The Minkowski Theorem for Max-plus Convex Sets
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. The Minkowski Theorem for Max-plus Convex Sets Stéphane Gaubert — Ricardo Katz. N° 5907 May 2006. N 0249-6399. ISRN INRIA/RR--5907--FR+ENG. Thème NUM. apport de recherche.
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(74) oqxakVUi «trtckjwoqfOckoWui ot«qV'ZNjf¤zot¡
(75) a ¤`j¦¥ ackRxVUoqi VUT± `_Ojwfx{bg3}Bckjwoqfozf<c RVG{bjwTWVUfxakjozfoJckRVG}Uoqf`«qVUsDa VUce§Q/Rjma
(76) {bo`Va/foJc
(77) ¡#oqi ¤=jf c RVGT=rts`¢£uvg3a}UrqakVq± Ve}erJgxakVOrJf¯VUs`cki VUTWVOuoqjwfzcoJ
(78) rÊrq}UV=T=r%_foqcªVDrJf¯VUsc ikVeTWVOu3ozjfcªoq/ckRVDakV5c±rqaGakRot¡f¯jwf qsbrqTWuv V 4§ bGVUvwot¡|I§ H
(79) Vefx}5Vz±bj¶c{bo`Vea#foqc/akVUVeT uoza akjvwV'ckogxa V H
(80) Vevxj3¥ a/rJuxuikorq}~RWc o{bVeikjw«qVckRV i Veakgvc a/oqckRxV|uikVa Vefc/u3rJuVUi§ ®°fÊrq}5ce±¡#VWqjw«qVWr<{bjwi Ve}Bc|ui o`oJoq#r Zjwf¤qot¡
(81) ak¤`j!cd_`uVckRVeoqi VUT IoqiT=rts`¢£uvwgxa}5oqf`«zV5s}5ozfVea ÆQ/RVUozikVeT 4x§ 5±IikozT ¡Rjm}~R=¡*V'{bVe{bg3}5Vc RVT=rts`¢£uvwgxaZjwf¤qot¡
(82) ak¤`jbckRxVUoqi VUT ÊQ/RVeoqi VUT 4§Í B±`rJf3{ jc a'qVefVUi~rJvw#j `ertc jozfDc oOc RV}erqakVGoJgf`oqgfx{Ve{N}Uoqf`«qVUs akV5c a ÊQ/RxVUoqi VUT 4§ 4 5r§ Vjfxrqvvw_q±»¡#Vª{bV{bgx}5V rzarGakuVe}5jmrJv»}Urza V
(83) ra vwjwqRckvw_TWozikV
(84) uxikV}5jma V
(85) «zVUi~a jwoqfWoqc R6V hdxrza jma@c RVUozikVeaT i
(86) oqZoqvwvVeri [ Zoqv bFb_^,rJf3{ rJzfVUgxKi [ rJzxU ^Iozni OxfjckVUvw_DzVUfVei rJckV{OT=rts`¢£uvg3a}5oqf`«zV5sD}5ozfVeae±3À*oqi oqvwvmrJi G_ 4x§ cx§ VjwfxrJvwv_z±b¡*V'foJc Vc Rxrtc#ckRxV¬T=rqjfOi Veakgvc a#oJckRV¬ui VeakVUfc*uxrqu3Veie±bQ/RVeoqi VUT=k a 4§w
(87) 4x§ 4`±bRxr%«zVVUVef rqffozgfx}5V{ Ê¡j¶c Roqgc
(88) uiko`oq #jf c RV|a gi «qVe_Ouxrqu3Vei [ m¿| z- ^¦§ . . . . . . . . n. . . . . . . . . . . v. "!.
(89) .
(90) !#"%$'&
(91) )(* ,+-. 0/213. x2 A e. k i j f. h g. d c. a b. x1. nA ?>d: '¨
(92) fDgxfoqgxfx{bVe{DT=rts`¢¦uxvgxa#}5ozf`«qV5s=akV5c#rqfx{Wc RikVeV¬akVUzTWVUfc ajf<qVUf¢ Vei rqvuozakjckjwoqf}5ozfzc~rJjwfVe{Djfjce§ . . b § C: A CA 7 =:9A ®°f=c RjmaakVe}Bc jozf±¡*V
(93) zj«zVxrqakjm}
(94) {bVO3fj¶c jozfxarqfx{WVea c rqvwjwakROa ozTWVVUvwVUTWVUfc rqik_vVeTWTWrzaU§Q!oGi jfx c o=vjwqRc
(95) c RVªrqfxrJvwoqz_=¡j¶c RN}5vmrqa a jm}Urqv}5oqf`«zV5s rqfxrJvw_ba jmae±b¡*VGakRxrJvwvgxakV|c RV|Ioqvwvwot¡jfWfoqc rJckjwoqf§ V {VUfoqckV¬_ R c RVTWrJs`¢¦uvwgxa/akVUTWji jwfx§ V|{bVefoJc V_ a ⊕ b := a ∨ b ckRVT=rts`¢£uvwgxa/rq{{bjckjwoqf± rqfx{=`_ ckRxV'T=rts`¢£uvwgxaTgv¶c juxvjm}UrJckjwoqf§ V'akV5c ± §Q/RxV'akV5c*oq«qV}Bc oqi~a oqakjT`eV nabot«z:=VUi aR+ b jwa{bVefoJc Ve{`_ R §@¨ «qV}Bckozi}5oqf3a jmadc jfx:= ¬oz−∞ fvw_GoJ ª:=Vefck0i jVa@jwa{bVefoJc Ve{`_ »§ Y#_=a }UrJvmrJi±q¡#VTWVrJfDrqfOVUvwVUTWVUfc#oJ §@®£ rJfx{ ±`¡*V¬a VUc ± rqfx{¡#V{bVefoJc VG`_ λu c RVª«zVe}5ckoqi'¡Rj¶c RNVUfc ikjwVeu,a vλ ∈+Ru §Zrts`¢¦uxλvgx∈a'R}5ozf«zV5sa VUc a¬rJufx{⊕}Uvoq:= fVa
(96) uRx∨r%«zvV VUVef {V OxfV{Djwf<ckRVjwfzc ikob{bg3}Bckjwoqf§®°fDckRV|akVelgVevƱ`Iozi#xikVe«jcd_q±c RV¬ckVUi T hk}5ozf«zV5s i
(97) gxakVe{<¡j¶c Roqgbc uxikV}5jma jwoqfxaa R3rJvwv/rqv¡/r%_baGV<gxfx{bVUi~a cko`ob{¯jfckRVT=rts`¢£uvwgxaakVUf3a Vz\ § Y# _ hk}5oqfxUV iB±¡*Va R3rJvwv#rqv¡/r%_ba TWVrJfr ÊT=rts`¢¦uxvgxa /}5ozf`«qV5sD}UoqfVz§ ¸ ʽ ?21 .Q GU& 21n j ¬}Uoqf`«qVUsORgxvn v j AA RF2 % 1uR G}5o R(A) 1Bp21n jp $ $ * 1u }Uoqf`«qVUsD}5oqTjwfxrtc jozfxaa j.
