Module 4 Basic Concepts OF Convexity PDF

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Module 4: ConvexityLesson 1: Basic Concepts _______________________________________________________________________________________________Learning OutcomesAt the end of the lesson, you should be able to: a. deine convex sets; b. distinguish the diference between convex and non- convex sets; and c. ...

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Modern Geometry

Modul e4:Convexi t y Lesson1:Basi cConcept s _______ _________ ________ ________ _________ ________ ________ ________ _________ ________ ________ _____ Learni ngOut comes Att heendoft hel esson,youshoul dbeabl et o: a.defineconvexset s; b.di st i ngui sht hedi ffer e ncebet weenc onvexandnon-conv exset s;and c. pr ov et hatt hei nt er sect i onoft woconvexset si saconvexset . _______ _________ ________ ________ _________ ________ ________ ________ _________ ________ ________ _____ Learni ngCont ent Thest udyofconv exi t yi n Eucl i dean geome t ryi saver ymodern dev el opment .Asi st r ue wi t ht heappr oach t hr ough t r ansf ormat i ons,t hest udyofconvexset sempl oysmanybasi ci deas ofmoder n mat hemat i c st o pr ovi deadded meani ngt o concept st hathavel ong been a par tof ge omet ry .The wor ds " convex"and " convexi t y"ar e common out si de ofmat hemat i cs.Conve x meanscurvi ngout war d.Forexampl e,aconvexl ensbul gesout war d.For t unat el y ,t heuseoft he wor dconvext odescr i bepar t i c ul arset sofpoi nt si n mat hemat i csi sonl yacar ef ulr efine mentof t hi scommon meani ng.St udyFi gur e3. 1,par t saandb.t odi scovert hebasi cdi ffer encebet ween convexse t sofpoi nt sandt hoset hatar enonc onvex.

Forac onvexsetofpoi nt s,suchasK i nFi gur e3. 2a,anysegmentwi t hendpoi nt sA andB i nt hesetl i eswhol l yi nt heset .Thi si snott r uef oral lpai r sofpoi nt si nt henonconvexseti n Fi gur e3. 2b.Forexampl e,l i nesegmentCD i ncl udespoi nt sout si desetK' .

Modern Geometry

DEFI NI TI ON.Ac onvexseti sasetofpoi nt sKsuc ht hati fA,B ϵ

´ ⊂ K,t hen AB

K.

Thesymbol ⊂ means" i sasubse tof . "Al t hought hi sdefini t i onofconvexse tmay seem r es t r i c t i ve,i ti s gener alenough t oi ncl ude many oft he common se t s ofpoi nt s st udi edi n Eucl i dean geomet ry .Exampl esar eaci r cul arr egi on,somepol ygonalr egi ons, t si nt er i or ,andas pher i calr egi on. segment s,anangl e( wi t hmeasur el esst han π )andi Theempt ysetandase tconsi st i ngorasi ngl epoi ntar ebot hconvexbyagr eement . Toshow t hatapar t i cul arse ti sconv exbydefini t i on i snotal wayseasy .I n addi t i on t o Theor em 3. 1 bel ow,anal yt i cgeomet ryi sof t en empl oyed.You shoul d assumeher eand el sewher ei nt hi sc hapt er ,unl esss t at edt ot hecont r ary ,t hatt hevar i abl esf orcoor di nat es ofpoi nt sar eel ement soft hese tofr ealnumber s. Thepost ul at i onalsys t em f orasecondar yschoolgeomet r yi nc l udest heassumpt i on t hat a hal f pl ane i s a conv ex se t .The de t er mi nat i on ofwhe t hera seti s conv ex ornoti s f aci l i t at edbyt hef ol l owi ngt heor em. THEOREM 3. 1.Thei nt er sect i onoft woconv exse t si saconv exse t .

A S

K1

I n Fi gur e 3. 3,l et K 1 and

B

K2

be any t wo convex set s and l e t se t S be t hei r

´ lpoi nt sof AB i nt er sect i on.ForA,B ϵ S,al ´ Al lpoi nt sof AB

K

ar eel ement sof K 1

because K 1

ar eal soel ement sof K 2 ,becausei ti sal soconv ex.Hence,

i sconvex.

