Convex Cones in Hilbert Spaces

ToappearinJOURNALOFMATHEMATICALANALYSISANDAPPLICATIONS
PartII:ConvexConesinHilbertSpaces NormalityandModulabilityIndices.
ofXandK=(BX+K)\(BXK).Thenormalityindex Abstract.LetKbeaclosedconvexconeinaHilbertspaceX.LetBXbetheclosedunitball A.IUSEM1andA.SEEGER2
aby-productoneobtainsaformulaforcomputingthemodulabilityindex isacoecientthatmeasurestowhichextenttheconeKisnormal.Weestablishaformulathat relates(K)tothemaximalangleofK.Aconceptdualtonormalityisthatofmodulability.As (K)=supfr0:rKBXg
beexpressedintermsofthesmallestcriticalangleofK. ofK.ThesymbolKstandsfortheabsolutelyconvexhullofK\BX.Weshowthat(K)can (K)=supfr0:rBXKg
KeyWords:Convexcones,normalityindex,modulabilityindex. MathematicsSubjectClassications:46B20,52A05,54B20.
Thisisacontinuationofourpaper[11]andthereforewekeepthesamenotationandterminology.Given 1Thereareseveralmetricsthatservetomeasuredistancesbetweenelementsoftheset
anormedspaceX,thesymbolsBXandSXrefer,respectively,totheclosedunitballandtheunitsphere. Introduction
Inthisworkweconsiderthestandardchoice Ì(X)nontrivialclosedconvexconesinX:
where haus(C1;C2)=maxsup %(K1;K2)=haus(K1\BX;K2\BX); z2C1dist[z;C2];sup z2C2dist[z;C1] (1)
iusp@impa.br) alberto.seeger@univ-avignon.fr) 1InstitutodeMatematicaPuraeAplicada,EstradaDonaCastorina110-JardimBotanico,RiodeJaneiro,Brazil(e-mail: 2UniversityofAvignon,DepartmentofMathematics,33rueLouisPasteur,84000Avignon,France(e-mail:

followingtwoconcepts: dierentfromf0ganddierentfromthewholespaceX.Forthesakeofcompleteness,werecallalsothe satisfyingK+KKandR+KK.SayingthataconvexconeKisnontrivialsimplymeansthatKis dist[x;C]referstothedistancefromxtothesetC.ByaconvexconeweunderstandanonemptysetK standsfortheclassicalPompeiu-HausdordistancebetweentwoboundedclosednonemptysetsC1;C2,and
Denition1.LetKbeaconvexconeKinanormedspaceX.Onesaysthat (a)Kisnormalifthereisaconstant>0suchthat (b)Kismodulableifthereisaconstant>0suchthat anyx2Xisexpressibleintheformx=uv (kuk+kvk)ku+vkforallu;v2K: (2)
Wearetakingk(u;v)k=kuk2+kvk2 withu;v2Ksatisfyingk(u;v)kkxk:
befoundforinstancein[12,14,19,22].Theconceptofnormalityisalsoclassicalanddoesn'tneedfurther nouniversalagreementwithrespecttotheterminology.Othernamesfortheconceptofmodulabilitycan isacceptable.Modulabilityisafundamentalconceptofthetheoryofconvexcones.Unfortunately,thereis 1=2;butanyequivalentnormintheCartesianproductXX
Theorem1.LetKbeaconvexconeKinanormedspaceX.Then, justication;seeforinstancethebooks[4,19,22]orthepioneeringworksbyKreinandcollaborators[3,15].
(a)KisnormalifandonlyifK=(BX+K)\(BXK)isbounded. Thenexttheoremisacombinationofseveralsources.Wetakethisresultforgranted.
(b)KismodulableifandonlyifK=co[(K[K)\BX]isaneighborhoodoftheorigin.
andthemodulabilityindex Oneofthemainissuesofthispaperconcernsthepracticalcomputationofthenormalityindex
(K)=supfr0:rBXKg (K)=supfr0:rKBXg (3)
previouswork[11].Itisnotalwayseasytocomputetheright-handsidesin(3)and(4).Theevaluationof ofanontrivialclosedconvexconeK.Themeaningsoftheseindicesareexplainedinfulllengthinour thesetsKandKisalreadyadiculttaskbyitself.Themainmeritofthepresentpaperisshowingthat (4)
isexpressibleintermsofthesmallestcriticalangleofK. Tobemoreprecise,weshowthat(K)canbeexpressedintermsofthemaximalangleofK,whereas(K) (K)and(K)admitniceandsimplecharacterizationformulaswhentheunderlyingspaceXisHilbert.
dimensionalHilbertspace: ity.ThepurposeofSection4isestablishingthefollowingtopologicalresultsinthecontextofaninnite i)WithrespecttothetruncatedPompeiu-Hausdormetric%,thepointedelementsofÌ(X)don'tform Anotherthemediscussedinthispaperhastodowiththelinkexistingbetweenpointednessandnormal-
openset.Equivalently,theunpointedelementsofÌ(X)don'tformaclosedset. 2