(98). . . max. . . n max. max. . n max. max. . max. i. . . . . . "!. . . n max #. %$. '&)(. +*. 2 $T2G2 1 Q %U2 J QIU 1 1E2 - k?. Q "(* 1uM21 k Q A# ⊕k∈K αk uk K u k∈K M $,$P ja2$T2G2 1 p j %R 2J $# p1B 1 2 1 -$ ⊕k∈K αk = # }UoqfVCD 2 %2 1uR A.$ RF{α 2 %k }1Ek∈K R .$
(99) }UoqfV n1B621 jr $ $/&)(* 1u+*K$' # rJ 1 3 a2$T)2 1 M j 2)A J. aIU 1 1u2 \ (A) ? M 0( 1u221 k u k∈K A# ⊕k∈K αk uk K r Q 1$ $P a$#) ,13p j R p2J. $T {αk }k∈K ' i Ve}UVea a jwoqf }5oqfxVP j A 1n " ,1 aRF (* # R%$32 A. v∈A. ki V} (A) := {u ∈ R | v ⊕ λu ∈ A M $ $ λ ∈ R } . * L"$T§ Q/RV ikV}5Vakakjozf´}5ozfV<oq
(100) ckRxV }Uoqf`«qVUsakV5c oqCVjzgi Vrtcrqf`_¯u3ozjfc jma i Ve} (A) = }5oqfxV ({(0, 1), (2, 0)}) §®£c/jma
(101) akRot¡f oqf<ckRV|i jzARc/R3rJfx{ a jm{bV|oq Vjzgi VG±b3Vevvot¡|∈§ A v. 4. #. v. v?v. ÈwxÜUxBÚ5Ý. n max. max.
(102) . /%1u" %L &2U1 C3J R G 1
(103). VVlgju R ¡j¶c Rc RVgxakgxrqvc oquoqvwoqq_z±/¡Rjw}~R }UrJf {bVOxfxVe{ `_©ckRVTWV5cki jm}X (x, y) → §GQ/RVakV5c jwa¬VelgjwuuVe{¡j¶c RNc RVuxikob{bgx}5c'c oquoqvwoqz_q§ V={bVefoJc V`_}5vwo (A) ckRV |e − e | }Uvoa gxikV¬oJrWa g3a VUc RA oq R § Q/RVIozvvwot¡jwfvwVUTWT=rqa
(104) qjw«qVa ozTVui oquVUikckjwVea/oq!i Ve}UVea a jwoqf }5oqfxVeae§ *¸ ´ ?21 a )J2$T -2RGU& 21 % 1 JUU3 J %a 1 J2$T -2R A R A v max. x. y. a $ $. n max. / . n max. . !. . n max #. v ∈ A#. n lV5c § VozirJvwv ±x{bV2OxfV|c RV|T=rJu ϕ : R → R `_ ϕ (u) = v ⊕ λu § jwa'rJfjwfzc VUi~a V}Bc jozf oq}5vwozakVe{akV5c ae±3rqfx{Nakox± b] jfx}UV ϕ jma¬v}5oq∈fc Ajf`gozgxaU±xikV}λ ∈(A)R = ∩ ϕ (A) jc
(105) jma}5vwozakVe{,§ *¸ ´ =21 ) J ,+-. PU& U2216 R p2 G 3 A R v, w ∈ A βv ≤ w β 6= 1B2 Oi Ve} i e V } (A) ⊂ (A) n lV5c i Ve} (A) § V=rqa a gxTVz±,¡j¶c Roqgc¬vwoza a¬oJzVUfVei rqvjcd_q±ckR3rtc β ≤ »§ªQ/RVef±»Ioqi|rJvwv u∈ ± λ∈R . max. #. λ. v. / . . λ∈Rmax. . !. v. . w. n max. λ. −1 λ. n max. λ. . . n max. #. #. . . v. #. max. −1. −1. ∈A akjwfx}5V v ⊕ β λu ∈ wA ⊕rqfxλu{ a =jwfx}5wV ⊕A βvjma
(106) }5⊕ozf`ββ«qV5s,§λu®£cI=oqvwvwot¡
(107) ⊕a#β(v c Rxrtc ⊕uβ∈ ikλu) V} (A) § ½ ½ ʽ º ¸ ƽ =21 .a )J$T 2.R)J + QU& 21 2 1 JUU3 R J %Q j 1 %RFu" RF2 16 j A QQRF2 % 1Ea 1 |i Ve} A v v∈A (A) n m¬jw«qVef ±b¡*VG{b2V OxfV¬ckRxGV &_"" U1oJ x ckoWVckRVGakV5c x∈R a guxu x := {1 ≤ i ≤ n | x 6= } . s 3a Veik«z V Oxi a c
(108) c Rxrtc¬j¶ ±3c RVUf rqfx{ckRxVakguuoqikcoJ jmackRxVªgfjwoqfoJckRxV akguuoqikc aªoJ u rqfx{ v §Nu,®£vcG∈IozvAvwot¡
(109) a|ckR3rtucª⊕c RvVUi ∈VOAjmaªrqf¯VUvwVUTWVUfc w ∈ A u¡⊕jckR´v T=rtsbjwTªgT akguuoqikce± TWVrJfjwfGc Rxrtc/a gxuu akguu Ioqi/rJvwv § HVUfx}UV
(110) Iozi*Ve«qVUi _ * ckRVeikV'V5sbjmadc~a =a g3}~R c Rxrtc λ v ≤ w §<Q/RVevik⊂VUIoqi Vq±`w_ lVUTWT=rv b∈§ cxA±!j¶ca g O}UVeac oNa Rot¡ v ck∈RxrJAcªi Ve} (A) ⊂ λikV} 6=(A) Ioqi Ve«qVeik_ v ∈ A § lV5c V ra VlzgxVUfx}UVDa g3}~R¯ckRxrJc lim β = ¯rJf3{ < β ≤ Iozirqvv {β } ⊂ R § £ ® k i V } q r x f { λ ∈ R ±bc RVUf r∈N u∈ (A) −1. . . . . w. . . . . . . / / 0!. . n max #. .$. #. . #. n max. #. . i. . v.
