ItisconvenientinthefollowingtouseasanauxiliarytoolalsotheEuclideanstructureinducedbythestandardscalarproductofRnInthiswaywecantalkofnormalconesandoftheSteinerpointforexampleandalsousetheEuclidea ID: 255034
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ByKnwedenotethesetofconvexbodies(nonempty,compact,convexsubsets)ofRn,equippedwiththetopologyoftheHausdormetricinducedbythegivennorm(oranyothernorm|thisyieldsthesametopology).Themetricnotionsdistance,diameter,widthrefertothemetricinducedbythenorm.Notionslike`convex',`starshaped'inthespaceKnrefertoMinkowskilinearcombinations,denedwiththeaidofMinkowskiadditionofconvexbodiesandmultiplicationwithnonnegativescalars.SinceitsucestostudycompletesetsinXonlyuptodilatations,wedeneD2asthesetofallcompletesetsofdiameter2(thediameteroftheunitball)inX.ItisthestructureofthissubspaceofKnthatinterestsusinthepresentpaper.TherstnaturalquestiontoaskiswhetherthesetD2isconvex.Itcertainlyiscon-vexinthosespaceswhich,likeEuclideanspace,havethepropertythateverycompletesetinthespaceisofconstantwidth(thisisproperty(A),inaterminologyintroducedbyEggleston[3]):ifK;Lareconvexbodiesofconstantwidthandofdiameter2,then(1)K+Lisofconstantwidthandofdiameter2,for2[0;1].Conversely,supposethatD2isconvex.ThentheproofofProposition3.1in[12]showsthatthespaceXhasproperty(A).Itisknown(Theorem3in[13]isastrongerresult)that,forn3,mostn-dimensionalMinkowskispaces(intheBairecategorysense)donothaveproperty(A).Thus,D2isgenerallynotconvex.Thenextnaturalquestion,whetherD2isalwaysstarshaped,alsohasanegativeanswer.AsimpleexampletothiseectisgiveninSection4(Example2).WeshowinSection3thatD2isstarshapedifandonlyifthespaceXhastheproperty,denotedby(F)in[13],thatthesetofallcompletesetsisclosedundertheoperationofaddingaunitball.Theresultfrom[13]mentionedabovestatesthat,forn3,mostn-dimensionalMinkowskispacesdonothaveproperty(F)ReaderswhobynowaregettingworriedwhetherD2mightevennotbeconnected,canbecomforted:weproveinSection3thatD2iscontractible.FairlysatisfactoryinformationonthestructureofD2isavailableforpolyhedralMinkowskispaces.Asourmainresult,weproveinSection4thatforspacesXwithapolytopalunitball,thespaceDXoftranslationclassesofbodiesinD2istheunionsetofanitepolytopalcomplex.Weconclude,inSection5,withsomeobservationsontheextremeelementsofD2,thatis,thoseelementswhicharenotanon-trivialconvexcombinationoftwootherelementsofD2.2PreliminariesRecallthataboundedsetKinXisdiametricallycomplete,orbrie ycomplete,ifdiam(K[fxg)diamKforallx2RnnK.Thecompletesetshaveausefulcharacterizationintermsofsupportingslabs.AsupportingslaboftheconvexbodyK2KnisanyclosedsetKthatisboundedbytwoparallelsupportinghyperplanesH;H0ofK.ThedistancebetweenHandH0isdenotedbyw()andcalledthewidthof.IfMisanotherconvexbody,wesaythatthesupportingslabofKisM-regularifthesupportingslabofMthatisparalleltohasthepropertythatatleastoneofitsboundinghyperplanescontainsasmoothboundarypointofM(thatis,aboundarypointthroughwhichpassesonlyonesupportinghyperplaneofM).WeusethisterminologyonlyinthecaseswhereMis2 Itisconvenientinthefollowingtouse,asanauxiliarytool,alsotheEuclideanstructureinducedbythestandardscalarproductofRn.Inthisway,wecantalkofnormalconesandoftheSteinerpoint,forexample,andalsousetheEuclideanvolume.Lemma3.LetK2Knbeaconvexbodyofdiameterd0.Forany0,thefollowingassertionsareequivalent:(a)K+Biscomplete.