Observe also that [u，-u]s={o，u，-u} foru∈Sn-1.（u，v）s，（u，v]sand [u，v）sare defined in similar manners.

Definition 1A setC⊂Sn-1is calleds-convex ifwheneveru，v∈C，0≤λ≤1 with λu+（1-λ）v≠o.

Ans-convex set is called proper if it contains no antipodes.

Remark 1（i）The equivalence of Definition 1 and the definition mentioned previously is proved in Ref. [10].

A subset of Sn-1is called closed if it is closed with respect to the usual Euclidean topology on Sn-1. The family of closeds-convex sets（calleds-convex bodies usually） in Sn-1is denoted by（Sn-1）. Allk-spheres，closed hemispheres and lunes are in（Sn-1）. The family of proper closeds-convex sets in Sn-1is denoted byp（Sn-1）.It is easy to check that ifCi∈（Sn-1） （resp：p（Sn-1）），i∈I，then so is the nonempty intersection ∩i∈ICi.

The following facts were known and proved repeatedly by different authors （see，e.g. Refs. [10，32] and the references therein，or check them directly）：let subsetC⊂Sn-1. ThenC=RC∩Sn-1andCis a （closed）s-convex set if and only ifRCis a（closed） convex cone；Cis a propers-convex set if and only if the convex coneRCis pointed，i.e.RC∩（-RC）={o}.

Thanks to above facts，for ans-convex setC⊂Sn-1，we denote intC:=int（RC）∩Sn-1，riC:=ri（RC）∩Sn-1，bdC:=bd（RC）∩Sn-1，rbdC:=rbd（RC）∩Sn-1and clC:=cl（RC）∩Sn-1，called the interior，relative interior，boundary，relative boundary and closure ofCrespectively.

For a subsetS⊂Sn-1，its spherically convex hull （s-convex hull for brevity） is defined as

It is easy to show that cosSis the intersection ofs-convex sets containingS，i.e. the smallests-convex set containingS.

2 Compositions and operations on Sn-1

One of the main reasons why the study of spherical convexity develops so slowly is there are very few suitable compositions and operations on Sn-1. The Euclidean spaces are linear space，i.e. they are of rich algebraic structures，however，the spherical spaces even have no suitable “addition” on them. One may check that the spherical addition defined above is just communicative but not associative，i.e. it is even not a group composition.Generally，we will mention an even uninspiring conclusion later.

“You must always remain with me,” said the emperor. “You shall sing only when it pleases you; and I will break the artificial bird into a thousand pieces.”

To overcome the above difficulty，some authors started to look for useful compositions or operations on Sn-1，or other families of subsets. Refs. [34] and [35] studied such a topic fully and they showed that no operations between sets on Sn-1can be as good as those in Euclidean spaces （will be specified in Theorem 2）. Ref.[20] investigates the potentially suitable（e.g.，s-convexity-preserving at least）transformations on Sn-1and their applications：

Definition 2GivenT∈GL（Rn）（the general transformation group on Rn），we define:Sn-1→Sn-1by（u）:=，u∈Sn-1，called a classical transformation on Sn-1. The set of classical transformations is denoted by CT（Sn-1）.

For general （not necessarily invertible） linear transformationsT，the correspondingcan be defined in a similar manner （see Ref. [20] for details，or refer to the following Definition 3）. Each（no matterTis invertible or not） iss-convexity preserving，however，we will not discuss them in this paper. Here，for the later use，we present only the so-called spherical projection.

Let SV:=V∩Sn-1be ak-sphere （0≤k≤n-1）. Then，the spherical projectionPSV:Sn-1→is traditionally defined by （see Ref. [36]）

i.e.PSV（u） is the set of points nearestu，whereds（u，v）:=arccos〈u，v〉 is the intrinsic distance betweenuandv.It is easy to show that ifu∉V⊥∩Sn-1，whereV⊥denotes the orthogonal complement space ofV，thenPSV（u） is a singleton. Such a definition is of intuition but inconvenient in application，so we adopt the equivalent one given in Ref. [32]：

Definition 3Let SV:=V∩Sn-1be ak-sphere andPVbe the orthogonal projection from RntoV. Then the mapPSV:Sn-1→defined by

is called the spherical projection from Sn-1to SV，where kerPV={w∈Sn-1|PV（w）=o}.

The following theory shows some nice properties of spherical projections （Ref. [32] for the proof and other properties）.

Theorem 1[21]Let SVbe ak-sphere. Then for each （closed）s-convex setC⊂Sn-1，PSV（C）:=∪u∈CPSV（u）⊂SVis （closed）s-convex. Moreover，ifCis proper andC∩kerPV=Ø，then so isPSV（C）.

