Convex polygons are cover-decomposable Dömötör Pálvölgyi

!#"$%
'&()
*,+!-.+#/0+#12'3#465!+!487#9:
;<>=>?.@A /0B
∗
†
EC DGFIH)JLKNMOH
PRQTSVUXWLY[Z\UX]^Z`_aWcb]^dfeWOg#Q)dihjWcd>kcQjlg#WOmne>ocWOd Z\UXQ)bpQrqsS]h)WOdXSVZp]Od>Z ) S\tXhuUEZ\UX]^Z]Odfe
h)WLkOQIb\qndNoxW^_ZpUNQTgNms]^dNQYqyZ\U,Z\bu]^dzSVms]LZpQIS{W^_ P hI]^d,|zPQwNQIhjWc}gzW>SVQ~wEqnd>Z\WxZY`k(P
Wh)WLkOQ)bpqndNocSI€
k(P )
v _aWOmsw
 ‚!ƒ„…‡†ˆ‰,Š‹„GŒj†ƒ
‹ Ž0
Ž’‘“L”f•–•aŽL“0—a”f˜™”>š›œ•6‘>˜‘>žjŽ0jžOŸ¡ \T—až‘
—sš{Ž0°NŽ0±›‡”f—–˜X—–˜.²³Ž›œ•6‘>˜³Ž
—až“L”fP˜z)=‘>—–˜³{ŽL´µP —–˜¶| i‘>T∈•aŽ^I‘fž}  k · Ž · Ž0jž”>š P Ÿ¡¸ 1¹ šº”f•a´µ“L”^°NŽ0—–˜œ»¼—ažk ž¢y£0— ·¤f¥–¦¨›œ•–±¨§j¤f“^©>‘>ª0•–«I•a¬®ŽL­z´.¯ ‘ §j¤f©fª~«I¬®­z¯ Ÿ
½¾N¿ÀÂÁ6ÃOÁ®ÄGÀŵ¸Æ›œ•6‘>˜‘>žjŽ0 P —až’ž)‘>—a´[j”  Ž(“L”O°NŽ0 ¹ ´œŽL“L” · ›G”ÇžI‘  •aŽÈ—sšT²³Žišº”f•–•a”^ÉT—–˜œ»¶²³”f•a´œž^Ÿ¼Ê²³Ž0j޼Ž~Ë#—ažjjžE‘
žÌ³“)²(²‘>¡Ž0°NŽ0± šÍ”f•a´È“L”^°NŽ0—–˜œ»”>š‡²³ŽT›œ•6‘>˜³ŽÉT—–²()‘>˜³ž•6‘>jŽLž¡”>š “^‘>˜ ŽT´œŽL“L” ›‡”ÇžjŽL´
“L”f˜³ž)‘>˜z
—–˜Xj”(VÉr”¼“Lk”^=°NŽ0k(P
—–˜œ»Nž^)Ÿ¡Î³Ÿ³Ï{‘f“j².›œj”f›‡”ÇžjŽLk´¨¹ ²³Ž›œj”  •aŽ · ”>š{´œŽ0jŽ0 · —–˜œ—–˜œ»¨‘>•–•‹“L”^°NŽ0 ¹ ´œŽL“L” · P›‡”Çž)‘   •aŽž)Ž0jžT—–· ˜ÑÐ^ÒÇÓÇÔ!Ÿ
Õ³”fjŽ0•6‘>jŽL´µ›œj”  •aŽ · ž^֓L”f˜f׎L“0ÌœjŽLž^Ö³žjŽLŽØÚÙÛÜÏ¡ÔÇÝ>ÞVÖßr²‘>›œjŽ0à!Ÿ–Ðǟ
á ÄâÀœã0¾zäÇÃOå`æc¾ÅÈçpèÂÅéxêœä>ëìí ¥®¥zîâ¥sïÇ­âïf«§j¤Ç­³©>ªð¼ñ^ª0ò®ñïf«jª§)¤f©>ª~«~¢p¦Xªj§j¤fóî!¤>ñLïXô~¥aª^õ
ʲœ—až“L”f˜f×uŽL“0Ìœj޲‘fž  ŽLŽ0˜µ°NŽ0—söŽL´µ—–˜µ²œjŽLŽ’ž›‡ŽL“0—6‘>•‹“^‘fžjŽLž^Ÿ
÷ ë`¾XÄâæc¾#øúù.Åû®—8üxØÚÏ¡ÓÇý>Þ{þ ©fª~«Iÿ¨§)ª~­òÍ«ïf¥®¥yÿÈñIÿfóióȪ0òÍ«I¬º§,¤I~­ §j¤Ç­³©>ªðEî!¤Ç¥yÿ^¯N¤f­™¬8ñE§j¤f©>ª~«~¢p¦zª§j¤fóî!¤>ñLïXô~¥aª^õ
û®—–—8üØÚÛÜÏ¡ÓÇý>Þ ªE¤I~­ z­³¬Íòr¦Ç¬6ñL§E¬8ñE§j¤f©>ª0«I¢u¦Xª§j¤fóî!¤fñ0ïXô~¥aª^õ
û®—–—–—8üØÚÊTÊÔ>ÞÂþ ©>ª~«~ÿ¨¤~~­[òÍ«I¬ºïf­z¯Ç¥–ª’¬6ñE§j¤Ç©>ª~«~¢p¦Xª§)¤fóî!¤>ñ0ïzô~¥–ªOõ
\˜²œ—ažT˜³”fjŽÉ'ްNŽ0—s𮱍²³Ž“L”f˜f׎L“0ÌœjŽšº”f”f›GŽ0˜µ“L”f˜z°NŽ~˵›G”f•–±#»N”f˜³ž^Ÿ
ø âÅTþ ©fª~«Iÿ¤I0­[§)¤f­³©>ªðEf¥yÿ^¯N¤Ç­Ü¬8ñE§j¤f©>ª0«I¢u¦Xª§j¤fóî!¤fñ0ïXô~¥aª^õ
÷ ë`¾XÄâæc¾#
ÎÇ̳žjE•–— N޼—–˜ ØÚÏ¡ÓÇý>ޑ>˜³´ —–˜ ØÚÊTÊÔ cÞVÖ