(111). v. w. . max. r r∈N. r→∞. v. . r. . r. max. w. v ⊕ λu = lim (v ⊕ βr (w ⊕ βr−1 λu)). mj a#r|vjwTWj¶c/oJVevVeTWVUfc a*oJ 3V}Urqgxa V Iozi#rqvv §]`jf3}5V jma*}Uvoa V{,±qjc*Iozvvwot¡
(112) a c Rxrtc v ⊕ λu ∈ A §Q/RVUi V5IAozikVz± u ∈ i Vew} ⊕(A)β ±xrJλuf3{ ∈akoxA±bi Ve} (A)r ∈⊂ NikV} (A) IAozi
(113) rJvwv v ∈ A § C) § Q/RV¬}Uvoa gxikVrzakakgTWubc jozfOjfDckRV¬ui VU«`jwoqgxa*ui oquozakj¶c jozfO}UrJfxfoJc#3V|{bjma uVUf3a V{=¡j¶c R§ À*ozfxakjw{bVei A = ([ , ] × { }) ∪ R = {(x , ) | x ≤ } ∪ R ⊂ R §Q/RVUf± ikV} (A) = (u , u ) ∈ R | u 6= ∪ {( , )} rJfx{<ikV} (A) = R . Q/RVIozvvwot¡jwfT=rJs¢£uvwgxa=rJf3rJvwoqqgxV<oq'c RVfoJc jozfoq'VUsc ikVeTWV uoqjwfcO¡#rza=rJvwikVrq{b_¯g3a V{`_ HVUvwjw [ H
(114) Ve#v bb^£§ r→∞. −1 r. v. w. v. . #. ( , ). . . . . 2. 1. 2. . 1. 2 max. 2. 1. . 2. . 2 max. . ( , ). 2 max. . v. "!.
(115) .
(116) !#"%$'&
(117) )(* ,+-. 0/213. ¸ ʽ 3qs`cki VUTWVGuoqjwfzc ?1 G J + PU& A WV5s`cki VUTWVuoqjwfcM j ? p $ $ %R A y, z ∈ A α, β ∈ R . . . . / . !. . " ."2U1 $) 2 1 ( R. . 21L j n U&J.01BR max 1 #. . α⊕β =. max. . x = αy ⊕ βz =⇒ x = y. . 2$T2G2 1 x∈A I1B $ $# I D. . x = z.. |VUsc (A) A Q/R`gxae±rªuoqjwfcoJ jma#V5s`cki VUTWV¬j¶@j¶c
(118) }erJffoqc#VUvwoqfxªc oWrakVUqTWVefzc/oJ gxfvVaka#jcjwarqfDVUf3{Doq c Rjmaa VeqTWVUfce§ V/¡/rJAi fGc RV/ikVrq{bVei!c Rxrtc{bgxV*c o¬ckRV/jm{bVUTWuoJckVefx}5_ªoJrz{{bAjckjwoqf±JckRV/ui oquVUikcd_ d B± ¡jckR α ⊕ β = rqfx{ α, β 6= DjmafoqcVelgjw«%rqvVefc#c o rqfx{ x = z. x = αy ⊕ βz =⇒ x = y C) b§ b ®£ jmaGrqf³VUs`cki VUTWV=u3ozjfcGoq A ±c RVUf x = αy ⊕ βz ±!¡j¶c R α ⊕ β = ³rqfx{ x ± j W T u w v j V 2xX ∈ A a y, z ∈ A ozi (x = z, β = ). (x = y, α = ) ®°f3{bVUV{,±rzakakgTWV|c Rxrtc x = y gbc α < »§
(119) Q/RVUf± β = »§¨'akakgTWVG`_}5oqfc i rz{bjw}5ckjwoqf ckR3rtc x 6= z § Q/RxVUf±¡#V|Rxr%«zV ±xIoqi
(120) akoqTWV 1 ≤ i ≤ n ±3rqfx{a o3± x = αy ⊕ z = αx ⊕ z < x ±x¡Rjm}~Rjwa fxoqfxakVUfxakVq§ x > z RVef jwarD}UoqfVz±3jcjwa¬}5vwVerqi
(121) ckR3rtcj¶c~aozfv_VUsc ikVeTWVªuoqjwfzcjma »§¬®°fNc Rjma'}erqakVq±ckRV i VUvwVU«trqfzcfCoqckjw⊂oqf Rjwa#c Rxrtc
(122) oqVUs`cki VUTWVqVUfxVUi~rtckozie§ ¸ ʽ 3qs`cki VUTWV=qVefVUi~rtc oqi ?1 0 J % 2$T)2 1 0 x∈C VUs`cki VUTWVqVUfxVUi~rtckozia j C 1B $ $# I Da" ."C2U⊂1 R 2 1 R 221n p 1 )k" ,13a . I $ $ . QRF2 % 1ER. .$. #. . . . . . . #. . . . i. . i. i. i. i. i. . . . . i. ". n max. . i. . !. $. n max (. . #. . x = y ⊕ z, y, z ∈ C =⇒ x = y. x = z.. 3 . 1 .)6D2 %2 1E 1B2 1BK221 =V5s`cki VUTWV'i r%_ | λ ∈ Rmax } x %a21k a. F1 2GCD2 %2 C1E p I $ $ RMmax R x = 1uR {λx %$|V5s`c ¢£ . C#. C. (C) #. qs`c ikVeTV|zVUfVei rJckozi a#rJi V|}UrJvwvwVe{2 UR&J2 $T
(123) VUvwVUTWVUfc~a/jfvmrtc c jw}UV¬ckRVeoqi _q§. C) b§we ®£c}Urqf¯3V<ikVrq{bjwv_³}~RVe}~¤zVe{ckR3rtcªVe«qVUi _VevVeTWVUfcªoJ
(124) rqf¯V5s`cki VUTWVOi~r%_oq jmaªrqf C VUs`cki VUTWVqVUfxVUi~rtckozi*oq C § * L"$T b§wq lVUc¬gxa}Uoqfxakjw{VUi'ckRV=}5vwozakVe{}5oqf`«zV5sa VUc a Rxot¡fNjf VjzgikV<q§|®£c|}Urqf R V=Verza jwvw_a VeVUfckRxrJcGj¶c~a|V5s`c ikVeTVWuoqjwfc a|rJi V a = (5, 2) ± Ab =⊂ (4, ± ± rqfx{ 0) c = (3, 2) d = (1, 3) O § / Q R = V U V ` s k c i U V W T W V i % r b _ ¬ a J o / k i V } q r k i V q r x f { Æ. a e V W V k c R = V i w j q R | c x R rJfx{ e = (2, 5) (A) R (0, 1) R (2, 0) akjm{bVoJI V!jwqgxikVG 5§ Q/RV¬Iozvvwot¡jfx}5ozfxa cki gx}Bc jozfD¡jwvvrqvvwot¡ gxa*cko={bVeikjw«qV'ikVa gvc a*Iozi/}Uoqf`«qVUsOa VUc a/rqa/}5ozfxakVelgVUf3}5Vea oq!i Veakgvc a#Ioqi
(125) }UoqfVaU§ *¸ ´ =21 .Q 0J + 0U& U221k 2 *1Bp21 A R . #. 4. 2 max. #. . max. . / . a PJ. %. v?v. !. . . n max #. max. . CA = {(λx, λ) | x ∈ A, λ ∈ Rmax } ⊂ Rn+1 max #. ÈwxÜUxBÚ5Ý.