Tothebestofourknowledge,theabovedensityresultisnew.Byusingadualityargument,weestablish alsoalinkbetweenalmostreproducibilityandmodulability. ii)Bycontrast,theabnormalelementsofÌ(X)doformaclosedset.Infact,thesetofabnormalelements ispreciselythe%-closureofthesetofunpointedelementsofÌ(X).
Asmentionedintheintroduction,thepracticalcomputationofthecoecient(K)oerssometimesaserious challenge.Alleortsincharacterizing(K)bymeansofalternativeformulasarenottobedespised.The 2nexttheoremsuggestsconsideringtheexpression
APreliminaryFormulaforComputing(K)
inadvancethedistancefunctiondist[;K]. astoolforevaluating(K).Ofcourse,thecomputationoftheinmum(5)isgreatlysimpliedifoneknows (K)=inf kzk=1maxfdist[z;K];dist[z;K]g (5)
insofar. Theorem2.ForanontrivialconvexconeKinanormedspaceX,onehas Theterm(K)appearsalreadyinreferences[5]and[9],butnoconnectionwith(K)hasbeenmade
Weclaimthat Proof.Westartbyprovingtheinequality(K)(K).Assumethat(K)>0,otherwisewearedone. (BX+K)\(BXK)[(K)]1BX: (K)=(K): (6)
Takeanonzerovectorxintheaboveintersection.Since x2BX+K=)dist[x;K]1; (7)
dist[;K],onegets onehasmaxfdist[x;K];dist[x;K]g1:Inviewofthepositivehomogeneityofthedistancefunction (K)maxdistx x2BXK=)dist[x;K]1;
kxk;K;dist
(K)(K).Ababsurdo,supposethat Thisprovesthatxbelongstotheright-handsideof(7)asneeded.Wenowtakecareofthereverseinequality kxk;K1x
kxk:
forsomes>0.Wemustarrivetoacontradiction.Take"2]0;s[andndavectorz"2Xsuchthat (BX+K)\(BXK)[(K)+s]1BX kz"k=1; (8)
maxfdist[z";K];dist[z";K]g

Thecondition(10)impliesthat
SinceKisaconvexcone,anelementaryrearrangementyields z"2((K)+")BXK: z"2((K)+")BX+K;
Inviewof(8)and(9),onegetsinsuchacase [(K)+"]1z"2(BX+K)\(BXK):
contradictingthefactthat"

