A convex hull characterization

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Pue d ppled Mthets Joul 4; (: 4-48 Pulshed ole My 4 (http://www.seepulshggoup.o//p do:.648/.p.4. ove hull htezto Fo Fesh Gov Qut Deptet DISG Uvesty of Se Itly El ddess: fesh@us.t (F. Fesh qut@us.t (G. Qut To te ths tle: Fo Fesh Gov Qut. Cove Hull Chtezto. Pue d ppled Mthets Joul. Vol. No. 4 pp. 4-48. do:.648/.p.4. stt: Codtos so tht veto elogs to ove hull e oted. Multle ove futos e osdeed. If these ps e defed o ove set t s oted the lge epesso. s pplto fte ges wth le ove pyoff e studed. Keywods: Cove Le Ge. Itoduto Soe popetes of the lss of the -le ove futos e osdeed. These ps see to e the tul eteto of -le futos f the do s ouded. Gve le ove ppg defed o ody t s possle to eted ts defto f guet s outsde of the do. Ths popety eles us to fd the lytl fo of -le ove futo φ : X Y f X s ove set whh ots the ull veto. ty eleet of ove set e epeseted thee dffeet wys usg fte sequee of eleets of the set; these epesettos e oted y the followg theoes: the Shpley-Fol le Cthéodoy s theoe d Fehel-But theoe. Fo eple y the lst theoe fo y eleet sof ove oeted set X sequee of eleets of X ests suh tht s s epessed y ove oto of the sequee. I spte of ths t s well ow tht ty ove set hs ot ove ss tht s t s possle to fd uque sequee wth fte ue of eleets of the set whh sp y le ove R otos y othe. I lge t s poved tht s fee ove the stdd ss ths ples soe popetes oe of these s tht ty sequee of odule y e wtte s le oto of the vetos R t ( e t( e fo uque ophs :. Ths oept of fee set s ow oved to ove sets y le ove futos. Oe esult oted ths ppe s tht ove oto fo eleet of ove oeted set s epessed y uque ove oto of vetos φ e e fo le ove futo φ tht ( s y eleet of detees φ suh tht ξ. e Codtos e oted ode tht veto elogs to ove hull f t s sped y fte set. The odtos gve ew epesso to the ove hull. It s ow see [7] d [8] tht two pesos fte syet ges wth opt ove set s sttegy spe d le ove futo s pyoff hve solutos. I the lst seto we ot the fo of the pyoff of these ges.. Fee Cove Sets Let S e set of vetos R wth the ull veto S the deote y X os the ove hull of S. Suppose the l ue of lely depedet vetos X d e set of lely depedet vetos of X. -le ove ppg : fo s defed y φ ( whee d R. Eple.Cosde the futo f ( y y y Let the f ( y y d ( y y eve f does ot est so f