É'Ž(šº”f ̜•6‘>jލ‘>˜³´ žj”f•–°NŽ(²³ŽÈ›œj” •aŽ —–˜ —–jž´!̑>•¡šÍ”f ŸµÊ²‘>—až^Ö
žÌœ›œ›‡”ÇžjŽ P —až’‘R›G”f•–±#»N”f˜[”>š n °NŽ0—a“LŽLžE‘>˜³´™ÉrŽi· ²‘^°NŽ(‘¨“L”f•–•aŽL“0—a”f˜ P = { P · | i ∈ I } ”>šr)‘>˜³· žj•6‘>jŽLž’”>š P Ÿ
‹Ž0 O  ŽE²³ŽE“LŽ0˜XjŽ0”>š»Ç)‘L°#—–V±Ü”>š P ŸTʲ³Ž,“L”f•–•aŽL“0—a”f˜ P —ažx‘ k¹ šÍ”f•a´¶“L”^°NŽ0—–˜œ»”>š²³Ž’›œ•6‘>˜³Ž—sšr‘>˜³´.”f˜œ•–±—sš
Ž0°NŽ0±R)‘>˜³ž•6‘>jŽ”>š P̄ Ö!²³ŽjŽ ŽL“0—a”f˜.”>š P ²œj”f̜»Ç²µ²³Ž”f—–»Ç—–˜‹Ö³“L”f˜z)‘>—–˜³ž‘>T•aŽ^‘fž k ›G”f—–˜zjž”>š`²³Ž“L”f•–•aŽL“0—a”f˜
Ÿ
O={O |i∈I }
i
i
i
i
i
∗
†
!#"%$'&()+*,-."/&0&120!345$/#7698:;=<?>A@B<DCFEGCFE
HI &98JKL&0"%$/JM$/$/NO0&PFQ4/JR'&S>TU#VW8X3Y+2TZJ[Z&0T##"#000!Z"%$U1=2\0!345$/U]698(:;=<?>A_^`G`aE]'&bc^'dFCe'dFC
Ð
Ê ²³Ž“L”f•–•aŽL“0—a”f˜ P = { P | i ∈ I } “^‘>˜  ŽR´œŽL“L” · ›G”Çž)ŽL´ —–˜Xj”™VÉ'”[“L”^°NŽ0—–˜œ»Nž—sš‘>˜³´ ”f˜œ•–± —sš²³Ž¨žjŽ0
“^‘>˜  Ž“L”f•a”fjŽL´.ÉT—–²ÜVÉ'”R“L”f•a”fjžOÖâžÌ³“)²¶²‘>Ž0°NŽ0±)‘>˜³ž•6‘>jŽ,”>š P̄ “L”f˜X)‘>—–˜³ž’‘(›G”f—–˜zx”>š
O={O |i∈I }
 ”f²µx—–“L°#”f—a•a´œ”fŽjž^Ÿ ²³Ž›œ•6‘>˜³Ž—–˜Xj”ž ‘>•–•jŽ0»Ç—a”f˜³ž^Ö#ž)‘^±NÖzž z̑>jŽLž^ÖzžÌ³“j²¼²‘>rŽ^‘f“j²Rž z̑>jŽ“L”f˜X)‘>—–˜³ž‘> ”Çžjr”f˜³ŽT°NŽ0jŽ~Ë
”>šÂ‘>˜z±()‘>˜³ž•6‘>jŽ”>š P̄ Ÿ{ š`‘E· )‘>˜³ž•6‘>jŽ”>š P̄ “L”f˜X)‘>—–˜³žTžjÌ È“0—aŽ0˜X•–± · ‘>˜z±È›‡”f—–˜Xjž”>š O Öz²³Ž0˜¨—–'· “L”f˜z)‘>—–˜³ž · ‘>˜X±
›‡”f—–˜XjžE”>š O —–˜ ”f˜³Ž(”>š²³Ž•–—–•aŽ(ž z̑>jŽLž^Ÿ юȓL”f•a”f’²³Ž›‡”f—–˜XjžE”>š O žjŽ0›‘>)‘>jŽ0•–±™—–˜ÑŽ^‘f“)² ”>š²³Ž(ž z̑>jŽLž^Ÿ
VšÉrŽR“L”f˜³“LŽ0˜X)‘>jŽ”f˜ ›‡”f—–˜Xjž,—–˜¶×p̳žj”f˜³ŽR”>š²³ŽÈ•–—–•aŽRž z̑>jŽLž^֏²³Ž0˜ —–˜³žjŽ^‘f´ ”>š)‘>˜³ž•6‘>jŽLži”>š P̄ ÉrŽR“^‘>˜
“L”f˜³ž—a´œŽ0)‘>˜³žj•6‘>jŽLž”>š n ´!— Ž0jŽ0˜XÉrŽL´!»NŽLž^Öâ“L”fjŽLž›G”f˜³´!—–˜œ»Èj”²³Ž n °NŽ0—a“LŽLž”>š P Ÿ
\˜ #ŽL“0—a”f˜[àÉ'ŽE›œj”O°NŽ,ž)” Ž’jŽLžÌœ•–jž‘ ”f̜“L”f•a”f—–˜œ»R›G”f—–˜zxžjŽ0jžÉT—–²ÜjŽLž›GŽL“0j”()‘>˜³ž•6‘>jŽLž”>šÉ'ŽL´!»NŽLž^Ÿ
\˜ #ŽL“0—a”f˜ ÉrŽxšº”f · ̜•6‘>jŽ²³· Žx›œj”  •aŽ · ›œj ŽL“0—ažjŽ0•–±¼—–˜¨²³Žx´!̑>• °NŽ0jžj—a”f˜‹Ö!‘>˜³´µ‘>›œ›œ•–±¼²³ŽxjŽLžÌœ•–jžT”>š #ŽL“0—a”f˜.à
j”i›œj”^°NŽÊ²³ŽL”fjŽ · Ðǟ
… T… T„âŒj†ƒ