inSection4. ofagivenofK2Ì(X).Theleast-distanceproblem(12)isofinterestforitselfandwillbestudiedindetail
Corollary3.IfKisanontrivialclosedconvexconeinaHilbertspaceX,then(K)nor(K). isthelargestnonexpansivemapthatvanishesexactlyovertheabnormalelementsofÌ(X). Thenextcorollaryisrecordedforthesakeoflateruse.Notice,parenthetically,thatnor:(Ì(X);%)!R
thepointwisemaximalityofnor()amongallthenonexpansivemapson(Ì(X);%)thatvanishexactlyover theabnormalelementsofÌ(X). Proof.IfXisaHilbertspace,thendand%areinfactidentical.ItsucesthentoapplyCorollary1and
3.1Comparing(K)and(K) Anumber>0satisfyingtheinequality(2)iscalledanormalityconstantofK.Thebestpossible 3 TheBestNormalityConstantandtheAngularCoecient
normalityconstantofKisofcourse(K)= Thenumber(K)2[0;1]canbeusedasatoolforquantifyingthedegreeofnormalityofK. (u;v)6=(0;0)ku+vk
Thepurposeofthissectionistocompare(13)withthemuchsimplerexpression u;v2K inf kuk+kvk: (13)
momenttotheinterpretationofthecoecient(K).First,westate: Wehavedividedby2intheright-handsideof(14)justtomakesurethat(K)2[0;1]:Weshallcomeina (K)= u;v2K\SXu+v inf 2: Proposition1.LetKbeanontrivialconvexconeinanormedspaceX.Then
theminimizationproblem(14).Thissimpleobservationshowsthesecondinequalityin(15).Therelation Proof.Ifoneaddstheconstraintskuk=1andkvk=1inthefeasiblesetof(13),thenonearrivesat 12(K)(K)(K): (15)
(K)2(K)isobtainedasaconsequenceoftheMassera-Schaerinequality[16]whichassertsthat
forallnonzerovectorsx;yinanormedspaceX: x kxk kyky
maxfkxk;kykg 2kxyk
ofnormalityofK.Bytheway,arewesurethat(K)and(K)aredierentnumbers?Theanswerisyes, butadierencecanbeobservedonlyinanon-Hilbertiansetting.Westartwithaneasyexampleshowing that(K)and(K)maydier. InviewofProposition1,theterm(K)isalsoanacceptablecandidateastoolformeasuringthedegree
5

oftwosegments,namelyK\SX=cof(1=2;1=2);(0;1)g nontrivialclosedconvexconeinR2givenbyK=fx2R2:x20;x1+x20g:ThesetK\SXisaunion Example1.LettheplaneR2beequippedwiththeManhattannormkxk=jx1j+jx2j,andletKbethe
| {z È1 }[cof(1;0);(0;1)g
Èi.Thus,forsolvingtheminimizationproblem(14)onemaysupposethatuandvareondierentsegments, Ifuandvareonthesamesegment,sayÈi,thenthemidpoint(u+v)=2hasunitlengthbecauseitremainsin | È2 {z }:
sayu2È1andv2È2.Ifonewritesu=t(1=2;1=2)+(1t)(0;1); thenoneisleadtominimize v=s(1;0)+(1s)(0;1);
Ontheotherhand,bychoosingu=(1=3;1=3)andv=(1=3;0),onegets withrespecttot;s2[0;1].Theinmumisattainedwitht=1andarbitrarys1=2.Thus,(K)=1=2: u+v 2=st2+2t2s
(K)ku+vk kuk+kvk=1=3 2
boundaryofK,evenifKispointedandhasnonemptyinterior.ThisphenomenoncannotoccurinaHilbert Incidentally,Example1showsthatthecomponentsofapair(u;v)solving(14)don'tneedtobeinthe 1