(b)Every(K+B)-regularsupportingslabofKhaswidthd.Proof.Clearly,diam(K+B)d+2.ThebodyKhasasupportingslabofwidthd.SincethewidthsofparallelsupportingslabsareadditiveunderMinkowskiaddition,andeverysupportingslabofBhaswidth2,itfollowsthatK+Bhassomesupportingslabofwidthd+2andhencehasdiameterd+2.Supposethat(a)holds.ByLemma1,every(K+B)-regularsupportingslabofK+Bhaswidthd+2.Itfollowsthatevery(K+B)-regularsupportingslabofKhaswidthd.Letbea(K+B)-regularsupportingslabofK.ThenthereisasupportinghyperplaneHofK+BthatisparalleltoandcontainsasmoothboundarypointzofK+B.ThenormalconeN(K+B;z)isofdimensionone.Wehavez=x+ywithsuitablepointsx2Kandy2B,lyinginsupportinghyperplanesofKandB,respectively,paralleltoH.ByTheorem2.2.1(a)of[14],N(K+B;z)=N(K;x)\N(B;y):SinceN(B;y)=N(B;y),wegetN(K+B;x+y)=N(K;x)\N(B;y)=N(K+B;z);whichisofdimensionone.Itfollowsthatx+yisasmoothboundarypointofK+B.Thus,thesupportingslabis(K+B)-regularandhencehaswidthd.Thisproves(b).Supposethat(b)holds.Letbea(K+B)-regularsupportingslabofK+B.Byasimilarargumentasusedabove,isalso(K+B)-regular,hencetheparallelsupportingslabofKhaswidthd,by(b).Therefore,haswidthd+2.ByLemma1,K+Biscomplete. TheessentialpointofLemma3isthatcondition(b)isindependentof.Therefore,thecompletenessofK+Bforonepositivenumber0impliesthecompletenessofK+Bforallpositivenumbers0(andhencealsothecompletenessofK).Werecallfrom[13]thatthespaceXissaidtohaveproperty(F)if,foranycompletesetK,alsotheMinkowskisumK+Biscomplete.Theorem1.ThesetD2isstarshapedifandonlyifthespaceXhasproperty(F).Proof.IfXhasproperty(F),thenitfollowsfromLemma3that,foranyK2D2andany2[0;1],theset(1)K+Biscomplete.ByLemma1,ithasdiameter2.Thus,D2isstarshapedwithrespecttoB.IfD2isstarshaped,thenitisstarshapedwithrespecttoB,byLemma2.AnycompletesetKofpositivediameterhasahomothetK02D2,andthen(1=2)K0+(1=2)B4 4PolyhedralnormsInthissection,weconsidercompletesetsonlyuptotranslations.ForK2Kn,thesetofalltranslatesK+tofK,wheret2Rn,isdenotedby[K],and[Kn]isthespaceofallthesetranslationclasses,withthequotienttopology.TheMinkowskioperationsofadditionandnonnegativescalarmultiplicationcarryoverfromKnto[Kn],bymeansofthedenitions[K]+[L]:=[K+L]and[K]:=[K]forK;L2Knand0.Therefore,theusualconvexitynotions,likesegment,starshaped,convexcombination,convexset,extremepoint,convexhull,polytope,etc.,makesensein[Kn].Amapping':C![Kn]fromaconvexsubsetCofarealvectorspaceinto[Kn]isMinkowskilinearif'((1)x+y)=(1)'(x)+'(y)forallx;y2Candall2[0;1].WedenotebyDXthesetoftranslationclassesofthebodiesinD2.ThenDXisacompactsubsetof[Kn].Itneednotbeconvex,butitstillmakessensetoconsideritsextremepoints.Theelement[K]ofDXiscalledanextremepointofDXifarepresentation[K]=(1)[K1]+[K2]withK1;K22DXand01isonlypossiblewith[K1]=[K2]=[K].ThesetDXisourobjectofstudyinthepresentsection,forthecasewherethespaceX=(Rn;kk)ispolyhedral,thatis,theunitballBisapolytope.Thisisassumedfromnowon.ThefollowinglemmaleadstoanintuitivedescriptionofDXinsimplecases.Lemma4.Let1;:::;kbetheB-regularsupportingslabsofthepolytopalunitballB.EachK2D2isoftheformK=k\i=1(i+ti)(1)withti2Rn,i=1;:::;k.Conversely,ifK6=;isgivenby(1),thenK2D2ifandonlyifeachi+tithatisK-regularisasupportingslabofK.Proof.Letu1;:::;ukbetheouterunitnormalvectorsofthefacetsofB,anddenotebyH(K;u)thesupportinghalfspaceofKwithouternormalvectoru.IfK2D2,thenK=Tx2K(2B+x)(thesphericalintersectionproperty,seeEggleston[3]),thereforeK=k\i=1(H(K;ui)\H(K;ui)):(2)TheboundinghyperplanesHiofH(K;ui)andH0iofH(K;ui)areatdistanceatmost2apart,sinceK2D2.Iftheirdistanceisequalto2,thenH(K;ui)\H(K;ui)=i+tiforasuitablevectorti.Iftheirdistanceislessthan2,thenbyLemma1noneofthehyperplanesHi;H0icontainsafacetofthepolytopeK,andin(2)wecanreplaceH(K;ui)\H(K;ui)byi+tiforasuitablevectorti,withoutchangingtheintersection.Thus,Kcanberepresentedintheform(1).Conversely,letK6=;begivenby(1).TheneachB-regularsupportingslabofKhaswidthatmost2,andeachK-regularsupportingslabofKisparalleltooneoftheslabs1;:::;k.TheassertionnowfollowsfromLemma1. ToobtainrepresentativesoftheclassesinDX,thatis,elementsofD2uptotrans-lations,wemayxnoftheB-regularsupportingslabsofBwithlinearlyindependentnormalvectorsandtranslateonlytheremainingones,toobtainrepresentationsoftheform(1).6 distance3=4fromtheorigin(seeFigure3).ThesupportingslabsofBare1;:::;4fromExample1and5,withnormalvectors(0;0;1).Againwex2;3;4,theneachtranslationclassofDXhasarepresentativeoftheformP(;):=2\3\4\(1+(1;1;1))\(5+(0;0;1))withsuitablenumbers;2R.InthiscasewendthatDXhaspreciselythesixextremepoints[E1];[E2];[E3],whereE1=P(2 3;1 4),E2=P(1 3;1 4)andE3=P(0;1 4)(seeFigure4).AsindicatedintherightpartofFigure3,thesetDXconsistsofthefollow-ingfourconvexpolytopesoftranslationclasses:convf[E1];[E2]g,convf[E1];[E2]g,convf[E2];[E3];[E3]g,convf[E2];[E3];[E3]g.Notethatneither[E1]+[E1]nor[E2]+[E2]iscomplete,sincetheyarenotintersectionsofballs.ThereisnoelementofDXtowhicheachof[E1]and[E1]couldbeconnectedbyasegment,henceDXisnotstar-shaped. Figure4showsthreeoftheextremepointsofDX,twopointsintherelativeinteriorsofone-dimensionalfaces,andonepointintheinteriorofatwo-dimensionalfaceofDX.ThemainpurposeofthissectionistoshowthatforapolyhedralMinkowskispace,thesetDXisalwaystheunionsetofanitepolytopalcomplex.Inparticular,DXistheunionofnitelymanyconvexsets,andithasonlynitelymanyextremepoints.RecallthatapolyhedralcomplexinsomeRdisafamilyCofconvexpolyhedralsetsinRdwiththepropertythattheintersectionofanytwopolyhedrainCiseitheremptyorapolyhedronofCandthatanyfaceofapolyhedroninCbelongstoC.Apolytopalcomplexisapolyhedralcomplexallofwhosepolyhedraarebounded.WecallafamilyFofsetsin[Kn]apolytopalcomplexifthereareapolytopalcomplexCinsomeRdandamapping':Rd![Kn]suchthatF=f'(P):P2Cgandtherestrictionof'toanypolytopeofCisMinkowskilinear.InExample2,wecantaked=2andforCthecomplexconsistingofthepolytopesQ1;Q2;Q3inR2givenbyQ1:=convf(1 