z̜›œ›‡”ÇžjŽÉrŽT²‘^°NŽTVÉ'”²‘>• ³—–˜³ŽLž^Ö e ‘>˜³´ f Ö  ”f²(”>š Ž0˜³´!›‡”f—–˜X O Ÿ{ʲ³Ž0˜²³Ž0±i´!—–°z—a´œŽ²³Ž›œ•6‘>˜³Ž—–˜Xj”\Ér”›‘>jž^Ö
ÖGÉT²œ—a“j²[ÉrŽ“^‘>•–•
Ÿ¸
“L”f˜X)‘>—–˜³ž’—–jž ”f̜˜³´³‘>±N֋‘>˜
´œ”#ŽLž˜³”f^ŸEÏ`”f—–˜z
W W
ÖO GÉT²³Ž0jŽ²³Ž\Ér”  ”f̜˜³´³‘>ª¦O±.¯XªI•–ñ —–˜³ŽLž · §0¥–¤>ŽLŽ0ñL^ªj¦Ö‡—ažª“^¦O‘>¯X•–•aª ŽL´Ü²³Ž ïIjð ”>𡐠²³ŽEÉ'ŽL´!»NŽLž^ŸÊ¤I³ª0Ž ­ ïf­zª¯f¦^¥a¯zª ª ”>š'‘ÈÉrŽL´!»NŽi—až²³Ž
‘>˜œ»Ç•aŽ  Ž0VÉ'ŽLŽ0˜¨—–jž¡\Ér”  ”f̜˜³´³‘>±²‘>• ³—–˜³ŽLž^Ö · Ž^‘fžÌœjŽL´¼—–˜³ž—a´œŽ²³ŽÉrŽL´!»NŽÇŸÊ²‘>r—až^Öz²³ŽžÌ · ”>š‹²³Žx‘>˜œ»Ç•aŽLžr”>š
‘>˜³´ —až Ÿ ”OɕaŽ0 Ž‘,ÉrŽL´!»NŽÇ֑>˜³´ Ž‘,›‡”f—–˜XT—–˜²³Ž›œ•6‘>˜³ŽÇŸ¸ )‘>˜³ž•6‘>jŽ”>š žÌ³“j²²‘>
W
—–jž‘>›‡Ž~˙W—až‘> X2πÖ`—ažE´œŽ0˜³”fjŽL´W ± W (X) Ÿ¼ÛܔfjŽ»NXŽ0˜³ Ž0)‘>•–•–±NÖ`šº”f’›G”f—–˜zjž X , X , . . . X Ö W (X W, X , . . . X )
´œŽ0˜³”fjŽLž²³Ž
)‘>˜³ž•6‘>jŽE”>š û8šÍ”fx“L”f˜X)‘>—–˜ Ž0˜zIüÉT²³”Çž)ŽE“0•a”Ǟ̜jŽ“L”f˜X)‘>—–˜³ž
Ÿrʲ³ŽžjŽ0
”>š‘>•–•‹)‘>˜³ž•6‘>óijŽL¬8ž­”>¬8ó(š Wïf¥ —až´œŽ0˜³”fjŽL´  W± T r Ÿ‹Ž0 −W· ´œŽ0˜³”fjŽE²³ŽjŽ ŽL“0—a”f˜¶”>š W X‘  ”f, X̜T,²³. .Ž’. ”fX—–»Ç—–˜‹Ÿ
½¾N¿ÀÂÁ6ÃOÁ®ÄGÀ Å z̜›œ›‡”ÇžjŽR²‘> W = { W | i ∈ I } —až(‘.“L”f•–•aŽL“0—a”f˜ ”>šÉrŽL´!»NŽLž^Ÿ W —ažž)‘>—a´ j”  Ž ­G¤f­³¢
§j¤f­ “^r‘>˜ ¬º§Lòͬ®Ž’­z¯ “L”f”f•a”fjžŽL— · ´Ü›œÉT•–±—–².VßxÉ'Ö³”¨—sš{“L”f²³•a”fŽ0jjžŽEž—ažÌ³“)‘²¶“L”f²˜³‘>žj)‘>‘>˜X˜X ±.k)ÉT‘>—–˜³ž².•6‘>j²³ŽŽ”>šºš¡”f•–‘È•a”^ÉrÉTŽL—–´!˜œ»N»RŽ ›œj”f›‡Ž0V±NŸT¸²˜X‘>x±R“Lö³”f˜œ˜z—–)j‘>ŽE—–˜³žjŽ0ž‘>”>xš{•a›‡Ž^‘f”fž—–˜X jž
S
W ∈W
k
›‡”f—–˜Xjž ”>š S ֜“L”f˜X)‘>—–˜³ž‘,›‡”f—–˜X”>š  ”f²µ“L”f•a”fjžOŸ
Õ³”f‘>˜z±ÈöœËœŽL´ ‘>˜³´ Ö#É'Ž’“^‘>˜µ‘>˜³´¨ÉT—–•–•
‘fžjžÌ ŽÉT—–²³”f̜•a”ÇžjžT”>šÂ»NŽ0˜³Ž0)‘>•–—–V±¨²‘>T²³Ž›‡”f—–˜Xjž”>š ‘>jŽ
—–˜Ü»NŽ0˜³Ž0)‘>•›‡”Çž—–—a”f˜ÜWÉT—–²¶jSŽLžj›GŽL“0xj” W Ö⏲‘>x—až^Ö⏲³· Ž0±.´œ”¼˜³”f´œŽ0jŽ0 · —–˜³Ž‘È•–—–˜³ŽE›‘>)‘>•–•aŽ0•{j”¼²³Ž  ”f̜˜³S´³‘>±
•–—–˜³ŽE”>š¡‘>˜z± W ∈ W Ÿ V˜³´œŽLŽL´ ֗sš²³Ž0jŽ‘>jŽ\Ér”RžÌ³“j²¶›G”f—–˜zjž^֞•–—–»Ç²z•–±›GŽ0Ìœ  ²³Ž›‡”f—–˜Xjž^Ÿ \—ažŽ^‘fž±¨j”(ž)ŽLŽ
²‘>Ž0°NŽ0±žÌ  ž)Ž0”>š²³Ž’”f—–»Ç—–˜‘>•
›‡”f—–˜XxžjŽ0²‘>“^‘>˜  Ž’“0̜x”  ±µ‘i)‘>˜³ž•6‘>jŽ”>š‘ W ∈ W Ö³“^‘>˜™‘>•ažj”  Ž