(126) b. /%1u" %L &2U1 C3J R G 1
(127). n lV5c rJf3{ ±¡jckR rJfx{ ¨'akakgTWVq±»¡jβckR∈ozgbRc'vwoza a
(128) oqqVe(λfVUi~xrJvwj¶,cdλ_z±),c Rx(λrtc xλ ,:=λ λ) ∈⊕Cλ 6= »§]`xjf3,}5xV A∈ jwAa¬}5ozf`«qV5λs,± ,λλ ±3rJfx{ a o3± λ x )∈A . max. #. 1. 1. 1. 2. 2. 2. 1. 2. 2. 1. A. 2. §. 2 ∈ −1. 1. . 2. Rmax (λ1 x1 ⊕. (λ1 x1 , λ1 ) ⊕ (λ2 x2 , λ2 ) = (λλ−1 (λ1 x1 ⊕ λ2 x2 ), λ) ∈ CA .. ZNoqi VUot«qVeie± C jma/oq`«`jozgxakv_=ui VeakVUi «qV{O`_=c RV|Tªgvckjwuvwjw}ertc jozf `_DrWa }Urqvwrqie§ *¸ ´ 3 M J +. P21n1 }5vwo M J ,+-. P21 A⊂R (A). . A. / . aJ %. . . n max. Q2 )a M1 U& #. /Q RxjwaIoqvwvot¡
(129) aIikozT c RV'}Uoqfckjwf`gjcd_oq,ckRV'Igfx}5ckjwoqfxa (x, y) → x ⊕ y rJf3{ (λ, x) → λx § V|fV5s`c
(130) Vadc~rJvwjwakRa ozTVui oquVUikckjwVea/oqc RVG}5ozfV C § ½ ½ ʽ 3 a PJ$T 2.R J ,+-. 021 1B2 A⊂R }5vwo (C ) = C ∪ (i Ve} (A) × { }). n lV5c }5vwo § ¨'akakgTWIV Ox(y,i~adc,α)c Rx∈rtc α 6=(C»§@)]`jwfx}5V (y, α) ∈ }5vwo (C ) ±ckRVeikVVUsbjwa c arakVelgVefx}5V {(λ x , λ )} ⊂ a g3}~RckRxrJc §NQ/RxVUf±@rqa lim λ = α 6= rJfx{ A jmaª}Uvoa V{,± C (λ x , λ ) = (y, α) ¡#V¤`fot¡.c Rxrtc lim 3Vevozfza c o A §\Q/RVeikVUIoqi Vq± x := lim x = lim § λ λ x = α y (y, α) = lim ¨'akakgTWVfot¡©c (λRxrtc xα, =λ )»I§ =l(αx, V5c x α)∈ A∈ rJCfx{ β ∈ R §Q!oGui ot«qV/c Rxrtc (y, α) ∈ (ikV} (A) × { }) jc#akg O}5Vac oªa Rxot¡c Rxrtc x ⊕ βy ∈ A §¨'a x ∈ A ¡#V
(131) ¤`fot¡ckRxrJc (x, ) ∈ C §Y'a jwf|c RV
(132) Êrz}Bcc Rxrtc }Uvo (C ) jma*rG}UoqfV I`)_ lVUTWT=rqa#b§w|rqfx{D§_ 4 B±`j¶c*Ioqvwvot¡
(133) ackRxrJc (x ⊕ βy, ) = (x, ) ⊕ β(y, ) ∈ }Uvo (C ) §@Q/RVUf±JckRVeikV#V5sbjmadc~ar¬akVelgVUf3}5V {(λ x , λ )} ⊂ C akgx}~RGc Rxrtc lim (λ x , λ ) = §Q/RxVUi V5Ioqi Vq± (x ⊕ βy, ) }5vwo (A) = A. x ⊕ βy = lim λ x = ( lim λ )( lim x ) = lim x ∈ Q/R`gxae±x}5vwo (C ) ⊂ C ∪ (i Ve} (A) × { }) § s `«`jozgxakv_ }5vwo § lVUc fot¡ ikV} (A) × { } § Q!rq¤qVrqf`_ x ∈ A § V ¤`fot¡µckRxrJc x ⊕C λy⊂∈ A (CIoqiO)rqvv λ ∈ R (y,§ Q/)RVe∈f±#j¶ {λ jwaOrakVelgVefx}5Va g3}~R c Rxrtc lim λ = »±*j¶cIoqvwvwot¡
(134) aªc Rxrtc (y, ) = lim (λ} (x⊂⊕ Rλ y), λ ) rJfx{´ckRVeikVUIoqi V }5vwo akjfx}UV Ioqi
(135) rqvv r ∈ N § (y, ) ∈ Q/R`gxae± C (C∪ ()i Ve} (A)(λ× { (x}) ⊕⊂λ}5vwoy),(Cλ ) § ) ∈ C ½ ½ Êb ¹ B a PJ. L" FJ216J. + 0221 *1B2 p J2$T -2R0J A⊂R C ⊂R n ®£ jma
(136) r=}5ozTWuxrq}5cakgxakV5c
(137) oq ± j c ª T x g. a
(138) c G V 3 z o g x f b { V < { IikozT rJot«qVz±rJfx{akoWikV} (A) = R § I Y#_OnAi oquozakj¶c jozfb§w2 c±}Uvo x ± J r 3 f < { k a x o ± jma
(139) }Uvoa V{,§ { } (C ) = C ∪ { } = C C *¸ ´ =21 A .Q 0J2$T R J +. U& 21n j R 2 V5s`c ¢£ (}Uvo (C )) ∩ (i Ve} (A) × { }) = V5s`c ¢£ (i Ve} (A)) × { } . n . #. . #. A. . . . . . . . . n max. . A. A. . #. . A. A. A. r→∞. r. r. r. r→∞. r→∞. r. r. r. r. r. r∈N. r→∞ r −1. −1 r r r. r→∞. r. r. . A. . . max.
