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1、Page1From:"AdvancesinDiscreteandComputationalGeometry",B.Chazelle,J.E.GoodmanandR.Pollack,eds.,ContemporaryMathematicsvol.223(1999).AMS,Providence,RI.Pp.163-199.BrankoGrünbaum:ACOPTICPOLYHEDRA1Abstract.Acopticpolyhedraarepolyhedrain3-space,withsimplepolygonsasfacesandwi
2、thnoselfintersections.Thesepolyhedraaregeneralizationsofconvexpolyhedra,andpresentavarietyofinterestingpropertiesandopenproblems.Amongthemostchallengingisthe"generalrealizabilityconjecture,"accordingtowhicheverycell-complexdecompositionofanorientable2-manifold(satisfyin
3、gsomenaturalconditions)isisomorphictoanacopticpolyhedron.Theknownpartitialresultsonthisconjecturearegiven.Definitionsandconceptsthatmaybeusefulinfuturestudiesarepresented,togetherwithavarietyofillustrativeexamplesandadditionalopenquestions.1.Generalintroduction.Thetheor
4、yofconvexpolytopeshashadaphenomenalfloweringduringthelastfiftyorsoyears,andisatpresentamaturefield2.Henceitseemstobetheappropriatetimetostartthesystematicstudyofmoregeneral,notnecessarilyconvex,polyhedraandpolytopes.Therearemanyreasonsforsuchactivity:(i)Thecollectionofo
5、bjectstobestudiedisvastlygreaterandmoreinterestingifnotrestrictedbyrequiringconvexity.Inparticular,suchpolyhedracanbeusedtomodelavarietyoforientableandnon-orientablemapsinavisuallyaccessiblemanner.(ii)Convexityisnotessentialformanyresultsintheformulationofwhichitisassum
6、ed.Butregardlessofwhetheracertainpropertycharacterizesconvexpolyhedraornot,theinvestigationofitsrangeofapplicabilityisboundtoproducenewinsights.Isconvexityjustaconvenientassumption,whichmakesitpossibletocarryout1ResearchsupportedinpartbyNSFgrantDMS-9300657.Manyofthenewr
7、esultsandinsightspresentedwereobtainedinlong-termcollaborationwithG.C.Shephard,buttheauthoraloneisresponsiblefortheviewsandstatementsformulatedinthepaper.2Anattractiveandup-to-dateintroductiontothistopicisZiegler'sbook[Z1].Page2acertainproof,ordothereexistsomelimitation
8、sonthevalidityofthetheoreminthenonconvexcase;ifso––whatarethelimitations,andwhathappensiftheyareexceeded?Agood
