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1、Ball-CoveringsofSpheresLecturesbyChengLixinSchoolofMathematicalSciences,XiamenUniversitypptCopyright○cLixinCheng,2012pptCopyright○cLixinCheng,20121/ChengLixin(XiamenUniversity)UndergraduateLectures:Ball-CoveringsofSpheres491TheunitsphereofR?coveredbyopenballsofftheorigin2Ananalyti
2、ccharacterizationofball-coveringsinnormedspaces3Minimumball-coveringsoffinite-dimensionalnormedspaces4Minimumball-coveringsofinfinite-dimensionalspacespptCopyright○cLixinCheng,20122/ChengLixin(XiamenUniversity)UndergraduateLectures:Ball-CoveringsofSpheres491TheunitsphereofR?covere
3、dbyopenballsofftheorigin2Ananalyticcharacterizationofball-coveringsinnormedspaces3Minimumball-coveringsoffinite-dimensionalnormedspaces4Minimumball-coveringsofinfinite-dimensionalspacespptCopyright○cLixinCheng,20123/ChengLixin(XiamenUniversity)UndergraduateLectures:Ball-Coveringso
4、fSpheres49Question1pptCopyright○cLixinCheng,20123/ChengLixin(XiamenUniversity)UndergraduateLectures:Ball-CoveringsofSpheres49Question1LetR?={?=(?(1),?(2),···,?(?)),?(?)∈R,?=1,2,·,?},?(?,?)={?∈R?:‖?‖
5、d?=?(0,1),√︁∑︀where‖?‖=??(?)2isthenormof?∈R?.?=1pptCopyright○cLixinCheng,20124/ChengLixin(XiamenUniversity)UndergraduateLectures:Ball-CoveringsofSpheres49Question1Problem1.Whatistheleastnumberofopenballsawayfromtheoriginwhoseunioncoverstheunitsphere??i.e.Isthereanaturalnumber?such
6、that(1)thereexist?openballs{?(?,?)}?notcontainingtheorigin0such???=1that?⊂∪??(?,?),and?=1???(2)if{?(??,??)}?=1are?openballsnotcontainingtheorigin0suchthat??⊂∪?=1?(??,??),then?≥??pptCopyright○cLixinCheng,20125/ChengLixin(XiamenUniversity)UndergraduateLectures:Ball-CoveringsofSphere
7、s49Question1References.linearspaces,differential(gradient)ofmulti-variablefunctions,convexanalysis(separationtheoremofconvexsets,subdifferentialofconvexfunctions),functionalanalysis(Hahn-Banachtheorem,separabilityofBanachspaces),generaltopology(compactsets)pptCopyright○cLixinCheng
8、,20126/ChengLixin(XiamenUniversit