(140). . . A. . . . . A A. r. r. . r. r→∞. r. r→∞. r. A. r∈N r. r→∞. r. r. r. r. r. r→∞. r→∞. . A. A. A. . max. . −1 r. r→∞. . . r→∞. −1 r . A. r. A. −1 r. . −1 r. r. A. . n max. A. . n max. #. A. / / . max. r r∈N −1 r. A. / . . . A. !. . n+1 max. #. . A. A. . . n max #. A. . . A. . v. "!.
(141) .
(142) !#"%$'&
(143) )(* ,+-. 0/213. n lV5c V5s`ck¢¦ }5vwo )) ∩ (ikV} (A) × { }) §Q/RVef±jwf uxrJikckjm}5gvmrJi± x ∈ i Ve} (A) § '¨ akakgTWVckRxrJ(x, c x )=∈ y ⊕ z ±( ¡jck(C R i Ve} § ¨'a}5vwo (C ) = C ∪ (i Ve} (A) × { }) _ nikozu3oa jckjwoqf § cb±¡#V¤`fot¡ c Rxrtc y,(y,z ∈), (z, (A) }Uvo (C ) §µQ/RVef x = y ozi x = z ±akjf3}5V ) ∈ q r x f { U V s k c ¦ ¢ 5 } w v o § Q/RVeikVUIoqi Vq± x ∈ V5s`c ¢£ (i Ve} (A)) rJfx{ (x, ) = (y, ) ⊕ (z, ) (x, ) ∈ ( (C )) U V ` s. c £ ¢ i e V } § (x, ) ∈ ( (A)) × { } lV5c|fot¡ V5s`c ¢£ i Ve} × { } §GQ/RVUf±oq`«`jozgxakv_ (x, ) ∈ i Ve} (A) × { } §¨a a gTWV (x, ) ∈ c Rxrtc (x, ) = (x , λ ) ⊕ (x( , λ(A)) ±¡j¶c R }5vwo (C ) § Q/RxVUf± x = x ⊕ x ∈ rqfx{ λ = λ = »§Q/RVeikVUIoqi Vq±@rq)aG}Uvo (C (x) =, λ C), (x∪ (, ikλV} )(A) `_ni oquozakjckjwoqf´§ c±@j¶c Iozvvwot¡
(144) ac Rxrtc x , x ∈ i Ve} (A) § VjwfxrJvwvw_q± x = x oqi x = x akj×f3}5{V x})∈ VUsck¢¦ (ikV} (A)) §NQ/R`gxae± VUs`c ¢£ (}5vwo (C )) § (x, ) ∈ *¸ ´ =21 .Q 0J + 0U& U221k 2 A R VUs`c ¢£ (}5vwo (C )) ∩ C ⊂ VUs`c ¢£ (C ). n s `«`jwoqgxa/akjfx}UV }5vwo (C ) § C ⊂ Q/RVIozvvwot¡jwfuxikozu3oa jckjwoqf ikVevwrJckVa*VUs`cki VUTWV|u3ozjfc~arJfx{<V5s`cki VUTWV|i~r%_baU§ ½ ½ ʽ =21 C ⊂ R J. % $T1 γ 6= %R $T21 ψ : R → R . . . . A. #. . . . . . A. . A. . . A. . . . . 1. 2. 2. 1. . 1. A. . 1. 2. 2. 1. A. 1. 2. 2. . . 1. A. 2. A. 1. 2. . A. / . . !. . . n max #. A. A. A. . A. #. . . . . / . . A. !. . n. . . ψ(x) 6=. . U&Smax ) 1 1. x∈ C \{ }. Σ := {x ∈ C | ψ(x) = γ} .. n . . n. max ) 2 !#"$'&
(145) P$' U M )
(146) D max 1B 1 2 ) ψ(x) = ⊕n ai xi a ∈ Rnmax # G M $ $ %RPRF (* p1BQJ i=1 +. )221 2. V5s`c (Σ) = V5s`c ¢£ (C) ∩ Σ . l V5c x, y ∈ Σ qr fx{ α, β ∈ R V|a g3}~R<c Rxrtc α ⊕ β = ,§Q/RVUf±rqa . . #. max. ψ(αx ⊕ βy) = αψ(x) ⊕ βψ(y) = γα ⊕ γβ = γ(α ⊕ β) = γ. rqfx{<oq`«`jozgxa vw_ αx ⊕ βy ∈ C ±xj¶cIozvvwot¡
(147) a#ckRxrJc αx ⊕ βy ∈ Σ §Q/RVeikVUIoqi Vq± Σ jma
(148) }Uoqf`«qVUs»§ lV5c VUsc (Σ) §D¨'akakgTWVWckR3rtc x = y ⊕ z ±IoqiGakoqTWV y, z ∈ C \ { } §OQ/RVUf± ψ(y) = γ ozi x∈. a w j x f }UV §]`gxuu3oa Vz±b¡j¶c Roqgc/vwoza a#oJqVUfxVUi~rJvwj¶cd_z±qc Rxrtc ψ(y) = γ § ψ(z) = γ = ψ(y) ⊕ ψ(z) ¨'a x = y ⊕ ψ(z)γγ = ψ(x) ± ¡ x R U V i ¬ V U } v V rJi v_ γψ(z) z ∈ Σ rqfx{ ψ(z)γ ≤ »±¡#V'¤`fot¡ c Rxrtc γψ(z) z ozi x = γψ(z) z. x=y ]bjfx}UV x 6= y jTWuvwjVa ψ(z)γ = Æa VeVy