interpretation.Indeed,onecanwrite Ifthenormkkderivesfromaninnerproducth;i,thenthecoecient(K)admitsaninterestingangular 3.2AngularInterpretationof(K)
with (K)=cosmax(K) 2
max(K)= u;v2K\SXarccoshu;vi sup (17)
seein[6,9,18],amongotherreferences. extensivelystudiedin[7]and[8].Thefunctionmax()hasfoundalargevarietyofapplicationsasonecan denotingthemaximalangleofK.Theanglemaximizationproblem(18)itofinterestforitselfandhasbeen
Byobviousreasons,wereferto(K)astheangularcoecient3ofK.Notice,incidentally,that
Kisequivalenttotheconditionmax(K)=.ThisfactisnolongertrueiftheHilbertspaceisinnite tosay,K\K=f0g:InanitedimensionalHilbertspace,pointednessofanontrivialclosedconvexcone RecallthataconvexconeKinanormedspaceissaidtobepointedifitdoesn'tcontainaline,thatis (K)=0ifandonlyifmax(K)=:
dimensional. Example2.Let`2(R)betheHilbertspaceofsquare-summablerealsequences.Noticethat
isaclosedconvexconebecauseitisexpressibleasintersectionofclosedhalf-spaces.Onecaneasilycheck thatKispointed.Now,foreachn1,weconstruct K=fx2`2(R):nXk=1xk0;8n1g
un= vn= p2n(1;1;1;1;:::;1;1 p2n(0;1;1;1;1;:::;1;1
1 | | {z {z };0;0;:::);
Noticethat(un;vn)isapairofunitvectorsinKand2nterms };0;0:::):
unitvectoranditsopposite. Bylettingn!1onegets(K)=0.ThisinmumisnotattainedbecauseotherwiseKshouldcontaina 0(K)un+vn 2 =1 2pn:
whatwecallthedegeneracyphenomenon.Weinsistonthefactthatthedegeneracyphenomenoncannot occurinanitedimensionalsetting. 3Inspiredbytherelation(17),onecouldusetheequalitymax(K)=2arccos[(K)]asdenitionofthemaximalangleofa Example2showsthatapointedclosedconvexconemaywellhaveamaximalangleequalto.Thisis
nontrivialconvexconeKcontainedinageneralnormedspace.Suchdenitionishoweverpurelyformal. 7

metric 3.3LipschitzBehavioroftheAngularCoecient Westartwithausefullemmaonthenonexpansivenessoftheangularcoecient()withrespecttothe
that#and%arenotthesamemetric. Lemma1.LetK1andK2benontrivialclosedconvexconesinanormedspaceX.Then, Thedenitionof#bearsastrongresemblancewiththedenition(1)thatwegaveof%.Beaware,however, #(K1;K2)=haus(K1\SX;K2\SX):
Proof.Consideranarbitrary">0.Pickupu";v"2K2\SXsuchthat j(K1)(K2)j#(K1;K2):
Selectthenacoupleofvectorsu";v"inK1\SXsuchthat ku"u"kdist[u";K1\SX]+"; u"+v" 2(K2)+":
Onegets 2[(K1)(K2)]ku"+v"kku"+v"k+2" kv"v"kdist[v";K1\SX]+":
2sup dist[u";K1\SX]+dist[v";K1\SX]+4" ku"u"k+kv"v"k+2"
Byletting"!0onearrivesat (K1)(K2)w2K2\SXdist[w;K1\SX]#(K1;K2):
supw2K2\SXdist[w;K1\SX]+4"
Theproofoftheinequality(K2)(K1)#(K1;K2)isanalogous. manneralsowithrespecttothemetric%. Remark2.Themetric#ismajorizedby2%,sotheangularcoecient()variesinaLipschitzcontinuous
ofagivennontrivialclosedconvexconeK. 4Weareinterestedinbetterunderstandingtheminimizationproblem(12)thatdenestheradiusofnormality
ThissectiontakesplaceinaHilbertspacesetting.Recallthatinsuchacontext,thetruncatedPompeiu- AntipodalityandDistancetoAbnormality
Hausdordistance%admitstheequivalentformulation %(K1;K2)=maxsup x2K1\SXdist[x;K2];sup 8 x2K2\SXdist[x;K1]: (19)