The projections from Sn-1to its closeds-convex sets can also be defined in a same manner.Definition 4ForC∈，the mapPC:Sn-1→defined by

is called the spherical projection from Sn-1toC.

We refer to Ref.[32] for the properties and the analytic description ofPC. Here，as an application of spherical projections，we turn to another important topic，the spherical support function of a closeds-convex sets. As known，the support function of a convex body（i.e. a compact convex set with nonempty interior） in Rnis a crucial tool in convex geometric analysis，due to the fact that many nice properties and meaningful compositions of convex bodies can be derived/realized through their support functions. So，it is natural to look for the counterpart for closeds-convex sets in Sn-1，including the definition of spherical support functions and their essential characteristics etc. Various definitions of support function of ans-convex set have been proposed，among which the one given in Ref. [17] or[34] should be the most suitable one. However，the definition given there is formulated in geometric languages，hence it is not so practical. Here，we reformulate their definition in terms of Definition 4.

Forw∈Sn-1，denote Sw:=Hw∩Sn-1（an （n-2）-sphere），Sw+:=Hw+∩Sn-1（an open hemisphere） and（Sn-1）:={C∈（Sn-1）|C⊂Sw+}.

Definition 5Letw∈Sn-1，C∈（Sn-1）. Then the functionhw（C，·）:defined by

is called a spherical support （s-support for brevity） function ofC，where sgn denotes the sign function，andds（w，P[u，w]s（v））:=min{ds（w，s）|s∈P[u，w]s（v）}.

Observe that sinceP[u，w]s（C） is compact，hw（C，·） is attainable，i.e. for eachu∈Sw，there isv0∈Csuch thathw（C，u）=arccos（〈w，P[u，w]s（v0）〉）. We refer to Ref. [17] for the applications，Ref. [32] and the references therein for the properties ofs-support functions.

It is expected that thes-support functions can be used to studys-convex sets. However，unlike the situation in Euclidean spaces，where it is known that a function is the support function of some convex body iff it is sublinear，we have not known the essential property ofs-support functions yet. So，we propose the following valuable question.

Question 1What’s the essential character ofs-support functions?

Ref. [21] has made some efforts on this question，however the results obtained there are not as satisfactory as expected.

Now，it is time to turn back to the existence of the binary operations on Sn-1or on K（Sn-1）. As it is wellknown，the core of classical Minkowski theory is to combine the geometric invariants （such as volumes，mixed volumes etc.） and the binary operations （such as the Minkowski addition，Lp-addition etc.），so to form a systematic method for solving the extremal sets or uniqueness problems related to convex bodies. Hence，it is natural to expect the same work ons-convex sets. Thus，it becomes crucial to find suitable binary operations in spherical spaces.

Noticing that the Minkowski addition is convexity-preserving and projection-covariant （see the definition below），Ref. [34] started to look for the binary operations with such two properties on Sn-1. Unfortunately，the conclusion obtained there is somehow depressing.

Definition 6[34]Let *:be a binary operation andv∈Sn-1. If for eachkspherev∈S（0≤k≤n-2），

wheneverC，D∈（Sn-1），then*is calledv-projection-covariant. If*isv-projection-covariant for allv∈Sn-1，it is called projection-covariant.

Now，we state the depressing result in Ref. [34].

Theorem 2[34]A binary operation *:（Sn-1）×（Sn-1）→（Sn-1） is continuous （w.r.t the Hausdorff metric） and projection-covariant iff it is trivial，i.e. it is one ofC*D≡C，C*D≡-C；C*D≡DandC*D≡-D.

Theorem 2 implies no hope to find nice binary operations on Sn-1：if a binary operation on Sn-1is continuous，s-convexity-preserving in the sense thatC*D:={u*v|u∈C，v∈D}∈（Sn-1） wheneverC，D∈（Sn-1） and projection-covariant，then *is trivial. Hence we have to drop some requirements or restrict the domain when look for binary operations. Clearly thes-convexity-preserving property should be kept，and which of the continuity and projection-covariance should be dropped depends on the problem one concerns with. The following operation provided in Ref. [34] is a nontrivial example without continuity.

Then it is easy to check the * defined above is projection-covariant but not continuous.

If restricted on C，Ref. [34] proved the following interesting result.

Theorem 3A map *:（Sn-1） ，i.e. a restricted binary operation on（Sn-1），is both continuous ands-projection-covariant iffC*D:≡cos（C∪D），（C，D）∈C orC*D:=-cos（C∪D），（C，D）∈C.