(149) VUT=rJi ¤§ b B±,jc'Iozvvwot¡
(150) a
(151) c Rxrtc x = y ozi x = z §|Q/RVef± VUsck¢¦ § x∈ lV5cGfot¡ (C) ∩ ΣV5s`c ¢£ §D]bguuozakVc Rxrtc x = αy ⊕ βz ±¡jckR y, z ∈ Σ rJfx{ α ⊕ β = »§ Σ ckRxrJc ]bjfx}UV x ∈ V5s`xc ¢£∈ (C) ±¡*(C) V¤fx∩ot¡ oqi x = βz §¨a a gTWVz±,¡j¶c Roqgbcvwoza aoqqVefVUi~rJvwj¶cd_z± x = αy c Rxrtc x = αy §#Q/RxVUf± γ = ψ(x) = ψ(αy) jTWuvwjVa/ckR3rtc α = »±3rqfx{a o x = y §*Q/RVeikVUIoqi Vq± = αγ U V s c § x∈ (Σ) XoJckV*ckR3rtcckRV/}5ozfx{bjckjwoqfªoqbckRV#ui VU«`jwoqgxa!ui oquozakj¶c jozf|jwa@a rtckjma OxVe{,±tjwfªu3rJikckjm}5gvmrJi±e¡RxVUf ψ(x) = Iozi
(152) a ozTWV a ∈ R § ⊕ ax . −1. −1. −1. −1. . −1. −1. . . . . . n i=1 i i. v?v. ÈwxÜUxBÚ5Ý. n.
(153) . /%1u" %L &2U1 C3J R G 1
(154). ½ ½ Êb ¹. ?1. . "!. a J ,+-. )U&21n j . 2. Rnmax #. . 5V s`c ¢£ (C ) ∩ (A × { }) = V5s`c (A) × { } . n À*ozfxakjw{bVeic RV/T=rts`¢¦uxvgxavwjfxVerJiIozikT oqf {bV2OxfVe{`_ λ) = λ ±tIozi@rqvv z ∈ R rqfx{ λ ∈ R ±#c~rJ¤qV γ := »±rJfx{ rquuvw_ψni oquRozakj¶c jozfb§w bcko´ψ(z, ckRVN}UoqfV § V {Ve{bgx}UV¬ckRxrJc/VUs`c ¢£ (C ) ∩ Σ = VUsc (Σ) ±¡RxVUi V Σ := {(xλ, λ) ∈ C | λ = }C= A⊂×R{ } §]bjfx}UV VUs`c (A × { }) = V5s`c (A) × { } ±`c RVG}5ozikozvvmrJi _Wjwa/uxikot«zVe{,§ lV5cWgxaikV}UrJvwv*ckRxrJcWr³}5ozfV m j a * , 1E2$ D 2 %2 1uRDj
(155) c RVUi V<VUsbjwa c a r O3fj¶c V akgxakV5c a g3}~RDc Rxrtc C = }UoqfV C(A)⊂ § R A⊂R *¸ ´ * , 1E2$ aD 1ERPJ. %Q j R pJ2$T R n lV5c rqfx{ C = }5ozfV (A) § V'rqa akgTWVq±q¡jckRxoqgbc#voakaoq,qVUfxVUi~rJvwj¶cd_z±JckR3rtc A = {u , . . . , u } L < I q o i q r v v § lV5c {VUfoqckVªrWvwjfxVerJi
(156) Ioqi T ±3a gx}~R c Rxrtc u 6= 1≤k≤m ψ(x) := IoziªrJvwv 1 ≤ i ≤ n §Q/RVUf± ψ(u ) 6= »±IoqirJvwv 1 a≤xk ≤ m § lV5c x = ⊕ λ u a 3>VDr akVelgVefx}5Voq!VUvwVUTWVefzc~aoJ C akgx}~R<ckRxrJc lim x = x Iozi
(157) a ozTWV x ∈ R § ]`jwfx}UV ±'rJfx{akjwfx}5V »± jmaDoqgxfx{bVe{ rqa c VUfx{xaOc o´jf Oxfjcd_q§ HVUfx}UVq±¡#λVD}Uψ(u rqfrq)a ak≤gTWψ(x Vq±@¡j¶)c Roqgcvwoza aGoq
(158) ψ(u qVUfxVU)i~rJ6=vwj¶cd_z±c RxλrtcckRVeikV<V5sbjwa c a λ r ∈ R akgx}~R¯c Rxrtc Iozi
(159) rJvwv k = 1, . . . , m c~rJ¤`jwfOa gxxa VlgVUfx}UVea#jfV}5Vaka rJi _ B§Q/RVUf± lim λ =λ A. . . A. . #. n+1 max. . max. . A. . n max. (. . . 1$. 1$. n max. 1. #. k. (. n max. / / . #. m. . k. r k. k. i i. i. r. r. r→∞ k. r. . r k. . . 3. lim xr = lim. r→∞. r→∞. m M. n . / . jp 1k) -U1. k=1. k=1. . ;.
(160) > G@;=7 S8/GA 7 8:
(161) "); =8:
(162) 0;?A 798/ ? 7d< d89:
(163) 0 7/:
(164) 8d;=:/.
