linear algebra and its applications convexity

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1、CHAPTER12ConvexityConvexityisaprimitivenotion,basedonnothingbutthebarebonesofthestructureoflinearspacesoverthereals.Yetsomeofitsbasicresultsaresurprisinglydeep;furthermore,theseresultsmaketheirappearanceinanastonishinglywidevarietyoftopics.Xisalinearspaceoverthereals.Forany

2、pairofvectorsx,yinX,thelinesegmentwithendpointsxandyisdefinedasthesetofpointsinXofformax+(1-a)y,0

3、set.(c)K={x},asinglepoint.(d)K=anylinesegment.(e)LetlbealinearfunctioninX;thenthesetsl(x)=c,calledahyperplane,(2)l(x)

4、ley&Sons,Inc.187188LINEARALGEBRAANDITSAPPLICATIONSConcreteExamplesofConvexSets(f)Xthespaceofallpolynomialswithrealcoefficients,Kthesubsetofallpolynomialsthatarepositiveateverypointoftheinterval(0,1).(g)Xthespaceofreal,self-adjointmatrices,Kthesubsetofpositivematrices.EXERCI

5、SEI.Verifythattheseareconvexsets.Theorem1.(a)Theintersectionofanycollectionofconvexsetsisconvex.(b)Thesumoftwoconvexsetsisconvex,wherethesumoftwosetsKandHisdefinedasthesetofallsumsx+y,xinK,yinH.EXERCISE2.Provethesepropositions.UsingTheorem1,wecanbuildanastonishingvarietyofc

6、onvexsetsoutofafewbasicones.Forinstance,atriangleintheplaneistheintersectionofthreehalf-planes.Definition.ApointxiscalledaninteriorpointofasetSinXifforeveryyinX,x+ytbelongstoSforallsufficientlysmallpositivet.Definition.AconvexsetKinXiscalledopenifeverypointinitisaninteriorp

7、oint.EXERCISE3.Showthatanopenhalf-space(3)isanopenconvexset.EXERCISE4.ShowthatifAisanopenconvexsetandBisconvex,thenA+Bisopenandconvex.Definition.LetKbeanopenconvexsetthatcontainsthevector0.WedefineitsgaugefunctionpK=pasfollows:ForeveryxinX,p(x)=infr,r>OandxinK.(5)rEXERCISE5

8、.LetXbeaEuclideanspace,andletKbetheopenballofradiusacenteredattheorigin:11x11