K1 ∩ AB ⊂ ¿

K2

K2 i ) ;t hus, K 1 ∩ sconvex.Theor em 3. 1 maybegener al i zed t omor et han t woset s,asi n Exer ci s e3 ofExer ci s eSet3. 1.Thet heor em mayal so beused t oshow t heconvexi t yofot her common set sofpoi nt s.Forexampl e,anangl e( measur el esst han π ¿ andi t si nt er i orcanbe definedast hei nt er sect i onoft wohal f pl anes,sot hatTheor em 3. 1appl i es. An al t er nat i veappr oach i st heuseofanal yt i cgeomet ryt opr ovet hatset sofpoi nt sar e convex.Thi sappr oachcanevenbeusedf ora hal f pl ane. EXAMPLE. Show t hat S= { ( x , y ) : x >0 }

S

i sapl aneconvexset .

i st hesetofpoi nt swi t hcoor di nat es( x,y)sucht hat x> 0 .

x 1, y 1 x y )and 2, 2 )beanyt wopoi nt sof S asi n Fi gur e3. 4.such t hat x 1> 0 , A¿ B¿ x ´ , x 2 ≥ x k ≥ x1 ,hence x k > 0 and oranypoi nt k ,y)of AB x 2> 0 ,and x 2 ≥ x1 .Then f P¿ ´ ⊂ S ,sot hat S i sconvexbydefini t i on. AB Let

Modern Geometry

y (

A

(

P B x FI GURE 3. 4 Be f or e expl or i ng mor e compl ex pr oper t i es ofconvex set s,i ti s necessary t oi nt r oduce sever albasi cconc ept sf r om moder nge omet ryt hatwi l lbeusef ulher e. s cont i nuousat x=a i fand onl yi f ,gi ven e> 0 ,t her e Recal lt hata f unct i on y=f ( x ) i exi s t sa δ sucht hat

|f ( x ) −f (a)|< e if |x−a|< δ Af unct i on i sc ont i nuousi fi ti scont i nuousate ver ypoi ntofi t sdomai n.Forexampl e,t he f unct i oni nFi gur e3. 5ai sc ont i nuous,wher east hef unct i oni nFi gur e3. 5bi snot . Acur vei st hegr aphofasetofequat i onsoft hef orm x=f ( t ) , y =g(t) ,f or f and g cont i nuousf unct i onsandt hedomai nof t ani nt ervalofr ealnumber s.

I ft hef unct i oni sone-t oone,exceptpossi bl yatt heendpoi nt s ,t hecurvei sasi mpl ecurve,andi f t hepoi nt sc orr espondi ngt obot h endpoi nt soft hedefini ngi nt ervalar ei dent i cal ,t hecurv ei sa cl osed cur ve.Fort hi s sect i on,you wi l lneed t or ec ogni zeexampl es oft hese vari ous t ypes of curvesf r om dr awi ngssot hatt heequat i on need notbedeal twi t h.I nt ui t i vel y ,asi mpl ecl osed curvei st houghtofasacur vet hatbegi nsandendsatt hesamepoi ntbutdoesnotcr ossi t sel f ; t hus,t her ei sonl yonei nt er i or .( SeeFi gur e3, 6. )Thesetofpoi nt son andi nsi deasi mpl ecl os ed curveoranangl ei napl anei scal l edapl aner egi on.

Modern Geometry

ASSESSMENT TASK Answert hef ol l owi ng:Submi tyourout put svi agoogl edr i ve.Thel i nkwi l lbef orwar ded. 1.Whi choft heseset sar econvex? a.Anangl e f .St r ai ghtl i ne b.I nt er i orofanel l i psoi d g.Ar ay c . As i ngl epoi nt h.At r i angl e d.At ri angul arr egi on e. Ar ect angul arr egi on 2.Whi choft heseset sar econvexse t s? a.Aci r cul arr egi onwi t honepoi ntont heboundaryr emoved. b.Ar ec t angul arr egi onwi t honev er t exr emov ed. c. Ar ect angul arr egi onwi t honepoi nt ,notaver t ex,ont heboundaryr emoved. d.Aci r cul arr egi onwi t honei nt er i orpoi ntr emoved. 3.Pr ovet hatt hei nt er sect i onofanycol l ect i onofconvexset si saconvexset . 4.I st heuni onoft woconvexset severac onvexset ? I si tal waysaconvexset ? 5.Pr ovet hatat r i angul arr e gi oni saconvexs et . saconvexset . 6.Show anal yt i cal l yt hat S={ ( x , y ) : x>3 } i

REFERENCES JAMESR.SMART Cal i f or ni aSt at eUni ver si t y ,SanJose

Modern Geometry

Pr epar edby:PAULJOHN M.AGCAOI LI ,LPT I nst r uct or...