Pue d ppled Mthets Joul 4; (: 4-48 4 f ( y s ot le futo. Whees fo < < f ( y ( ( y ( y ( y f ( y ( f ( y tht s f ( y s ove le futo of eh vle septely. The followg s ow s stteet Poposto. Let φ : X R e -le ove ppg d X ove suset of φ ( X s ove R. R. The ge Poof. The ge ude φ of evey ove oto of eleets of X s ove oto R. pot of ove hull hs thee ow epesettos y sequees of eleets of the se ove set. The fst epesetto s fouded o the Shpley-Fol le see [] [] d [].Cosde fte fly " of suset of R the Mowsy su # # s defed y eleet-wse ddto of vetos # $# % # : # & The Mowsy su hs edte popety: fo y oepty fte fly of sets of R ( (. So f # ( d # # % # wth # (. The epesso of sy # % # depeds o the se pot s. Fo the epesetto # # % # the spe R holds Le (Shpley-Fol If the ue of s gete th the deso of the spe the pot s hs the epesso # # # ** -** whee # ( d #. I othe wods # ** ( -**. By the Cthéodoy s theoe y s os R y e epeseted s ove oto of eleets of S. I ptul f S s oeted the Fehel-But theoe see [] d [8] poves tht y s os e epessed s ove oto of eleets of S. Thesethee epesettos of eleet do ot estlsh the estee of ove ss fo o S euse the eleets y deped o the ptul s to e oputed see [6]. Nevetheless the et defto ssets the odto ode tht ove set s fee. Fo defto of fee odule o suset see [4]. I the followg we peset dffeet epesetto fo eleets of ove set sed o the defto of fee ove set. Defto. Let K e suset of ove set X R d let : K X e the seto of K X. Deote y suset of R the X s fee ove K f fo evey futo f : K uque le ove ppg φ : X ests suh tht φ f. Eple.. Let. $ & / R e suset d (. ts ove hull. We wt to pove tht (. s fee ove..e. fo evey :. uque le ove futo : (. ests suh tht. Uty. The odto es tht d. The fo evey :. we hve to pove the uty of : (. suh tht d. Fo y " (. d fo evey le ove futo holds " so s uvolly defed. Estee. The futo defed y s le ove ft fo evey (. Epltly the epesso of the uque le ove s gve y " 4 5 6 7 8 9 6 7 Whee t s supposed d : R fo y " (. wth fed d. The φ s the t 8 9. The udelyg popety of the eple ove wll e poved oe geel settg. Soe popetes of the ppg φ : X follow. Se X s ove X < < ( X. I ptul X. Moeove Φ ( supposed ( ( d Φ sewsyet t follows ( opg d t follows Φ sewsyet d Φ ( ( ( Φ ( (

4 Fo Fesh d Gov Qut: Cove Hull Chtezto Let X R d N {} suh tht < y the ( t holds φ ( (4 Poposto. Let R ( R d X > the X ( X fo < <. Poof. If < < stsfes the ( ( so X. If the ( ( d X. The et poposto eteds the ( d defes le ove ppg f ts guet s outsde the ody Poposto. Let e vetos X d δ R the le ove ppg φ : X stsfes φ ( δ ( δ φ ( δφ ( Poof. Let < µ ν < suh tht @ B C D (5 E d δ <. It s lwys possle to detete suh ues µ ν. Ideed f < the elto F @ B C D E ees δ µ ( ν the fo eple suppose E fo > δ t holds δ < µ <. The y the Poposto. the vetos X d the ppg φ s defed o the. φ µ µ ν e ( δ ( ν ( δ ( ν tht s δ ( ν y the ( µ δ µ ( µ ( ν ( ν µ ( µ ( ν ( ( ν µ ( ν ( ( ( ( ( µ ν δ ( δ Theoe. Let φ : X Y e -le ove futo d X ove set wth X the φ y e epessed y le oto of ( φ ( whee X sp the vetos X. Poof. The oull vetos X y e wtte s G % G G G γ ( γ N {} γ γ whee. By ppltos of the Poposto. to the guets of φ t follows φ ( φ ( ( ( ( ( ( ( φ (

Pue d ppled Mthets Joul 4; (: 4-48 4 ( ( < < φ ( ( ( ( (6 whee s the opleet of D wth espet to the set " d the su (6 s ove ll the susets Now use the ove lety o ll the vetos φ ( whee y the ( φ ( ( φ ( ( φ ( s s ( ( < < φ ( ( ( (7 s the opleet of s s whee wth espet to the set d the su (7 s ove ll the susets. So φ s oted y le epesso φ ( ( whee soe I the -desol veto spe e the ull veto. R deote y S the ove hull of the vetos e e } of the stdd { ss. S s opt oeted ove set d ts eleets y e epessed y ove otos of the ut vetos e e }. { Theoe. The set S s fee ove the stdd ss e e } of { R. Poof. Let K { e e } the odto φ f ples y φ ( e f ( e d the futo φ s defed φ ( ξ e ξe ξ ξ R ξ Uqueess. If φ s le ove futo wth φ ( e the fo y S ξ e ξe so φ s uque. Estee. ξ e ξ ξ ξ defes futo φ : S. Let us pove tht φ s le ove futo. let ξ e ξ ξ the e S φ ξ e ξ e ξ ξ e ξ e ( By the Fehel-But theoe y eleet of opt oeted ove set s epessed s ove oto of the sequee of vetos of tht s ξ ξ. By the theoe. ests uque le ove futo φ suh tht φ ξ ξ ξ e ( e e ξ so y eleet y e epessed s ove oto of the vetos φ e e. I othe wods ( y detees le ove futo φ suh tht ξ e. Eple. Let e ove oeted set R. If ξ ξ ξ ξ ( s eleet of the y the theoe. t follows φ ( ξe ξe ξ ξ ξφ ( e ξφ ( e whee φ : S s le ove. Ths ples e e d so φ ( ( S. Cove Hull Chtezto Let e fte set of vetos R the ts ove hull s defed y