problem(18).Forthesakeofconvenience,wereformulate(18)intheequivalentform of%Ȧsexplainednext,theleast-distanceproblem(12)turnsouttoberelatedtothetheanglemaximization K2.Forcomputationalpurposes,itispreferabletoworkwith(19)andnotwiththeoriginaldenition(1) Theexpressionontheright-handsideof(19)issometimesreferredtoasthegapdistancebetweenK1and
Asinreference[7],onesaysthat(u0;v0)2XXisanantipodalpairofKif cos[max(K)]= u;v2K\SXhu;vi: inf (20)
approximateantipodality. Sincetheinmumin(20)isnotnecessarilyattained,itishelpfultointroducealsoasuitableconceptof u0;v02K\SXandhu0;v0i=cos[max(K)]: (21)
Denition2.LetKbeanontrivialconvexconeinaHilbertspaceX.AnantipodalpairofKwithin atolerancelevel"0isapair(u";v")2XXsuchthat
antipodalpairsmaynotexistfornontrivialclosedconvexconesininnitedimensionalHilbertspaces. Inthesequel,Rw=ftw:t2Rgdenotesthelinegeneratedbyanonzerovectorw2Xandw?indicates Antipodalityinthesense(21)isrecoveredbysetting"=0.Example2nicelyillustratesthefactthat u";v"2K\SXandhu";v"icos[max(K)]+": (22)
thehyperplanethatisorthogonaltothisline. pairofKwithinatolerancelevel"0.Assumethatu"6=v"anddene Lemma2.LetKbeanontrivialclosedconvexconeinaHilbertspaceX.Let(u";v")beanantipodal
Then,Q"isanunpointedclosedconvexconesuchthat dist[x;K]r1+hu";v"i Q"=(K\(u"v")?)+R(u"v"):
dist[x;Q"]1+ 1hu";v"ir1+hu";v"i 2"
2 8x2Q"\SX; 8x2K\SX: (23)
thatXisnitedimensional.Importantadjustmentsintheproofareneededinordertotakecareofthe Proof.Thisresultwasobtainedin[10]fortheparticularcase"=0andundertheadditionalassumption (24)
anddividetheproofinthreesteps. generalcase.Forconvenience,weintroducethenotation
Step1.TheconvexconeQ"isclosedbecauseitisexpressibleassumofalineRw"andaclosedset w"=ku"v"k1(u"v")
containedinw?".ObservethatQ"isunpointedbecauseQ"\Q"containstheunitvectorw". 9

ydenedas andt2R.Clearlyt=hx;w"i,andthereforejtj1bytheCauchy-Schwarzinequality.Considerthepoint Step2.Weestablishtheinequality(23).Takeanyx2Q"\SX,sothatx=z+tw",withz2K\w?"
NotethatinbothcasesybelongstoK,becausez;u";v"2K.Weproceedtoestimatethedistancebetween xandy.Considerrstthecaseoft0.Onehas y=(z+(t=2)ku"v"ku"ift0; z(t=2)ku"v"kv"ift0:
kxyk2=t2w"r1hu";v"i =t2"1+1hu";v"i 2 u"2
Pluggingintheabovelinethedenitionofw",oneendsupwith 2 2r1hu";v"i 2 hu";w"i#:
Hence,dist[x;K]kxykp(1+hu";v"i)=2:Thecaseoft0isdealtinasimilarway. kxyk2=t21+hu";v"i 2 : Step3.Weprovetheinequality(24).Takex2K\SXandconsiderthevector
Wedecomposeyintheformy=xhx;w"iw"+jhx;w"ij y=x+jhx;w"ij ku"v"k(u"+v"): (25)
Clearlyy2Rw".Weclaimthaty2K\w?".Forcheckingthaty2w?",notethat | ku"v"k(u"+v") {z y } +hx;w"iw" |{z}
:
hy;w"i=hx;w"ihx;w"ikw"k2+jhx;w"ij= ku"v"k2hu"+v";u"v"i=0;
ku"v"k2hu"+v";u"v"i
usingthefactthatkw"k=ku"k=kv"k=1.Forcheckingthaty2K,rewriteyas y=x =(x+2ku"v"k1jhx;w"ijv"ifhx;w"i0; ku"v"k(u"v")+jhx;w"ij
x+2ku"v"k1jhx;w"iju"ifhx;w"i0: ku"v"k(u"+v")
10

