凸体间几种仿射不变距离的等价性与估计

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1、数学杂士，I：．、Vo1．35(2015)J．ofMath．(PRC)NO．2THEEQUIVALENCEANDESTIMATEOFSEVERALAFFINEINVARIANTDISTANCESBETWEENCoNVEXBoDIESSHAOYong-chong，GUOQil、Departmento}MathematicslSuzhouUniversity0}ScienceandTechnology}Suzhou215009,China1Abstract：Inthispaper，theequivalencesofseveraldif

2、erentdefinitionsoftwotypesofBanach—Mazurdistancebetweenconvexbodiesareshownrespectively，theconditionsunderwhichthesetwotypesofBanach—Mazurdistancecoincidearediscussed，andtheBanach·Mazurdistancesbetweenpolarbodiesofspecialconvexbodiesarestudiedaswel1．Theresultsobtained

3、herewillplaysomeroleinestimatingthebestupperboundofBanach—Mazurdistances．Keywords：afineinvariantdistance；convexbody；Banach—Mazurdistance；polarset2010MRSubjectClassification：52A38Documentcode：AArticleID：0255—7797(2015)02—0287-071IntroductionDenoteby／C”thefamilyofallcon

4、vexbodiesfi．e．theconvexsetswithnonemptyinterior)intheEuclideanspace．Othernotationarereferredto[15]．DenotebyAff(N)(GL())thefamilyofallaffine(1inear)mapsfromtoandbyaff(R”)thefamilyofallaffinefunctionalson．Asarule，elementsofaredenotedbylower—caseletters，subsetsbycapitals

5、andrealnumbersbysmallGreekletters．GivenC∈．thenbywemeanthehomotheticcopyofCofratiowiththecenterattheoriginO，andwewrite∞C：：(C’一)+．Inthewell—knownpaper[9]，JohnprovedthatforeverycentrallysymmetricconvexbodyC∈”withtheoriginasitscenter，thereisauniqueellipsoidE(i．e．anaffinei

6、mageoftheunitballin)suchthatECCcE，whichinsomesensedescribesthesimilaritybetweenCandE．Lateron，itwasrealizedthattheJohn’Sapproachprovidedactuallyawaydescribingthediferencesbetweenconvexbodies，therefore，asaconsequence，several(similarly，translationoraffineinvariant)distan

7、cesbetweenconvexbodies，suchastheSO—calledBanach—Mazurdistanceetc，wereintroducedandstudied(see[1-3，5，8，10—141)．Itturnsoutthatthesedistancesdefinedforconvexbodiesplaysomerolesinconvexgeometricalanalysisandotherrelatedmathematicsareas(cf．【4，6，71)．Receiveddate：2012--10-30

8、Accepteddate：2013--04-11Foundationitem：SupportedpartiallybyNationalNaturalScienceFoundationofChina(11271282)；partiallybytheG