3;1 4);(0;1 4)g;Q2:=convf(2 3;1 4);(1 3;1 4)g;Q3:=Q2andtheirfaces,andfor'themapping(;)7![P(;)].8 Theorem3.IfXisapolyhedralMinkowskispace,thenDXistheunionsetofanitepolytopalcomplex.Theproofofthetheoremrequiresanumberofpreparations.Asalreadymentioned,itisconvenienttousethestandardEuclideanscalarproducth;iasanauxiliarytool.ThisallowsustouseEuclideannormalvectorsofhyperplanesand,later,toemploythesupportfunctionhKofK2Kn,asafunctiononRn.Inparticular,aclosedhalfspacecanbewrittenintheformH(u;):=fx2Rn:hx;uigwithu2Rnnfogand2R.WeshalldeduceTheorem3fromaresultofMcMullen[10],whichisbasedonthetechniqueofdiagramsandrepresentationsofpolyhedra.Wereferto[10]forthistechnique,butwemustexplainMcMullen'sapproachwiththenecessarydetailsthatweneedtoapplyhisresult.Letu1;:::;uNbetheouterunitnormalvectorsofthefacetsoftheunitballB.WewriteUforthek-tuple(u1;:::;uk),k=2N,withui+N:=uifori=1;:::;N.LetbearepresentationassociatedwithU,thatis,alinearmappingfromRkintoRknwithkernelker=f(hb;u1i;:::;hb;uki):b2Rng:(3)Wereferto[10]fortheconstructionofsuchamappingbymeansofa`linearrepresen-tation'.If(e1;:::;ek)isthestandardbasisofRk,wedene ui:=(ei)fori=1;:::;kanddenoteby Uthek-tuple( u1;:::; uk).Thek-tuple UiscalledarepresentationofU.9 Forgiven(1;:::;k)2Rk,thesetP:=k\i=1H(ui;i)(4)iseitheremptyorapolytopewhosefacetshaveonlynormalvectorsfromfu1;:::;ukg.Thesetofallpolytopesoftheform(4)isdenotedbyP(U).IfP6=;,thepointp=(1;:::;k)=kXi=1i uiissaidtobeassociatedwithP.Invirtueof(3),thepointpisassociatedpreciselywiththepolytopesofthetranslationclass[P].Thefollowingresultsareprovedin[10].LetP2P(U).Thenp2pos U,andintP6=;ifandonlyifp2intpos U.ThevectoruiisthenormalvectorofsomefacetofPifandonlyifp2intpos( Unf uig).ForagivenP2P(U),thenumbersiin(4)areingeneralnotuniquelydeterminedbyP,butwecanalwaysassumethati=i(P):=h(P;ui)(5)fori=1;:::;k;hereh(P;)isthesupportfunctionofP.Thepointp(P)=Pki=1h(P;ui) uiiscalledtherepresentativeofP.AlltranslatesofPhavethesamerepresentative.ApointpassociatedwithPisitsrepresentativeifandonlyifitbe-longstotheclosedinnerregion,whichisdenedbyclir U:=k\j=1pos Unf ujg:Thecontinuous,surjectivemapping%:P(U)!clir U;P7!p(P);whichassociateswitheachpolytopeofP(U)itsrepresentative,hasthepropertythat%1(p)2PT(U)forp2clir U,wherePT(U)(usingMcMullen'snotation)denotesthesetoftranslationclassesofpolytopesfromP(U)(asasubspaceof[Kn],hencewiththequotienttopology).AsdenedbyMcMullen[10],atype-coneisanysubsetofclir UwhichisanonemptyintersectionofsetsoftheformrelintposeVwitheV U.Thus,clir Uispartitionedintothetype-cones.Thebijectivemapping%1:clir U!PT(U)hasthefollowingproperties.Itiscontinuous.Restrictedtotheclosureofanytype-cone,itisMinkowskilinear.And,mostimportant:pointspandqofclir