estimatenextthedistancebetweenxandy.Directlyfrom(25)onegets Inbothcases,y2Kbecausex;u"andv"belongtoK.Weconcludethaty=y+ybelongstoQ".We
Inotherwords, kxyk=jhx;u"v"ij ku"v"k2ku"+v"k=jhx;u"v"ij
kxyk=r1+hu";v"i 2(1hu";v"i)p2(1+hu";v"i):
Tocompletetheproofof(24)wemustcheckthat1+(1hu";v"i)1";i.e., 2 with=jhx;u"v"ij
jhx;u"v"ij1hu";v"i+": 1hu";v"i0:
Since(u";v")satisestheapproximateantipodalitycondition(22),itisclearthat hu";v"ihx;v"i+"; (26)
and,aposteriori, hu";v"ihx;v"i+"+(1hx;u"i); hu";v"ihx;u"i+":
Thecombinationofthelasttwoinequalitiescanbewritteninthecompactform maxfhx;u"ihx;v"i;hx;v"ihx;u"ig1hu";v"i+"; hu";v"ihx;u"i+"+(1hx;v"i):
butthisisprecisely(26). Keepinginmindthecharacterization(19)of%,oneseesthat(23)and(24)produce
i.e.,onegetsanupperestimateforthetruncatedPompeiu-HausdordistancebetweenKandQ".Thisobservationhassomenoteworthyconsequences.Forexample,itallowsustoestablishthefollowingtopological
%(K;Q")1+ 1hu";v"ir1+hu";v"i " 2 ;
resultrelatingthesets
Theorem3.LetXbeaHilbertspace.Then, Abn(X)=fK2Ì(X):Kisabnormalg;
nor(K)= U(X)=fK2Ì(X):Kisunpointedg:
Inparticular,Abn(X)istheclosurewithrespecttothemetric%ofthesetU(X). Qunpointed%(K;Q) Q2Ì(X) inf 8K2Ì(X):
11

inAbn(X),onegetscl%[U(X)]Abn(X)and Proof.ThesetAbn(X)isclosedbecauseitscomplementisopen(cf.Corollary2).SinceU(X)iscontained
Weclaimthat nor(K)Qunpointed%(K;Q): Q2Ì(X) inf (27)
sequencef(un;vn)gn1fortheantipodalityproblem(20),i.e.,un;vn2K\SXaresuchthat IfKisaray,then(K)=1and(28)holdstrivially.AssumethenthatKisnotaray.Consideraminimizing Qunpointed%(K;Q)(K): Q2Ì(X) inf (28)
ofLemma2,theset goesto0asn!1.SinceKisnotaray,thereisnolossofgeneralityinassumingthatun6=vn.Inview "n=hun;vnicos[max(K)]
belongstoU(X)andinf
Qn=(K\(unvn)?)+R(unvn) (29)
But"n!0andhun;vni!cos[max(K)]6=1.Hence, Qunpointed%(K;Q)%(K;Qn)1+ Q2Ì(X) 1hun;vnir1+hun;vni "n 2 :
1+ Abn(X)cl%[U(X)]:Hence,U(X)isdenseinAbn(X)and(27)isinfactanequality. Thistakescareof(28).By-the-way,ifKisabnormal,then(K)=0and%(K;Qn)!0.Inotherwords, 1hun;vnir1+hun;vni
2 !r1+cos[max(K)] 2 =(K):
space(Ì(X);%). eachQnisunpointedbutthelimitKispointed!ThisconrmsthatU(X)isnotaclosedsetinthemetric by(29).Sincef(un;vn)gn2Nisaminimizingsequencefor(20),itfollowsthat%(K;Qn)!0.Observethat Remark3.LetKandf(un;vn)gn2NbeasinExample2.WeknowthatKisabnormal.LetQnbedened
5Thecomputationof(K)isnotalwaysaseasyasonemaythingatrstsight,soweconsidernowan
alternativecharacterizationofthenormalityindex(K).ThistimeouranalysisisrestrictedtoaHilbert spacesetting. Relating(K)totheMaximalAngleofK
Theorem4.ForanontrivialclosedconvexconeKinaHilbertspaceX,onehas (K)=(K)=nor(K): 12 (30)