44 Fo Fesh d Gov Qut: Cove Hull Chtezto whee { o } { ( Σ Σ s the ut sple R }. The et poposto lys dow odto so tht veto elogs to ove hull. Theoe. Let o } e ove hull whee { ( s t wth the o{ } ff (. (. The detets hve the se sg. Poof. Suppose o{ } the fo ( Σ tht s. The s the soluto of the syste. Ipose we ot d so the (. Covesely f ( d ( hold ovg wd we get suh tht the o{ }. Net step s the eteso of the Theoe. to ove hull gve y the vetos R wth > d (. Fo ths pupose t s eessy to todue soe geelztos of eleety deftos le lge. Fo oe detls see [5]. The ultde I I of legth s defed y {( : < < } esdes we defe fo fed tul ue ( I {( p : whee } < < p < < Let F e feld wth h ( F the fo ty t osde the le fo the petes F gve y Defto. Let : F ( F e p defed y I I If ( the e the Plüe oodtes of the suspe sped y the ows of. Eple. Fo the t wth ( The defto of ofto ty t. Defto.. e eteded to Let F e t wth ( the ofto of the ety < oly f s defed y : fo ( of fo d th ow e whee the lst detet the -th ow s e ll (.e. the -th ut veto. The su s ove ( I d ( I Slly fo Eple. > Fo 4 F. d (

Pue d ppled Mthets Joul 4; (: 4-48 45 4 4 4 ( 4 4 ( 4 4 s the eple t wll e useful to wte the ues of the ows (olus volved the epso of the ofto le pees (dees. Let C ( the -ofto t of. The tspose of C.e. the -dots of wll e deoted d y. Theoe. y osstet le syste whee fo y F ( hs s set of solutos d F suh tht (. d Poof. ( e the (-veses of the the solutos of d e (. The Theoe. s evdet Ce s ule geelzto. ll tht s eessy to the et poposto s ow edy. Theoe. Let o } e ove hull whee { ( s t wth ( the o } ff { ( γ γ γ whee the oftos fo. γ e oted y the t C ( The sus wth hve the se sg of. Poof. Suppose o } the { 4 ( Σ tht s. The y Theoe s the soluto of the syste. Ipose usg lety y ows t follows d so the (. By If we ot the (. s ull the soluto of the syste the o. s spel se f oly ( e oull the soluto of d e etee pots the s etee pot of the ove hull. Eple. veto s o } wth { R ( f t stsfes the followg odtos ( ( Gve y t t suh tht ( ( ( s lled (-vese of. Let d C d (. The ( s the set of (-veses of ff (-vese spe N s the t (. (. d ( see [9]. ( RN wth pesed ge R d ull d ( suh tht d ( R d ( y y N By the et poposto t wth Σ s oted. Theoe.4 Let s gve y suh tht C d (. The t ( Q( I ( ( Q C Q. whee ( deotes the Mooe-Peose vese of. Poof. It s ow tht the t equto XB C hs soluto ff CB B C whh se the geel soluto s X CB Q( I BB whee Q s ty t of ppopte ode. Ipose the