Ubelongtothesametype-coneifandonlyiftheirimagesunder%1aretranslationclassesofstronglyisomorphicpolytopes.ThepolytopesP1;P2arestronglyisomorphic(see[14],pp.100)ifdimF(P1;u)=dimF(P2;u)10 isahyperplane.Therefore,theset(Hi)isahyperplaneinRkn(itisnotthewholespace,sinceHi\ker=;).LetS+i;SiRknbethetwoopenhalfspacesboundedby(Hi).CallingtwopointsofRknequivalentiftheybelongtothesameelementsof(H1);:::;(HN);S+1;S1;:::;S+N;SN,weobtainapartitionofRknintotheequiva-lenceclasses.Theyarerelativelyopenconvexpolyhedralsetsandarecalledthecells.Byatype-cellweunderstandanynonemptyintersectionofatype-coneandacell.Sincetheclosuresofthetype-conesformapolyhedralcomplex,byMcMullen'sresult,andclearlyalsotheclosuresofthecellsformapolyhedralcomplex,wededucethattheclosuresofalltype-cellsformapolyhedralcomplex.Lemma6.LetTbeatype-cell,andsupposethatC:=T\%(D2)containsmorethanonepoint.ThenrelbdCrelbdT:Proof.ByLemma5,thesetCisconvex.Suppose,contrarytotheassertion,thatthereexistsapointp2relbdCwithp2relintT.SinceCcontainsmorethanonepoint,wecanchooseapointp02relintCwithp06=p.Wecanfurtherchooseanumber0suchthatq:=p+(pp0)2T,whichispossiblesincep2relintTandp02T.LetP2%1(p),P02%1(p0)andQ2%1(q).Sincep;p0;q2clir U,wehavep=kXi=1h(P;ui) ui;p0=kXi=1h(P0;ui) ui;q=kXi=1h(Q;ui) ui;hencekXi=1[ih(Q;ui)] ui=owithi:=(1+)h(P;ui)h(P0;ui):Thisimplies(1h(Q;u1);:::;kh(Q;uk))2ker:Therefore,thereexistsavectorb2Rnwithih(Q;ui)=hb;uiifori=1;:::;k;whichmeansthath(Q+b;ui)=(1+)h(P;ui)h(P0;ui);i=1;:::;k:(8)SinceP;P0;Qbelongtothesametype-cell,theybelongtothesametype-coneandhencearestronglyisomorphic.AsintheproofofLemma5,wemayassume,afterrenumbering,thatu1;:::;umarepreciselythenormalvectorsoftheP-regularsupportingslabsofP;thenu1;:::;umhavethesamemeaningforP0andQ.Sincep;p02%(D2),relations(6)and(7)arevalid,andwededucefrom(8)thath(Q;ui)+h(Q;ui)=2h(B;ui)fori=1;:::;m:Supposethatforsomejmwehaveh(Q;uj)+h(Q;uj)2h(B;uj):12 SinceB=N\i=1H(ui;hp;uii)\H(ui;hq;uii);K0=N\i=1H(ui;hp0;uii)\H(ui;hq0;uii);wenowdeducethatB=K01 2(p0+q0).Similarly,K1isatranslateofB.Thisshowsthat[B]isanextremepointofDX.Toshowthesecondannouncedpropertyofthisexample,letidenotetheB-regularsupportingslabofBwithnormalvectorsui,i=1;:::;N.Weassume,withoutlossofgenerality,thatuNn+1;:::;uNarelinearlyindependentandconsiderthepolytopesP(1;:::;Nn):=Nn\i=1(i+iui)\n\j=1Nn+j:ThereisaneighbourhoodUof(0;:::;0)2RNnsuchthatfor(1;:::;Nn)2UthepolytopeP(1;:::;Nn)hasthesamenumberoffacetsasBandbelongstoD2.Further,wecannd(01;:::;0Nn)2UsuchthatthepolytopeP(01;:::;0Nn)issimple.ByLemma2.4.12of[14],thereexistsaneighbourhoodVof(01;:::;0Nn)inUsuchthatfor(1;:::;Nn)2VthepolytopesP(1;:::;Nn)belongtothesameequivalenceclassofstronglyisomorphicpolytopes.IfP1;:::;PkarestronglyisomorphicpolytopesinD2,thenthepolytope1P1++kPk(1;:::;k0,1++k=1)belongstoD2(asusedintheproofsofLemmas5and6).Weconcludethatthereisan(Nn)-dimensionalconvexsubsetCofVsuchthatfor(1;:::;Nn)2CthepolytopesP(1;:::