nor(K)(K):Ontheotherhand,Theorem2andCorollary3yield(K)=(K)and(K)nor(K), respectively.So,wejustneedtoprovethat(K)(K): Proof.Mostoftheheavyworkhasbeendonealready.FromtheproofofTheorem3weknowalreadythat
InageneralHilbertspacetheprooffollowsasimilarpattern,exceptthatnowtheinmuminthedenition Theinequality(31)isestablishedin[9,Proposition1],butonlyinanitedimensionalHilbertspacesetting. of(K)isnotnecessarilyattained.Oneworksthenwithanapproximatesolutionz"asin(9)-(10),and,
again. ofcourse,attheveryendoftheproofonelets"!0.Itisnotworthwhilewritingdownallthedetails
(orChebyshev)normkxk=maxatbjx(t)j:Asindicatedin[11,Example6],fortheclosedconvexcone Example3.ConsiderthespaceC([a;b];R)ofcontinuousfunctionsx:[a;b]!Requippedwiththeuniform examplesshowthattheequality(K)=(K)isnotnecessarilytrueinageneralnormedspace. ThatkkderivesfromaninnerproductisanessentialassumptioninTheorem4.Thefollowingtwo
t2[a;b]besuchthatu(t)=1.Then,ku+vku(t)+v(t)1;gettinginthisway(K)1=2.This lowerboundisattainedbychoosingforinstance onehas(K)=1:Weclaimthat(K)=1=2.Toseethis,takeanypairofvectorsu;v2K\SX.Let K=fu2C([a;b];R):u(t)08t2[a;b]g
possible. Intheaboveexampleonegets(K)>(K),butobtainingthereverseinequality(K)

(a)(K)=(K+)and(K)=(K+).
proofoftherstformulain(b)runsasfollows: IsometryTheorem(cf.[23])whichassertsthatQ7!Q+isadistance-preservingoperationon(Ì(X);%).The Proof.Part(a)isestablishedin[11,Theorem6].Part(b)isobtainedbyexploitingtheWalkup-Wets (b)mod(K)=nor(K+)andnor(K)=mod(K+).
mod(K)= = Qnotmodulable%(K;Q)= Pabnormal%(K+;P)=nor(K+): P2Ì(X) Q2Ì(X) inf inf Qnotmodulable%(K+;Q+) Q2Ì(X) inf
Thesecondformulain(b)isproveninasimilarway.
andonlyifK+ispointed. subspaceofX.RecallthataclosedconvexconeKinareexiveBanachspaceXisalmostreproducingif Thenexttheoremrelatesthesets AconvexconeKinanormedspaceXiscalledalmostreproducingifspan(K)=KKisadense
aswellasthefunctionsmod()and(). Nmod(X)=fK2Ì(X):Kisnotmodulableg; Nar(X)=fK2Ì(X):Kisnotalmostreproducingg;
Theorem6.LetXbeaHilbertspace.Then, (a)ForallK2Ì(X)onehas
(b)TheradiusofmodulabilityofK2Ì(X)admitsalsotherepresentation Inparticular,Nmod(X)istheclosurewithrespecttothemetric%ofthesetNar(X). mod(K)= Q2Nar(X)%(K;Q): inf
Proof.Part(a)isamatterofrephrasingTheorem3.Wejustneedtokeepinmindthedualityformulasof Theorem5(b)andthefactthatNar(X)=fK2Ì(X):K+isnotpointedg;
mod(K)=(K):
Part(b)isalsoamatterofusingdualityarguments.ByrecallingTheorems5(b)and4,inthatorder,one getstheequalities Nmod(X)=fK2Ì(X):K+isabnormalg:
Theorem5(a)yields(K+)=(K)andcompletestheproofofthetheorem. mod(K)=nor(K+)=(K+):
14