“0̜” ™²³Ž›GŽ0Ìœ  ŽL´›‡”f—–˜Xž)Ž0^Ÿ
Õ—–jžžÌœ›œ›‡”ÇžjŽ²‘>É'޲‘L°NŽ”f˜œ•–±R”f˜³ŽE”f›GŽ0˜É'ŽL´!»NŽÇÖ³²‘>—až^Ö W = {W } Ÿ
¾#ø ø ê Åí ñI¬®­z¯f¥aª ªj¦^¯Xª¬8ñ õ
é’æ>Ä Ä 'Ä ¾#ø ø ê âÅ'‹Ž0 S  Ž‘’ö³˜œ—–jŽ›‡”f—–˜XTžjŽ0T‘>˜³´ W ‘ÉrŽL´!»NŽÇŸ ю›œj”^°NŽ²³Žxž)‘>jŽ · Ž0˜zTÉT—–² k = 3 Ö
²‘>—ažOÖ S “^‘>˜  Ž’“L”f•a”fjŽL´µÉT—–².VÉr”R“L”f•a”fjžžj̳“j².²‘>x‘>˜X±)‘>˜³ž•6‘>jŽ”>š W ²‘>“L”f˜z)‘>—–˜³ž‘>•aŽ^‘fž 3 ›‡”f—–˜Xjž
”>š S ÖG“L”f˜X)‘>—–˜³ž‘(›‡”f—–˜X”>š  ”f²¶“L”f•a”fjž^Ÿ z̜›œ›G”ÇžjŽ’ö³jžx²‘>²³Ži‘>˜œ»Ç•aŽ”>š W —až‘>•aŽ^‘fžj π ŸÊ²³Ž0˜ W —až²³Ž
̜˜œ—a”f˜.”>šVÉ'”²‘>•sšÍ›œ•6‘>˜³ŽLž^Ö A ‘>˜³´ B ŸÊ`0‘ NŽ²³Ž’)‘>˜³ž•6‘>jŽE”>š A û®jŽLžj›‹Ÿ B ür²‘>“L”f˜X)‘>—–˜³žxŽ~˜‘f“0•–±RVÉ'”(›‡”f—–˜Xjž
”>š S ֋ž)‘L±NÖ A ‘>˜³´ A û®jŽLž›‹Ÿ B ‘>˜³´ B ü~Ÿ,ʲ³Ž0jŽ · —–»Ç²X  Ž“L”f—–˜³“0—a´œŽ0˜³“LŽLž  Ž0VÉ'ŽLŽ0˜ A Ö A ‘>˜³´ B Ö B Ö
à
i
i
1
2
1
2
1
2
1
k
1
2
2
k
k
W
i
"!$#
&% &%'
(
1
2
1
2
1
2
1
2
 ²³Ž0jŽ~šº”fjŽÈ²³ŽRžjŽ0 {A , A , B , B } “L”f˜X)‘>—–˜³ž,VÉ'”œÖ²œjŽLŽÇ֔fEšº”f̜´!— Ž0jŽ0˜Xi›‡”f—–˜Xjž^Ÿ¶Ù'̜E—–˜ ‘>˜X± “^‘fžjŽÇÖÂÉ'Ž
“^‘>˜Ñ“L”f•a”f {A , A , B , B } žÌ³“j²[²‘> A ‘>˜³´ A û®jŽLžj›‹Ÿ B ‘>˜³´ B ü‘>jŽ”>š'´!— Ž0jŽ0˜XE“L”f•a”fjž^Ÿ x”^ɒÖ
—sš‘
)‘>˜³ž•6‘>jŽ’”>š W “L”f˜X)‘>—–˜³ž²œjŽLŽ’›G”f—–˜zjž^֜—–“L”f˜z)‘>—–˜³žŽ0—–²³Ž0 A ‘>˜³´ A ֜”f B ‘>˜³´ B Ö³‘>˜³´µÉrŽE‘>jŽ’´œ”f˜³ŽÇŸ
z̜›œ›‡”ÇžjŽT˜³”OÉ ²‘>²³Ž‘>˜œ»Ç•aŽ”>š —až•aŽLžjž²‘>˜ Ÿ юT»Ç—–°NŽVÉr”’›œj”z”>šºž—–˜(²œ—až“^‘fž)ŽÇÖXž—–˜³“LŽÉ'ŽÉT—–•–•â‘>›œ›œ•–±
²³Ž,—a´œŽ^‘fž”>š  ”f²™›œj”z”>šºžx•6‘>jŽ0^ŸE \˜¶Wš‘f“0^ևÉrŽiž²³”OÉ π²‘>—–˜™²œ—až“^‘fžjŽE²³Ž ß ›œj”f›GŽ0\±.²³”f•a´œžÉT—–² k = 2 Ÿ
ю,“^‘>˜Ü‘fžjžÌ · Ž²‘>x²³Ž’›‡”Çž—–—–°NŽ x¹ ‘cË#—až—až—–˜ W Öâ‘>˜³´¶²‘>˜³”¼VÉ'”È›‡”f—–˜Xjžx²‘L°NŽ²³Ž,ž)‘ · Ž y¹ “L”z”fj´!—–˜‘>jŽÇŸ
ٔf².”>š`²³ŽLž)Ž’“^‘>˜  Ž’‘f“)²œ—aŽ0°NŽL´  ±‘>˜.‘>›œ›œj”f›œ—6‘>jŽEj”f)‘>—a”f˜‹Ÿ ю’žI‘L±R²‘> X < Y —sš{²³Ž y¹ “L”#”fj´!—–˜‘>jŽ’”>š
—ažž · ‘>•–•aŽ0²‘>˜µ²³Ž y¹ “L”#”fj´!—–˜‘>jŽE”>š Y ŸÊ²œ—až”fj´œŽ0—–˜œ»È—až“^‘>•–•aŽL´²³Ž y¹ ”fj´œŽ0—–˜œ»³Ÿ'¸ žjÌ  žjŽ0 I ”>š S —až‘>˜
X
¬®­òVª~Ê«~©c²³ïfŽ ¥ ”>š S —sš ∀X”>š < YÉT—–<²¨jZŽLž›‡∈ŽL“0S':j”X, ZÖ ∈ I → Y ∈ I.