(165) . !. . n. . 1 2)rDmax 1E a j. $. . %RG2. . }5oqfxV (V5s`ck¢¦ (C)). C= l V5c x ∈ C § VoziVerq}~R i ∈ {1, . . . , n} {bVO3fVckRVGakV5c. #. max. k. m M λk uk ∈ C. λrk uk =. Xot¡ ¡#V|uikot«zVc RVT=rJjwf i Veakgvc a/oq!c Rjwa/u3rJuVUi§ p Q ,! r"%1 QJ2$T -2R)J ?21 ¸`½ ¸ C⊂R . m r k k=1 k r∈N. n max. k. Q/RxVUi V5Ioqi Vq± C jwa}5vwozakVe{,§ 4 § : 798
(166) 89AB;?7 . 1≤i≤n. . . r k. r→∞. U&S. A. . A. n max. n+1 max . n. C. #. 2 . +2 G $ 2$T2G2 1 . C. C1B. Si = {u ∈ C | u ≤ x, ui = xi } = C ∩ {u ∈ Rnmax | u ≤ x, ui = xi } .. ¨'a {u ∈ R | u ≤ x, u = x } jwa}UoqTWuxrq}5c¬rqfx{ C jma}5vwozakVe{,±,¡#Vª¤`fot¡c Rxrtc S jma¬r }5ozTWuxrq}5c akgxakV5c*oJ R ¡Rxjw}~R=jmafxoqfb¢£VUTWubcd_3V}Urqgxa V x ∈ S §Q/RVeikVUIoqi Vq± S Rxrza*r|TWjwfjwTWrqv3VevVeTWVUfc u § V}5vmrJjwT c Rxrtc jmarJfWVUsc ikVeTWVqVUfxVUi~rtckozi@oJ §@¨'akakgTWV/ckR3rtc IoqiakoqTWV § Q/RxVUf± u = y oqi uu = z §*lV5c/gxa#rqa a gTWVz±¡jckRxoqCgbc/voaka*oJ!qVUfxVUi~rJvwuj¶cd_z=±qc yRxrt⊕c zu = y §@Q/y,RxVUzi V5∈Ioqi CVq± akjwfx}5V rqfx{ §rH
(167) Vefx}5Vz± akjwfx}5V rJfx{ jma¬r=TWjfjwT=rJv y∈S VevVeTWVUfc'oq S y§¬≤Q/Rug3aU≤± ux jwa¬rJfyVUs`=cki VUuTW=VªzxVUfVei rJckoqioq uC =§®£yc¬jwa¬}5vwVerqyi
(168) ≤ckR3rtuc x = ⊕u u ±,rJfx{Nakox± }UoqfV VUs`c ¢£ § x∈ V|Rxr%(«qV|akRot(C)) ¡fOc Rxrtc C ⊂ }UoqfV (V5s`ck¢¦ (C)) §Q/RxV¬oqckRVeijf3}5vwgxa jwoqfjma#cki j«`jmrJv¦§ n max n max. i i. i. i. i. i. i. i. i. i. i i i. i. i i. i. i. i. i i. i. i. i. i i. i. i. n i i=1. . v. "!.
(169) z.
(170) !#"%$'&
(171) )(* ,+-. 0/213. ¸`½ ¸. ÊZNrJs`¢¦nvwgxa<Zjfx¤qot¡
(172) ak¤j'Q/RVUozikVeT. . !=21 \ ,! r"%1 $eJ r" J1 J ,+-. U&21) j n 2 0+2 e $ $#) ,1 j 1 J A+ WJ 1 3 j 1QG 1 A n+1 . 1 .)k" R ,max 13a# j k RG2 . }5o (V5s`c (A)).. A. A=. n l V5c § V2xO fV´c RV}UoqfV max lVUTWT=# r=b§w`x§Q/∈RVeAf±b`_À*ozikozvvmrJi _Ob§w± CA jma=r=}5{(λz, vwozakVe{ λ)}5oz|fV|z rJ∈fxA, {DckλR`gx∈a Rmax} ⊂ Rn+1. }5ozfV (VUs`c ¢£. rqajwf. CA. `_DQ/RxVUoqi VUT 4§wJ§ ¨'a (x, ) ∈ C ±!_³Q/RVUozikVeT 4 §wW¡*VW¤`fot¡ c Rxrtc|c RVUi V=V5sbjmadc ±fxrqTWVUvw_ (λ u , λ ), . . . , (λ u , λ ) ±xakgx}~R<ckRxrJc C CA =. (CA )). . 1. A 1. A. 1. n+1. n+1. V5s`cki VUTWVWqVefVUi~rtc oqi~aoq. n+1. . (x, ) =. HVUfx}UVq±. n+1. n+1 M. (λk uk , λk ).. k=1. ¡RVeikV M λ = . Y#_À*ozikozvvmrJi _Nb§we`±¡#V¤`fot¡ ckRxrJc jwa|rqfVUsc ikVeTWVzVUfVei rJckozioJ j¶d±rJfx{oqfvw_ jd± u ∈ A jwarJf¯VUsc ikVeTWVOu3ozjfcoJ (uA §, Q/) R∈jwaCa Rxot¡
(173) a|ckR3rtc x jwaGckRV }5ozf«zV5s}5oqTCjwfxrtc jozf¯oJ
(174) rtc TWoadc V5s`cki VUTWV=uoqjwfc a|oq A §<®£cGIozvvwot¡
(175) a¬ckR3rtc A ⊂ }Uo (V5s`c (A)) §Q/RV=oJc RVUiGjwfx}Uvgxakjwoqfjwa c ikjw«`jwrqvÆn§ + 1 ?1 M ) % ! 2L"1 J2$T -2R0J + 21 k+2 0$T2) ,1k j ¸`½ ¸ 1BU&S 1 aJ ,A+-.⊂ 0JR 2 % 1 3 j . 1 .) " 1 p j %R . 1 .)LD 1E A p A q /ikV} I 1B RG2 (A) p+q ≤n+1 }5o V5s`c (A)) ⊕ i Ve} (A). A= ( HVUi Vq±¡*V
(176) {bVefoJc V
(177) `_ ckRxVT=rts`¢£uvg3arJfxrqvozqgV/oqc RVZjwf¤zot¡
(178) a ¤`jxakgT oJ»cd¡*oGakgxakV5c~aU±z¡Rjw}~R jma#{bV OxfxVe{ rqackRVa VUc*oq⊕T=rts`¢¦uxvgxa#akgT=a*oq!rG«zVe}5ckoqiIikozT²ckRCV Oxi~adc/a VUc/rqfx{Ooqrª«zVe}5ckoqiIikozT²ckRV akVe}Uoqfx{<oqfxVq§ § 2V OxfV´c RV}UoqfV C = {(λz, λ) | z ∈ A, λ ∈ R } ⊂ R rqajwf n lV5c lVUTWT=r=b§w`x§Q/∈RVeAf±b`P _ lVUTWT=rW§_ 4`±x}5vwo (C ) jma
(179) rW}5vwozakVe{ }5ozfV|rJfx{DckR`gxa }Uvo (C ) = }5ozfV (VUs`c ¢£ (}5vwo (C ))) `_DQ/RxVUoqi VUT 4§wJ§ Y#_Dnikozu3oa jckjwoqfb§w2 cW¡*V¤`fot¡ ckRxrJc }5vwo (C ) = C ∪ (i Ve} (A) × { }), rqfx{DckRVef lVUTWT=rqa
(180) §rqfx{b§w%GjTWuvw_ X V5s`ck¢¦ (}5vwo (C )) = [VUs`c ¢£ (}5vwo (C )) ∩ C ] ∪ [V5s`ck¢¦ (}5vo (C )) ∩ (ikV} (A) × { })] V5s`ck¢¦ (C ) ∪ (VUs`c ¢£ (i Ve} (A)) × { }). ⊂ x=. n+1 M. n+1. λk uk ,. . k. k=1. k=1 k . A. A. k. . . . n max . !. . . $. . . 32. . . $. #. max. A. #. n+1 max. A. A. A. . A. A. . A. A. A. v?v. ÈwxÜUxBÚ5Ý. A. . A.