Along this stream，Ref. [20] provided a nontrivial binary operation on：forC，D∈，define

It is easy to check that the binary operation “+s” is neither continuous nor projection-covariant，however “+s” allows the identity map onto be a valuation in the sensewheneverC，D，C∪D，C∩D∈（see Ref. [20] for the proof）.

Naturally，in the same manner one may define “+s” onfor generaln，however，it turns out that such a “+s” is even nots-convexity-preserving forn≥3 （Ref. [20] for counterexamples）. Even so，“+s” on K（R1）suggests a new criterion of “nice” binary operations：allowing the identity to be a valuation. Hence，we propose the following question.

Question 2Are there any binary operations on（resp.or other subfamilies of）allowing the identity to be a valuation? If yes，classify them.

3 Combinatorial property of s-convex sets

The combinatorial property ofs-convex sets describes the behaviour ofs-convex combination together with other operations or compositions such as the intersection and union etc. Such a property is fundamental in convex，discrete and combinatoric geometry and is fully studied for convex sets in Rn. In this section，we collect some facts which show thats-convex sets share many nice combinatorial properties with convex sets，and also mention some valuable unsolved problems.

First，we state the following three fundamental results.

Theorem 4（Ref. [10]，Radon-type Theorem） LetS⊂Sn-1be a set with at leastn+1 points ando∉cosS.Then there areS1，S2⊂Ssuch thatS1∪S2=S，S1∩S2=Ø and cosS1∩cosS2≠Ø.

Remark 2The conditiono∉cosScannot be omitted as shown by：in S1，letS={u，-u，v} withv≠±u，then it’s easy to see Theorem 4 is invalid for such anS.

Theorem 5（Helly-type theorem） LetC1，C2，…，Cm⊂Sn-1be propers-convex sets （m≥n+1）. If

Remark 3（i） If someCiare not proper，Theorem 5 may not hold：in S1，chooseC1={u，-u}（an impropersconvex set） for any fixedu，C2=[u，v]sand [v，-u]swith any fixedv≠±u，then any two of them have nonempty intersection，howeverC1∩C2∩C3=Ø.

（ii） Theorem 5 has been proved repeatedly （even for various kinds of so-called spherically convex sets，Refs. [1-6]）. However，the first analytic proof （with the help of Theorem 4） is given in Ref. [10].

Observe that in the counterexample shown in （i） of Remark 2，C1is a 0-sphere and 0-sphere appears in all other known counterexamples. So，we propose the following question.

Question 3If the family of （not necessarily proper）s-convex setsC1，C2，…，Cmis 0-sphere free，is Theorem 5 valid?

Theorem 6（Ref. [10]，Carathéodory-type Theorem） LetS⊂Sn-1be a set witho∉cosS. Then for eachu∈cosS，there existu1，u2，…，un∈Sand λ1，λ2，…，λn∈[0，1] withi.e. any element can be expressed as ans-convex combination at mostnpoints inS.

Theorem 6 was frist proved in Ref. [10] where only propers-convex sets were considered，so the conditiono∉cosSappears in the theorem. We guess this condition could be dropped and so propose the following question.

Question 4Does Theorem 6 hold for any subsetS?

In the following，we mention more results related to Theorem 5. Since there are numerous generalizations，extensions，ramifications and applications of Helly’s theorem for Euclidean convex sets，it is reasonable and valuable to consider the same work fors-convex sets. As a try，Ref. [37] made some efforts on this topic. More precisely，the theorems of Helly type related to the so-called spherical translations are established in Ref. [37]. First we recall the concept of gnomonic map （Ref. [17]）.

Definition 7Letv∈Sn-1. Then the gnomonic mapgv:Sv+→Rv:={x∈Rn|〈v，x〉=0} is defined as

wheregvis a bijection from Sv+to Rv（≌Rn-1），andgv-1（x）=ρ（v+x），∀x∈Rv. Meaningfully，bothgvandgv-1are convexity-preserving （see Refs. [17，32] for different proofs）.

Next，in terms of gnomonic map，we define the spherical translations.

Definition 8[37]Letv∈Sn-1. For each translationLb（·）=·+bon Rv，whereb∈Rvthe spherical translation Lbon Sv+is defined byFinnaly，we present the following extensions of Theorem 5 as illustrations.

Theorem 7[37]LetC，C1，C2，…，Cm∈Kvp（Sn-1）. If for any {i1，i2，…，in}⊂{1，2，…，m}，there isb=bi1，…，insuch thatCij∩Lb（C）≠Ø，j=1，2，…，n，then there isb*such thatCi∩Lb*（C）≠Ø，i=1，2，…，m.