”fjŽx²‘>‘)‘>˜³ž•6‘>jŽ
S = ∅}.
”>š W ‘>•–Éô¤ ‘L#±!­âž¨¦NïǗ–˜X«Ijÿ Ž0jžjŽLS“0jž¨²³Ž  ”f̜˜³´³‘>± W—–˜Bd
‘>˜ —–˜X(S)
jŽ0°Ç=‘>•Ÿ {XÕ³”f∈PŽ^‘f:“jW² X(X)∈∩Bd
²³Ž ñ ïN¦N¤ ”>š X —až
(S)
žjŽ0°NŽ'²‘>
Sh (X) = {Y ∈ S : W (Y )∩Bd (S) = X}. 
Õ³”fE²³Žiö³jž›œj”#”>šuÖ`“L”f•a”f’²³Ž›‡”f—–˜XjžE”>š²³Ž  ”f̜˜³´³‘>±™‘>∀X,
•–jŽ0˜Y‘>∈—–˜œ»³Bd֑f“L“L(S)
”fj´!:—–˜œSh»¶j”(X)∩Sh
²³Ž”fj´œŽ0 (Y< )Ÿ=Õ³”f∅.
Ž0°NŽ0±  ”f̜˜³´³‘>±.›‡”f—–˜X X և“L”f•a”fŽ0°NŽ0±µ›G”f—–˜z—–˜¶²³Ž,ž²‘f´œ”^É ”>š X j”¼²³Ž,”f²³Ž0“L”f•a”f²‘>˜ X Ÿ’ß'”f•a”f²³Ž
jŽLžE”>šT²³Ž›‡”f—–˜Xjž,‘> —–)‘>—–•–±NŸ¨¸˜z±Ü)‘>˜³žj•6‘>jŽÈ”>š ²‘>“L”f˜X)‘>—–˜³ž‘>’•aŽ^‘fž’\Ér”.›G”f—–˜zjž^Ö`“L”f˜z)‘>—–˜³ž”f˜³Ž¼”f
VÉ'”  ”f̜˜³´³‘>±È›‡”f—–˜Xjž^ Ÿ Vš
—–T“L”f˜X)‘>—–˜³ž”f˜³Ž  ”f̜˜³´³‘>W±¼›G”f—–˜z^֜²³Ž0˜¨²³Ž”f²³Ž0T›G”f—–˜zT—až—–˜¨—–jžTž²‘f´œ”^ɒֳž)”²³Ž0±
²‘L°NŽÈ´!— ‡Ž0jŽ0˜z’“L”f•a”fjž^Ÿi š¡—–’“L”f˜z)‘>—–˜³žVÉ'”  ”f̜˜³´³‘>±¶›‡”f—–˜Xjž^֋²³Ž0˜[²³Ž0±™‘>jŽ“L”f˜³žjŽL“0̜—–°NŽi›‡”f—–˜Xjž‘f“L“L”fj´!—–˜œ»
j”i²³Ž y¹ ”fj´œŽ0^Ö³žj”i²³Ž0±¨²‘L°NŽ’´!— Ž0jŽ0˜Xx“L”f•a”fjž‘>»X‘>—–˜‹Ÿ¡Ê²œ—ažTö³˜œ—až²³ŽLž²³Žxö³jž›œj”z”>šŸ
1
2
1
2
1
2
1
2
1
2
1
1
2
2
1
2
y
y
y
W
W
W
W
W
W
W
y
Z
Z
Z
Y
Y
X
Y
X
Õ—–»Ç̜jŽ(Ð Z jŽ0›œ•6‘f“LŽLž X —–˜
X
W (2; y)
Ÿ
Ž²³Žx)‘>˜³ž•6‘>jŽ”>š ÉT²œ—a“j²ÑûuÐOür“L”f˜X)‘>—–˜³ž‘> ”Çž
Õ³”f²³ŽžjŽL“L”f˜³´R›œj”#”>šuÖzšº”f‘>˜z±(öœËœŽL´ ÖX•aŽ0
VÉ'”›G”f—–˜zjž”>š S Ö³ûºàNü
—–jž‘>›‡Ž~ËE²‘fž y¹ “L”#”fj´!y—–˜‘>jŽ Wy ÖN(2;‘>˜³´y)û N ü`—–jž‘>›‡Ž~˲‘fž · —–˜œ— W· ‘>• x¹ “L”#”fj´!—–˜‘>jŽÇŸ{ V{—až{Ž^‘f·ž±j”
žjŽLŽx²‘>Tšº”f‘>˜X± Ö
—ažÌœ˜œ— X̳Ž0•–±¨´œŽ~ö³˜³ŽL´ Ÿ{˳‘ —–˜³ŽÇ֜²³”OÉ
“j²‘>˜œ»NŽLž‘fž Ìœ˜³ž”^°NŽ0T²³ŽjŽ^‘>•
˜XÌ · Ž0jžOŸ Vš y —ažy°NŽ0W±.(2;ž · y)‘>•–•'û͞ · ‘>•–•aŽ0²‘>˜Ü²³Ž y¹ “L”z”f·j´!—–˜‘>jŽ,”>šW²³Ž(2;›G”fy)—–˜zjžx”>š S ü~Ö⏲³Ž0y˜ W (2; y) “L”f˜z)‘>—–˜³ž