;Nn)belongtoD2.Itfollowsthatanyneighbourhoodof[B]inDXcontainsaconvexsubsetofdimensionNn.Asmentioned,Theorem3impliesthatforapolyhedralMinkowskispaceX,thesetDXhasonlynitelymanyextremepoints.OnemayconjecturethatpolyhedralMinkowskispacesarecharacterizedbythisfact,butwehaveonlybeenabletoprovethisinthetwo-dimensionalcase.Inhigherdimensions,wecanshowthatinMinkowskispaceswithastrictlyconvexnorm,thesetDXhasinnitelymanyextremepoints.Thisrequirestodealwithtwotasks:toconstructextremecompletesetsofgivendiameter,andfordierentonestoshowthattheyarenotequivalentbytranslation.Firstweconsiderstrictlyconvexnorms.Lemma7.LetXbeaMinkowskispacewithastrictlyconvexnorm.Let;6=SXbeasetofvectorsofnorm2withdiamS2,andletCbeacompletionofminimalvolumeofSo:=S[fog.Then[C]isanextremeelementofDX.Proof.Sincetheset (So)ofcompletionsofSoiscompact,itisclearthatacompletionC2 (So)ofminimalvolumeexists.WeassertthatCisextreme,inthesetofcompletesetsofdiameter2(noticethatdiamC=diamSo=2).Fortheproof,letK;MXbetwocompletesetsofdiameter2suchthatC=(1)K+M(10)14 andM1:=\t2M0(2B+t);thenM1isaMeissnerbodywiththeroundededgeshavingoascommonvertex.Moreover,M1istheonlycompletionofM0andhence,byLemma7,thetranslationclass[M1]isanextremeelementofD`32.InordertodealwiththesecondtypeofMeissnerbodies,considerthehalfspaceH+:=f(x1;x2;x3)2R3:x32p 2=3ganddenethenewsetsM00:=(2B+a1)\(2B+a2)\(2B+a3)\H+andM01:=\t2M00(2B+t):ThenM01istheonlycompletionofM00andisaMeissnerbodyofthesecondtype,withtheroundededgesformingatriangle.WestatethatM01isextreme,uptotranslations,inthefamilyofcompletesetsofdiameter2.Infact,supposethatM01isaconvexcombinationofcompletesetsAandBofdiameter2.ThenthesetwosetscontaintranslatesofM00(asfollowsfromfact(ii)above),whichimpliesthat[M01]=[A]=[B]. Afterthisdigression,wereturntotheextremepointsofDXingeneral.Theorem4.LetXbeaMinkowskispacewithastrictlyconvexnorm.ThenthesetDXhascontinuummanyextremepoints.Proof.Inthefollowingproof,thenotionsofunitnormalvectorandsphericalimagereferagaintoanauxiliaryEuclideanmetriconX.LetHbeatwo-dimensionalplanethroughoandletJ:=H\@2B,whereBistheunitballofX.Wechooseapointx12Jandlety1beoneofthetwopointsinJ\(J+x1).Thenkx1y1k=2(thisistheknownargumenttoconstructanequilateraltriangleinaMinkowskiplane;seeThompson[15],Theorem4.1.1).Letz6=x1;y1beapointofthearcofJbetweenx1andy1.Thenkzx1k2andkzy1k2.Letx2beapointoftheopenarcofJbetweenx1andz,anddeterminey22Jsuchthatky2x2k=2andy1belongstotheopenarcofJbetweenzandy2.Wecanchoosex2soclosetox1thatstillkzy2k2;ofcourse,ky2x1k]TJ/;ø 1;.90; T; 11;.515; 0 T; [0;2.ThereisaneighbourhoodNofzinXsuchthatkwxik2andkwyik2forallw2Nandfori=1;2.Nowwechoosenpointsw1;:::;wn2N\@2Bsuchthatthereareouterunitnormalvectorsuiof2Batwi,i=1;:::;n,whicharelinearlyindependent.Thatthisispossiblefollowsfromthefactthat2Bisstrictlyconvex.