46 Fo Fesh d Gov Qut: Cove Hull Chtezto equto B d C se ( the equto hs the geel soluto ( Q( I ( Q C Q. Theoe.5 Let o ( e ove hull. The followg fou stteets e equvlet (. o ( (. the le syste Σ (. o ( o( (d.. hs the soluto Poof. It s edte tht ( d ( e equvlet. Let us show tht ( d ( e equvlet. Suppose o( o( the so wth Σ. Covesely suppose wth Σ tht s d o( ut t lso whee ( Σ the o(. Moeove o( s δ δ δ δ δ δ ( ( δ δ ( δ δ se δ δ δ δ ( δ δ δ ( ( d δ δ δ ( δ δ e oegtve the o(. ( (d. Let suh tht Σ the thee ests so. Covesely the elto s stsfed y the thee ests o(. Eple.4 Cosde the ove hull o ( R d let ( Σ the ( q q ( q q ( q q { ( q q ( q q ( q q } q q ( q q ( q q { q q ( q q ( q q } ( ( q q q q q q { ( ( q q q q q q } the odto so ( ples fo ( o { : R ( } 4. Syet Two Pesos Ges Deote y Γ ( X X the tplet of two pesos Φ fte syet ge ol fo wth X the full sttegy spe see [7] d [8]. X os s the ove hull of suset S of Eulde -desol le spe R wth the ull veto S d let Φ : X X R e the le sewsyet ove pyoff futo. Epltly pose the followg os If X whee X d the y y y y X lewse f yδ yδ y Y whee y X d δ δ the y δ y δ y X Fo evey y X X t holds Φ ( y y The pyoff futo followg popetes Φ : X X R stsfes the X y the seod o Φ ( tht s Φ (. y X < < se X s ove ( X. Moeove y ( y y ( y tht s y y ( y (8

Pue d ppled Mthets Joul 4; (: 4-48 47 y y t follows y y ( y (9 opg (8 d (9 t follows Φ ( y y y y ( I the se wy of theoe. the lyt fo of the pyoff Φ s ow oted. Fo y two vetos X let ( the y the poposto. t follows Φ ( Φ ( ( ( ( ( usg ove lety ( ( ( ( ( ( ( ( ( ( ( ( y the ( ( ( ( ( ( ( ( ( ( ( ( ( By the syety of the ge Φ (. The el vlue Φ s so deteed y Φ d ( ( Φ (. Deotg y ( y soluto of the ge the hoe of the Φ d Φ ( ust stsfy the followg equltes d d ( Φ ( Φ ( Φ ( y Φ ( y Φ ( y y y y Suppose X e opt ove set d defe * y { : X y y} { y * : y X y y} * * y X X the y the posed odtos o Φ d X y d e ot epty d opt ove sets. The p Φ : X X { y} ( y X s uppe seotuous d we use the followg poposto see [8] Theoe 4. (Kut s fed-pot theoe If Φ ( y { } s uppe seotuous : y p o opt ove ove set the t lest eleet suh tht ( y y. X X d { Φ ( y} s ( y X X ests By the theoe 4. eleet ( y ests wth y d y. Se y t holds y y d y y y the y X Φ ( y Φ ( y Φ ( y tht s ( y s soluto of the ge Γ. Refeees [] M.Cte Foudtos of thetl eoos Cdge MIT PRESS. [] J.W.S. Cssels Mesues of the o-ovety of sets d the Shpley-Fol-St theoe Mthetl Poeedgs of the Cdge Phlosophl Soety Vol. 78 pp. 4-46 975. [] H.G. Egglestoe Covety Cdge Uvesty pess Lodo 958.

48 Fo Fesh d Gov Qut: Cove Hull Chtezto [4] C.Fth lge I Rgs Modules d Ctegoes Spge-Velg 98. [5] F.Fesh R.Gett dots of Mt Joul of Itedsply Mthets. Vol. No. pp.9-65. Tu Pultos 8. [6] J.B.Ht-Uuty C.Leéhl Fudetls of Cove lyss Spge-Velg. [7] T. Iu T. Wte Estee of pue sttegy equlu fte syet ges whee pyoff futos e teglly ove Dsete ppled Mthets Fothog. [8] S. Kl Mthetl Methods d Theoy Ges Pogg d Eoos - The Theoy of Ifte Ges ddso-wesley 959. [9] Y. Wey htezto d epesetto of the ( geelzed vese RN d ts pplttos Le lge ppl. N. 8 pp 87-96 998. [] L Zhou sple poof of the Shpley-Fol theoe Eoo Theoy Vol. pp. 7-7 99.

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