VÉ'”’›‡”f—–˜Xjž^ÖXž)‘L± X ‘>˜³´ Y ÖX‘>˜³´(”f˜³Ž · ”fjŽ”f˜—–jž  ”f̜˜³´³‘>±NŸ¸xž{ÉrŽT—–˜³“0jŽ^‘fž)Ž y ÖǏ²³Ž‘>›‡Ž~˔>š W (2; y) “)²‘>˜œ»NŽLž
“L”f˜X—–˜z̳”f̳ž•–±NŸ x”^É “^‘>˜.²³Ž’žjŽ0 {X, Y } Öâ”>š²³ŽVÉ'”È›‡”f—–˜Xjž—–˜ W (2; y) “)²‘>˜œ»NŽ ÑÕ³”f‘“LŽ0)‘>—–˜Ü°f‘>•–Ì³Ž’”>š y Ö
”f˜³Ž’”>š²³Ž · ֞I‘L±NÖ X Ö · ”^°NŽLžj”²³Ž  ”f̜˜³´³‘>±NŸ¸²œ—až›‡”f—–˜XÉrŽE²‘L°NŽ Y —–˜³ž—a´œŽÇÖG‘>˜³´VÉ'”È›‡”f—–˜XjžOÖ X ֑>˜³´
”f˜™²³Ž  ”f̜˜³´³‘>±NŸE š¡É'Žž•–—–»Ç²X•–±.š®Ìœ²³Ž0—–˜³“0jŽ^‘fžjŽ y ÖG²³Ž0˜ Z «ª\îâ¥sïX§)ªIñ X և²‘>—až^Ö Y ‘>˜³´ Z ÉT—–•–•  Ž—–˜
Z
û͞)ŽLŽEÕ—–»Ç̜jŽÈÐOü~Ÿ¸xž y —–˜³“0jŽ^‘fžjŽLžxj”(—–˜!ö³˜œ—–\±NÖG²³ŽEžjŽ0 {Z, Y } ÖG“L”f̜•a´Ü“j²‘>˜œ»NŽ,žjŽ0°NŽ0)‘>•
— · ŽLž^Ö  ̜²³Ž
W (2; y)
ž)‘ ŽÉ'‘^±NŸ Ž~ö³˜³Ž‘E»Ç)‘>›œ²¨ÉT²³”ÇžjްNŽ0—a“LŽLž‘>jŽ²³Ž›G”f—–˜zjžT”>š ֜‘>˜³´È\Ér”,°NŽ0—a“LŽLž^Ö ‘>˜³´ ‘>jŽ“L”f˜œ˜³ŽL“0jŽL´
— · V jŽ0›œ•6‘f“LŽL´ U ´!̜—–˜œ»¨²³ŽE›œj”#“LŽL´!̜jŽÇŸ юE»NŽ0xVÉ'”¼›‘>²³žOÖSP ‘>˜³´ P ŸÊ²³Ž›‘>U—– (P , PV ) —až“^‘>•–•aŽL´¶²³Ž
î!ïÇò ¦X˜ ªj§jÕ¤f—–ó»Ç̜î!j¤>ŽRñ~¬®òÍà!¬ºÖ¤f­VÉ'¤u”¶£ Ž~S˳‘ ¬®ò ›œ •aŽL«žª~ñ‘>jªŽÈ§^òT»Çò—–¤°NŽ0W˜‹Ÿ¶¡¤Êu£E²³Ž(¤f«°N¦XŽ0ª~«,—aò “LŽL¤ žiŸ ”>š ‘>jŽ
֑>˜³´ ²³Ž(°NŽ0—a“LŽLži”>š
‘>jŽ Y , Y , . . . Ÿ ˜  ”f²[Ž~·Ë³‘ · ›œ•aŽLž^և²³Ž,)‘>˜³ž•6‘>jŽLž’”>š W ‘>jŽ,P»Ç—–°NŽ0˜[—–˜™X²³,Ž,X›G”Ç, .žj—–..—a”f˜³žÉT²³Ž0˜Ñž)” · Ž›‡”f—–˜XP—až
jŽ0›œ•6‘f“0—–˜œ»R‘>˜³”f²³Ž0^Ÿ
ß'”f•a”f²³Ž°NŽ0—a“LŽLžr”>š P jŽL´ ÖX²³ŽT°NŽ0—a“LŽLž¡”>š P  •–̳ŽÇŸ  žjŽ0°NŽT²‘>¡Ž^‘f“)²È)‘>˜³ž•6‘>jŽ”>š W ²‘>r“L”f˜X)‘>—–˜³ž
‘>•aŽ^‘fžjVÉ'”È›‡”f—–˜XjžOÖâ“L”f˜z)‘>—–˜³ž‘>•aŽ^‘fž”f˜³Ž°NŽ0jŽ~Ë.”>š  ”f² P ‘>˜³´ P ŸTʲœ—ažx“L” · ›œ•aŽ0jŽLžx²³ŽEžjŽL“L”f˜³´.›œj”z”>šŸ
1
1
2
1
1
1
2
2
2
2
1
2
1
2
Y2
X4
X4
X3
X2
X3
Y3
Y2
Y1
X2
Y1
X1
X1
Õ—–»Ç̜jŽEà Ï{‘>²¶´œŽL“L” · ›‡”Çž—–—a”f˜³ž”>š{”fj´œŽ0\Ér”œŸ
Ö
P1 = X 1 X2 . . . P 2 = Y 1 Y2 . . .