(IfKisastrictlyconvexbody,thenthesphericalimageofarelativelyopensubsetof@Kisarelativelyopensubsetoftheunitsphere.)WedeneSi:=fo;xi;yi;w1;:::;wngfori=1;2,thendiamSi=2.LetCibeaminimalvolumecompletionofSi.ByLemma7,[Ci]isanextremepointofDX.SupposethatC1andC2weretranslatesofeachother.Forj2f1;:::;ngwehaveCi2B+wj,henceo2@Ci,andujisanouternormalvectortoCiato.ThepointoistheonlypointofCiatwhichthenormalconecontainsthelinearlyindependentvectorsu1;:::;un.ItfollowsthatC1andC2,whicharetranslates,mustbeidentical.Butthenx1;y22Ci,whichisacontradiction,sincekx1y2k]TJ/;ø 1;.90; T; 12;.284; 0 T; [0;2.Thisprovesthat16 closedconvexsubsetofB\(B+x)\(B+y)isnotofconstantwidthone.Similarly,wehaveM=C.Thus,Cisextreme.NowweassumethattheunitballBofXisnotapolygon;thenithasinnitelymanyextremepoints.Wesaythatanextremepointxissuitableif@Bdoesnotcontainasegmentoflengthatleastonethatisparalleltox.Clearly,atmostnitelymanyoftheextremepointsarenotsuitable.Letx0beanaccumulationpointofextremepoints.Thereisapointz2@B\int(B+x0)suchthearcof@Bbetweenx0andzcontainsinnitelymanyextremepoints.Wechooseacyclicorderon@B,andwetaketwodierentsuitableextremepointsx1andx2fromtheopenarcbetweenx0andzsuchthatx0;x1;x2;zfolloweachotherincyclicorder.Wechoosey1astherstpointin@B\(@B+x1)afterzincyclicorder,andy2isdenedsimilarly,forthestartingpointx2.LetCibethecompletionoffo;xi;yig,i=1;2,asconstructedabove,withstartingpointxi.Asshown,C1andC2areextremebodiesofconstantwidthone.SupposethatC2isatranslateofC1,sayC2+t=C1witht2R2.Clearly,t6=o.Sincex2+t2C2+t=C1Bandx2t2C1t=C2B,thepointx2isnotanextremepointofB,whichisacontradiction.Thus,C1andC2arenottranslatesofeachother.Itfollowsthatthereareinnitelymanytranslationclassesofextremebodiesofconstantwidthone. Example3.ThefollowingexampleshowsthatMinkowskiReuleauxtrianglesarenotalwaysextreme.LettheunitballBofXbeananelyregularhexagon,withverticesv1;v2;v3;v1;v2;v3,incyclicorder.Performingtheconstructionofthepreviousproofwiththestartingpointx1=v1,weobtainaMinkowskiReuleauxtriangleC1whichisanordinarytriangle.Startingwithx2=v2,wegetadierentMinkowskiReuleauxtriangleC2,whichisalsoanordinarytriangle.ThehexagonC:=1 2(C1+C2)istheMinkowskiReuleauxtrianglewhichisobtainedwhenwestartwiththe(non-extreme)point1 2(x1+x2).Thus,Cisanon-extremeMinkowskiReuleauxtriangle.Remark.Kallay[7]hasprovedanecessaryandsucientconditionforaconvexbodyofconstantwidthinatwo-dimensionalMinkowskispacetobeextremeinthesetofallconvexbodiesofthesameconstantwidth.Wefoundtheproofgivenabovemoredirectthananapplicationofthiscriterion.(Notealsothatin[7],Theorem2,thecondition2[0;2)hastobereplacedby2[0;),asshownbytheexampleofthespace`21,andalsobytheproofgivenin[7]).References[1]T.BonnesenandW.Fenchel,TheoriederkonvexenKorper.Springer-Verlag,Berlin,1934.Englishtranslation:TheoryofConvexBodies,editedbyL.Boron,C.ChristensonandB.Smith,BCSAssociates,Moscow,ID,1987.[2]G.D.ChakerianandH.Groemer,Convexbodiesofconstantwidth.InConvexityandItsApplications(P.M.GruberandJ.M.Wills,eds.),pp.49{96,Birkhauser,Basel,1983.[3]H.G.Eggleston,SetsofconstantwidthinnitedimensionalBanachspaces.IsraelJ.Math.3(1965),163{172.18