(181) %. /%1u" %L &2U1 C3J R G 1
(182). Xot¡|±zrqa ) ∈ C ⊂ }5vwo (C ) ±`_Q/RVeoqi VUT 4x§¡#V/¤fxot¡c Rxrtcc RVUi VV5sbjmadc*rCOxfxj¶c V
(183) f`gTªVUi qo 3VevVeTVefc a(x, oJV5s`c ¢£ ±qfxrJTWVUvw_ ¡jckR ±zrJfx{rLO3fj¶c Vf`gTªVUi@oJVUvwVUTWVUfc~a oq!VUsck¢¦ (ikV} (A)) × {(C} ±)f3rJTWVUvw_ (y(λ, u) ,¡λjckR) 1 ≤ h1 ≤≤ qk±x≤a g3p}~RDc Rxrtc . A. . A. A. k h . . (x, ) =. ¡jckR. §Q/RVeikVUIoqi Vq± p+q ≤n+1. M. k. 1≤k≤p. k. M (λk uk , λk ) ⊕ (y h , ) , 1≤h≤q. ¡ RVUi V M λ = rqfx{ i Ve} IoziªrJvwv 1 ≤ h ≤ q § Y#_¯À*ozikozvvmrJi _³b§we¡#VO¤`fot¡ c Rxrtc (u , ) ∈ C jmaªrqf VUs`cki VUyTWV∈zVUfVei (A) rJckoqioJ j¶d±brqfx{=oqfxv_Wjd± jwa#rJfDV5s`cki VUTWV
(184) uoqjwfc#oq §Q/Rjma*akRot¡
(185) ackRxrJc jma#ckRxVGa gT oJc RVG}5ozf`«qCV5sD}UoqTjf3rtckjwoqf<oJ up VU∈s`ckAi VUTWVu3ozjfc a/oq A rJf3{Doq Aq V5s`cki VUTWV¬qVUfxVUi~rtckozi xa oqikV} ¡j¶c R p + q ≤ n + 1 § H
(186) VUf3}5Vq± A ⊂ }5o (V5s`c (A)) ⊕ ikV} (A) §Q/RVOoJc RVUiªjfx}Uvg3a jwoqf¯jma c ikjw«`jwrqvÆ(A) § ¨'a
(187) r=}5ozikozvvmrJi _WoJQ/RVUozikVeT 4§w¬¡#V|qVUc
(188) rWuikV}5jma V«zVUi~a jwoqf<oJckRrV h xrqakjmac RVUozikVeMT i'IoqLi OxfjckVUvw_ zVUfVei rJckV{N}5ozfVeae§Q/RQV Oxi~a ci Veakgvc a¬oJ*ckRjma¤`jf3{¡#VUi Vozbc rqjfxVe{_NZNoqvwvVeQi [ Zoz#v bb-^rJfx{ rJJ¢ fxVUgi [ rqzxU ^¦§]bVU«qVei rqv»«%rqikjmrJfc~a*oqc Rjwa
(189) ikVa gvc
(190) Rxr%«zV¬rquu3VrJi Ve{Dj\ f []mrJgx bb± ]b c±»nÀ mY#F c^£§ * , 1E2$ 0D . 1u.R J %R ½ ½ Êb ¹ Y#rza jmac RVUozikVeT ?1 C U } q o f V I %R) $ M JC ,⊂ 1E R K 1I$T -U1* %K 2 G2$T)2 1* j FJ A 1 ⊂) C= (A) A x=. M. 1≤k≤p. . M λk uk ⊕ yh ,. k. 1≤k≤p. 1≤h≤q. k . h. A. k. A. . . . . . . . . . $0 j C# n Q/Rxjwa#Iozvvwot¡
(191) a/ikVrq{bjwvw_WIikozT. 1$. !. . n max. (. 1$. #.
(192). /Q RVeoqi VUT 4§w|rJfx{<Ii oqT lVUTWT=r=b§ÍJ b§ * L"$T4§Í ¨a*rJfjvwvg3adc i rJckjwoqfWoJ,Q/RVUozikVeT 4§ 4b±zvVUcgxa}5ozfxakjw{bVeioqfx}UV
(193) rJrJjwfªc RV
(194) }5vwozakVe{}5ozf«zV5s akV5c A ⊂ R {bVUujm}Bc Ve{¯jwf Vjzgi VNJ§ VORxr%«qVOrqvi Verz{b_akVUVef BqsrJTWuvwVeaWb§ÍrJf3{´b§wq c Rxrtc VUs`c (A) rJfx{<ikV} (A) = }5ozfV {(0, 1), (2, 0)} §Q/RVef± = {a, b, c, d, e} }5o {a, b, c, d, e} ⊕ }UoqfV {(0, 1), (2, 0)} A= `_DQ/RxVUoqi VUT 4§ 4b§Q/RV|a VUc a
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