Ÿ
x”^É ÉrŽ¡“^‘>˜Ìœ˜Ej”T²³Žr“^‘fžjŽÉT²³Ž0˜Ér޲‘^°NŽ¡)‘>˜³žj•6‘>jŽLž
”>š ò ¤ É'ŽL´!»NŽLž{‘>
²³Ž¡ž)‘ · Ž— · ŽÇŸ ю¡´!—ažj—–˜œ»Ç̜—až²
žjŽ0°NŽ0)‘>•
“^‘fž)ŽLž‘f“L“L”fj´!—–˜œ»Èj”i²³ŽjŽ0•6‘>—–°NŽ’›G”Çž—–—a”f˜¶”>šÂ²³Ž\Ér”É'ŽL´!»NŽLž^Ö V ‘>˜³´ W Ÿ
÷ ¾ çÁ ì
˜³Ž’”>š{²³ŽÉrŽL´!»NŽLž²‘fž‘>˜œ»Ç•aŽE‘>•aŽ^‘fž π Ÿ
Õ³”f²³Ž’”f²³Ž0“^‘fžjŽLž^Ö³ÉrŽ’“^‘>˜¶‘fž)žÌ · ŽÉT—–²³”f̜•a”Çžjž”>š{»NŽ0˜³Ž0)‘>•–—–\±²‘> W “L”f˜X)‘>—–˜³ž²³Ž’›G”Çž—–—–°NŽ x¹ ‘cË!—až^Ÿ
{Ë!jŽ0˜³´(²³Ž ”f̜˜³´³‘>±²‘>• ³—–˜³ŽLž'”>š
j”’•–—–˜³ŽLž^Öz²³Ž0±(´!—–°z—a´œŽ²³Ž›œ•6‘>˜³Ž—–˜Xj”EšÍ”f̜r›‘>jž^֛œ›GŽ0^Ö#‹”OÉrŽ0^Ö!‹Ž~šÍ^Ö
W

‘>˜³´ —–»Ç²X^֜ÉT²œ—a“)²µ—až W —–jžjŽ0•sšuŸ #ŽLŽEÕ`—–»Ç̜jŽ !Ÿ
¾ ç ê ®ê³À
¾³
ì ˜³Žž—a´œŽ,”>š —až—–˜ —–»Ç²X‘>˜³´.²³Ž”f²³Ž0x”f˜³Ž—až—–˜Ü‹Ž~šÍ^ŸÊ²‘>—ažOÖ³²³ŽÌœ˜œ—a”f˜
÷ ”>š{²³ŽÉrŽL´!»NŽLž“L”^°NŽ0x‘,²‘>•sšÍ›œ•6‘>˜³ŽÇŸ #ŽLŽ’Õ—–»ÇV̜jŽ ³Ÿ
¾ ç á ÄGÀGÃ^ê³ÁÍÀ
ì —–²³Ž0(û®—8ü”f˜³Žiž—a´œŽi”>š V —až—–˜ ›œ›GŽ0^ÖG²³Ži”f²³Ž0”f˜³Ž,—až—–˜Ü‹”OÉrŽ0^Ö`”fû®—–—8ü  ”f²
÷ ž—a´œŽLž‘>jŽ’—–˜ —–»Ç²X”fEû®—–—–—8ü  ”f²¶ž—a´œŽLž‘>jŽ—–˜.‹Ž~šÍ^Ÿ #ŽLŽÕ—–»Ç̜jŽEÝ!Ÿ
¾ ‹Å™ç êœæ ì ˜³Žžj—a´œŽE”>š —až—–˜Ü‹Ž~šÍ‘>˜³´.²³Ž”f²³Ž0”f˜³Ž’—až—–˜ ›œ›‡Ž0”f
”^ÉrŽ0^Ÿxʲœ—ažÉT—–•–•  Ž
÷ ²³Ž²‘>j´œŽLž“^‘fžjŽÇŸ #ŽLŽ’Õ—–»Ç̜jŽEý!Ÿ V
%
Upper
W
Left
Right
O
Lower
Õ—–»Ç̜jŽ юL´!»NŽ
W
W
W
V
V
Õ—–»Ç̜jŽ ʱ#›‡Ž’àµû x‘>•sšÍ›œ•6‘>˜³ŽOü
¾ ‡Åç ¾XäXÁ®êVì
—–²³Ž0û®—8ü{”f˜³Žž—a´œŽ”>š V —až—–˜ —–»Ç²X‘>˜³´(²³Ž”f²³Ž0r”f˜³Ž—až¡—–˜ ›œ›GŽ0'”f¡‹”^É'Ž0^֜”f
÷ û®—–—8ü ”f²µž—a´œŽLž‘>jŽ—–˜ ›œ›‡Ž0^֜”fû®—–—–—8ü ”f²µž—a´œŽLž‘>jŽ—–˜¶‹”^É'Ž0^Ÿ¡Ê²‘>—ažOÖ!²³ŽÌœ˜œ—a”f˜.”>š{²³ŽÉrŽL´!»NŽLž—ažT—–˜¶‘>˜
”f›‡Ž0˜¼ ²‘>•sšÍ›œ•6‘>˜³ŽÉT²³”ÇžjŽ  ”f̜˜³´³‘>±R“L”f˜X) ‘>—–˜³žT²³Žx”f—–»Ç—–˜‹Ö  ̜˜³”f˜³Žx”>š`²³Ž · “L”f˜z)‘>—–˜²³Ž”f²³Ž0^Ÿ #ŽLŽxÕ—–»Ç̜jŽ #Ÿ