Theorem 8[37]LetC，C1，C2，…，Cm∈Kvp（Sn-1）. If for any {i1，i2，…，in}⊂{1，2，…，m}，there isb=bi1，…，insuch that，then there isb*such that

Theorem 9[37]LetC，C1，C2，…，Cm∈. If for any {i1，i2，…，in}⊂{1，2，…，m}，there isb=bi1，…，insuch that，then there isb*such that

The spherical translations defined here are not satisfactory since they “move” sets only within an open hemisphere. So，we would like to see the more meaningful definition of spherical translations. More eagerly，we want to see the spherical analogues of various generalizations，extensions and ramifications of Helly’s theorem for Euclidean convex sets.

4 Structural property of s-convex sets

The structural property describes the structure of ans-convex set，namely，showing how to express ans-convex set by its smaller subsets together with suitable compositions and operations，such as takings-convex hull，intersection or union etc. The structures of （closed） convex sets in Rnhave been fully known （see Ref. [33] for details）. Here，in this section，we collect some known results fors-convex sets which show as well thats-convex sets share many nice structural properties with convex sets，and propose also some valuable unsolved questions.

The first candidate for expressing a （closed）s-convex set is its boundary. The following theorem shows that it works in most cases.

Theorem 10LetS∈，thenC≠cos（rbdC） iffCis ak-sphere or ak-hemisphere，where rbdC:=rbdRC∩Sn-1denotes the relative boundary ofC.

To prove Theorem 10，we need some preparations.

Lemma 1LetC⊂Sn-1be ans-convex set. ThenC=cos（rbdC） iffRC=co（rbdRC）.

ProofFirst，rbdC:=rbdRC∩Sn-1by definition implies rbdRC=R+（rbdC）.

Thus，ifRC=co（rbdRC），then for anyu∈C（⊂RC） impliesu∈co（rbdRC）. So，u=for some nonzerox1，x2，…，xm∈rbdRCand λ1，λ2，…，λm∈[0，1] withObserving thatxi=tivifor somevi∈rbdCandti＞0，i=1，2，…，m，we have

Denotingwe havewhere μi:=aλiti（≥0） （noticing，and further

So，C⊂cos（rbdC） which，together with the trivial fact that cos（rbdC）⊂cos（C）=C，leads toC=cos（rbdC）.

Conversely，ifC=cos（rbdC），then for anyx∈RC，x=tuwith somet∈R+andu∈C. SinceC=cos（rbdC），we havefor somevi∈rbdC（⊂rbdRC） and λi∈[0，1] with，we have

fortbvi∈RC，whereHence，RC⊂co（rbdRC） which，together with the trivial fact that co（rbdRC）⊂co（RC）=RC，leads toRC=co（rbdRC）.

Lemma 2[33]LetA⊂Rnbe a closed convex set. ThenA≠co（bdA） iffAis ak-affine space or ak-affine half-space.

Proof of Theorem 10SinceCis closed ands-convex，RCis closed and convex. Thus，C≠cos（rbdC） iffRC≠co（rbdRC） by Lemma 2 and further iffRCis ak-space or ak-half space（observingo∈RC） which is equivalent toC=RC∩Sn-1is a （k-1）-sphere or a （k-1）-hemisphere.

The other candidates are extremal subsets of ans-convex sets.

Definition 9[32]LetC∈. A pointu∈Cis called ans-extreme point ifu=λu1+（1-λ）u2for someu1，u2∈Cand λ∈（0，1），thenu1，u2∈{u，-u}. The set ofs-extreme points ofCis denoted by extsC.

Remark 4Letube ans-extreme point. Ifu1，u2are not antipodal，then one may choose λ=1/2 all the time andu1，u2∈{u，-u} meansu1=u2=u；ifu1，u2are antipodal，then λ≠1/2 for sure andu1，u2∈{u，-u} meansu1=u2=uoru1=u，u2=-uor the other way around.

Thes-extreme points and the extreme rays （or lines） of convex cones have a very close relation.

Proposition 1LetC∈. Thenu∈extsCiff there is an extremal ray or an extremal lineLofRCsuch that {u}∈L∩Sn-1. Moreover，in such a case，L:={tu|t∈R+} or {tu|t∈R}.