6.1Relating(K)totheSmallestCriticalAngleofK WhatmeanssuchathingasthesmallestcriticalangleofK?Whatisacriticalangleanyway?These
Denition3.LetKbeanontrivialclosedconvexconeinaHilbertspaceX.AcriticalpairofKis questionsneedtobeclariedbeforeformulatingthedualversionofTheorem4. antipodalityproblem(20).Itreadsasfollows(cf.[7,8]): anypair(u;v)2XXofvectorssatisfying Theconceptofcriticalanglederivesfromtherst-orderstationarity(orcriticality)conditionsforthe
vhu;viu2K+; uhu;viv2K+: u;v2K\SX;
Theangle(u;v)=arccoshu;viformedbyacriticalpairiscalledacriticalangle.Theadjective
simple: anglesofKandthoseofK+arerelatedbyacertainreexionprinciplewhoseformulationisastonishingly properisaddedwhenuandvarenotcollinear,thatistosay,jhu;vij6=1. Impropercriticalanglesareirrelevantandusuallyleftasidefromthediscussion.Thepropercritical
sionalityasssumptioncanbedropped. Thisprinciplewasestablishedin[8,Theorem3]inanitedimensionalHilbertspace,butthenitedimen-
Inaninnitedimensionalcontextthereisnoguaranteeabouttheattainabilityofcriticalanglesbecause isapropercriticalangleofK()isapropercriticalangleofK+:
theunitsphereSXisnolongercompact.Despitethisfact,itmakessensetorefertothenumber
Theorem7.ForanontrivialconvexconeKinaHilbertspaceX,onehas asthesmallest(proper)criticalangleofK.Weareperhapsabusingoflanguage,butnottooseriously. min(K)=max(K+)
Proof.BycombiningTheorems4and5(a),onegets (K)=sinmin(K) 2 : (32)
(K)=(K+)=(K+)=cosmax(K+) 2 =cosmin(K) 2 =sinmin(K) 2 :
ofattainabilityofthesmallestcriticalangleisnotaproblematall. Formula(32)appliesevenifKdoesn'tadmitacriticalpair(u;v)formingtheanglemin(K).Thelack
15

7Itisnottheintentionheretodisplayafulllistofconclusionsthatcanbedrawnfromtheseresults,butat
Theorems3and4,andtheirdualcounterparts,areperhapsthemostsignicantcontributionsofthispaper. leasttwocorollariesdeservetobeproperlyrecorded. ByWayofConclusion
closedconvexconeKinaHilbertspaceXissaidtobe Therstcorollaryconcernsthemodulabilityandnormalityindicesofinfra-dualcones.Recallthata 8

Inviewof(33),wejustneedtoprovethat
Tocheckthisinequalityitsucestoguaranteetheexistenceofavectorxsuchthat kxk=1fdist[x;K]+dist[x;K]gp2: sup
Toseethatthissystemissolvable,westartwithavectorwoflengthp2=2lyingintheboundaryofK.We takethenaunitvectorh2Xsuchthathh;w0wi08w02K:
dist[x;K]=p2 2;dist[x;K]=p2 2;kxk=1: (34)
w0=(1=2)w,oneseesthathh;wi=0,i.e.,hisorthogonaltow.Anotherusefulobservationisthis:wisa hisusuallyreferedtoasthenormalconetoKatw(cf.[20,Section2]).Bytakingw0=2wandthen Geometricallyspeaking,theinequality(35)meansthathisnormaltoKatw.Thecollectionofallsuch
pointinKatminimaldistancefromw+(p2=2)h.So,itisnotdiculttocheckthatx=w+(p2=2)h solvesthesystem(34).Indeed, kxk2=kw+(p2=2)hk2=kwk2+(p2=2)2khk2=1; dist[x;K]=pkxk2(dist[x;K])2=q1(p2=2)2=p2=2; dist[x;K]=kxwk=k(p2=2)hk=p2=2;
anddist[;K](cf.[17]). therstequalityin(36)beingaknownPythagoreanformulathatrelatesthedistancefunctionsdist[;K]
Part(b).IfKissupra-dual,thenK+isinfra-dual.Part(a)yields(K+)p2=2(K+):Theorem5(a) doestherestofthejob.
Corollary6.SupposethatK2Ì(X)isnotaray.Thefollowingstatementsholdtrue: Part(c).Itisobtainedbycombining(a)and(b).
(a)Iff(un;vn)gn2N,withun6=vn,isaminimizingsequencefortheantipodalityproblem(20),then Ourlastcorollaryconcernsapracticalalgorithmforsolvingtheleast-distanceproblem(12).
(b)IfKadmitsanantipodalpair,say(u0;v0),then thesequencefQngn2Ndenedby(29)isminimizingfortheleast-distanceproblem(12).
isanexactsolutionto(12),i.e.,anabnormalelementofÌ(X)lyingatminimaldistance fromK. Q0=(K\(u0v0)?)+R(u0v0)
Proof.CombineTheorem4andtheargumentdevelopedintheproofofTheorem3. AgreementinMathematics. Acknowledgements.ThisresearchwascarriedoutwithintheframeworkoftheBrazil-FranceCooperation 17

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