Extending Lipschitz functions via random metric partitions

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1、ExtendingLipschitzfunctionsviarandommetricpartitionsJamesR.Lee¤AssafNaorU.C.BerkeleyMicrosoftResearchjrl@cs.berkeley.eduanaor@microsoft.com1IntroductionManyclassicalproblemsingeometryandanalysisinvolvethegluingtogetheroflocalinforma-tiontoproduceacoherentglobalpictur

2、e.Inevitably,thedifficultyofsuchaprocedureliesatthelocalboundary,whereoverlappingviewsofthesamelocalitymustsomehowbemerged.Itisthereforedesirablethattheboundariesbe“smooth,”allowingagracefultransitionfromoneviewpointtothenext.Forinstance,onemaypointtoWhitney’suseofpart

3、itionsofunityinstudyingwhatisnowknownastheWhitneyextensionproblem[36,37].Inthepresentwork,weconsiderwhatisperhapsthemostbasicWhitney-typeextensionproblem,thatofextendingaLipschitzfunctionsothatitremainsLipschitz.Oftensuchamapisextendedbyfirstproducingacoverofthenewdom

4、ain,extendingthemappinglocally,andthengluingtogethertheindividualpieces.Ourmainobservationisthatinmanycases,ifonechoosesarandomcoverfromtherightdistribution,theboundarycanbemade“smooth”onaverage,evenwhenthelocalmapsareindividuallyquitecoarse.Thisinsightleadstotheunifi

5、cation,generalization,andimprovementofmanyknownresults,aswellastonewresultsformanyinterestingspaces.1.1TheLipschitzextensionproblemLet(Y;dY),(Z;dZ)bemetricspaces,andforeveryXµY,denotebye(X;Y;Z)theinfimumoverallconstantsKsuchthateveryLipschitzfunctionf:X!Zcanbeextended

6、toafunctionf˜:Y!Zsatisfyingkf˜kLip·KkfkLip.(IfnosuchKexists,wesete(X;Y;Z)=1).Wealsodefinee(Y;Z)=supfe(X;Y;Z):XµYgandforeveryintegern,en(Y;Z)=supfe(X;Y;Z):XµY;jXj·ng.Estimatinge(Y;Z)isaclassicalandfundamentalproblemthathasattractedalotofattentionduetoitsintrinsicintere

7、standapplicationstogeometryandapproximationtheory.ItisaclassicalfactthatforeverymetricspaceY,e(Y;`1)=1,andKirszbraun’sfamousextensiontheorem[19]statesthatwheneverH1andH2areHilbertspaces,e(H1;H2)=1.Werefertothebooks[3,35]foradetailedaccountofthecasee(Y;Z)=1.Typically,

8、proofsofthefactthate(Y;Z)=1involveshowingthatitispossibletoextendanarbitraryLipschitzfunctiontoanadditionalpointwhilepreservingtheL