ProofSupposeLis an extremal ray or line such that {u}∈L∩Sn-1（naturallyL={tu|t∈R+} or {tu|t∈R}now）. Ifu=λu1+s（1-λ）u2for someu1，u2∈Cand λ∈（0，1），then λu1+s（1-λ）u2=||λu1+（1-λ）u2||u∈L. SinceLis an extremal ray or an extremal line，we haveu1，u2∈Land in turnu1，u2∈L∩C. Thus，in the case whereLis an extremal ray，u1=u2=uis clearly. In the caseLis an extremal line，we denoteL=L+∪L-whereL+:={tu|t≥0}，L-:={tu|t＜0}. Observe thatu1，u2∈L-cannot happen for otherwise we would haveu=λu1+s（1-λ）u2∈L-. Thus，u1，u2∈L+oru1∈L+，u2∈L-（or the other way around）. Ifu1，u2∈L+，thenu1=u2=uis clearly. If，say，u1∈L+，u2∈L-，then naturallyu1=u，u2=-u. In conclusion，uis ans-extreme point ofC.

Conversely，supposeuis ans-extreme point ofC. DenoteL:={tu|t∈R+} （if -u∉C） or {tu|t∈R} （otherwise）.

If there arex1=t1u1，x2=t2u2∈RC，whereu1，u2∈C，t1，t2∈R+，and λ∈（0，1） such thaty:=λx1+（1-λ）x2∈L.

In the casey:=λt1u1+（1-λ）t2u2∈L=o，we may assumex1≠o，x2≠o（for otherwisex1，x2∈Lobviously），sou2=-[（t1λ）/（t2（1-λ））]u1=-u1（noticingt1λ=t2（1-λ）） derived by ||u1||=||u2||=1）. Now we claimu1=uor -u（consequently，u2=∓u） in this case：for otherwiseC⊃[u1，u]∪[u，-u1] would implies thatuis not ans-extreme point.Thus，Lis a line （foru，-u∈L） and （without loss of generality，assumeu1=u）x1=t1u，x2=-t2u∈L. SoLis an extremal line.

In the casey≠o，we may assumex1≠o，x2≠oas well. Thus，by the property of the redial projection，

which leads tou1=u2=uoru1=±u，u2=∓usinceuis ans-extreme point ofC. In the caseu1=u2=u，we havex1=t1u，x2=-t2u∈L，and in the caseu1=±u，u2=∓u，we havex1=t1（±u），x2=t2（∓u）u∈Las well （observing thatLis a line for sure in such a case）. So，Lis an extremal ray or extremal line ofRC.

Since a closed convex coneKhas extremal rays iff it is pointed，i.e.K∩（-K）={o} （Ref. [33]），andC∈Sn-1is proper iffthe closedRCispointed （Ref. [32]），the following conclusion is straightforward.

Corollary 1[10，32]LetC∈. T he n e xts C≠Ø.

Next corollary can also be derived from Proposition 1 easily.

Corollary 2[10，32]LetC∈andu∈extsC. For anyk，if there areu1，u2，…，uk∈Cand λ1，λ2，…，λk∈[0，1] with，such thatu=，then allui∈{u，-u}.

The situation for non-proper closeds-convex sets looks uninspired，but is actually interesting.

Proposition 2LetC∈andu，-u∈C. Then

（i）u∈extsCiff-u∈extsC.

（ii） for anyv≠±u，v∉extsC.

Proof（i） It follows from the fact that in this case，theLin Proposition 1 is necessarily an extremal line.

（ii） Sincev∈[u，v]s∩[v，-u]s⊂C，it is not ans-extreme point.

As a consequence，we have the following result.

Corollary 3LetC∈Then

（i） ifCis non-proper，then the number of elements in extsCis 0 or 2.

（ii） ifCcontains more than one pair of antipodes，then extsC=Ø.

（iii） ifu∈extsCbut-u∉C，thenCis proper.

Now，we propose the following question.

Question 5Which closeds-convex sets are of exact twos-extreme points?

At the end，we state the spherical analogy of structural theorem of compact convex sets in Rn（see Ref. [10]）with a proof included.

Theorem 11LetC∈. ThenC=cos（extsC）.

ProofC∈implies thatRCis pointed and in turn is line free. Thus by Theorem 1.4.3 in Ref. [33]，RC=co（extrRC），where extrRCdenotes the union of extremal rays ofRC.

Hence，for eachu∈C⊂RC，for some nonzeroxi∈Li，whereLiis an extremal ray ofRC，i=1，2，…，m，and λi∈（0，1） withTherefore，we have by the property of radial projection，

The structure of non-proper closeds-convex sets is somehow complicated. We will discuss it in forthcoming papers.

AcknowledgementsThe author is grateful to the referee for his valuable comments and pointing out the errors and typos.