\—až˜³”f²‘>j´¶j”ÈžjŽLŽE²‘>²³Ž0jŽi‘>jŽ’˜³”R”f²³Ž0›‡”Çžjž—  —–•–—–—aŽLžOŸ  žjŽ0°NŽ²‘>\Ér”¼ÉrŽL´!»NŽLž“L”fjŽLž›‡”f˜³´!—–˜œ»
j”i²³Ž°NŽ0—a“LŽLž”>š‘,“L”f˜X°NŽ~˛‡”f•–±z»N”f˜µ“^‘>˜œ˜³”f  Ž”>š{V±#›GŽÐûÍÙ'—–»zü”f”>š{V±#›GŽ’Ý¨û z›‡ŽL“0—6‘>•8ü~Ÿ¡Ê²œ—ažTÉT—–•–•  ŽÌ³žjŽL´
—–˜²³Ž›œj”#”>š{”>šÊ²³ŽL”fjŽ · Ðǟ
\—ažž²³”^ÉT˜Ü—–˜ ØÚÏ¡ÔÇÓfÞ`²‘>x—sš W = {V, W } —ažx‘(žjŽ0x”>šVÉ'”ÈÉ'ŽL´!»NŽLžx”>š\±z›‡ŽÝ¶û z›GŽL“0—6‘>•8ü~Ö⏲³Ž0˜ W —až ­â¤Çò
ßxŸœ V˜.‘ž)Ž0—aŽLž”>š{
Ž ·· ‘fžÉrŽEž²³”^É ²‘>x‘>•–•‹”f²³Ž0›‘>—–jž‘>jŽ ßxŸ
V
V
W
W
W
W
V
Õ—–»Ç̜jŽEÝ Ê¡±z›‡Ž ûß'”f˜X)‘>—–˜âü
Ý
V
V
W
W
V
Õ`—–»Ç̜jŽý ʱ#›GŽ . û x‘>j´ü
W
V
W
W
V
V
W
V
Õ`—–»Ç̜jŽ ʱ#›GŽ’Ý.û z›GŽL“0—6‘>•8ü
¾#ø ø ê‡Å ª0ò W = {V, W } ôjªiï(ñLª0ò¤u£’ò ¤ ª¦^¯XªIñ ¤u£’òÍÿI ¤f­âòïf¬8­Çõ ª~­ W ¬6ñ õ
z̜›œ›‡”ÇžjŽ²‘> U —až‘¨É'ŽL´!»NŽ(²‘>E“L”f˜z)‘>—–˜³ž,²³Ž›G”Çžj—–—–°NŽ x¹ ‘cË!—ažE‘>˜³´Ñ²‘fž‘>˜œ»Ç•aŽ(•aŽLžjž²‘>˜ π ŸÈÕ³”f‘>˜z±
֑>˜³´›‡”f—–˜Xž)Ž0 Ö!•aŽ0
Ž²³Ž’žjŽ0”>š{)‘>˜³ž•6‘>jŽLž”>š ²‘>“L”f˜z)‘>—–˜¶Ž~˳‘f“0•–± ›‡”f—–˜Xjž”>š Ÿ
k>0
ÎÇ̳žjE•–— NŽ—–˜ ²³Ž›œSj”z”>š”>Tšr‹Ž ·(S)· ‘  ÐÇ֋šº”f‘>˜X±™öœËœŽL´ y Ö`‘>˜³´ kU> 0 Ö`•aŽ0 U (k; y)  Ž²³kŽ)‘>˜³ž•6‘>j޼S”>š U
ÉT²œ—a“j²[ûuÐOü¡“L”f˜X)‘>—–˜³žT‘> · ”Çžj k ›‡”f—–˜Xjž”>š S ևûºàNü{—–jž‘>›GŽ~Ë(²‘fž y¹ “L”#”fj´!—–˜‘>jŽ y Ö!‘>˜³´¶û Nü—–jž'‘>›‡Ž~Ë(²‘fž · —–˜œ— · ‘>•
“L”#”fj´!—–˜‘>jŽÇŸ Vš y —až°NŽ0±µž · ‘>•–•Ö³²³Ž0˜ U (k; y) “L”f˜X)‘>—–˜³ž k ›‡”f—–˜Xjž^ÖGž)‘L± X , X , . . . , X ÖG‘>˜³´.”f˜³Ž · ”fjŽ,”f˜
x¹
—–jž  ”f̜˜³´³‘>±NŸ¸xžrÉrŽx—–˜³“0jŽ^‘fžjŽ y Öz²³Žx‘>›‡Ž~˼”>š U (k; y) “)²‘>˜œ»NŽLžT“L”f˜X—–˜z̳”f̳ž•–±NŸ¡Õ³”f‘“LŽ0)‘>—–˜¨°f‘>•–̳Žx”>š y Ö#”f˜³Ž
”>š
Ö³ž)‘L±NÖ Ö ”^°NŽLžj”²³Ž ”f̜˜³´³‘>±NŸ'¸²œ—až›‡”f—–˜XÉ'Ž’²‘^°NŽ
—–˜³ž—a´œŽÇÖâ‘>˜³´VÉ'”
›‡”f—–X˜Xjž^,Ö XX, .֋.‘>.˜³, ´X X ”f˜™X²³Ž · ”f̜˜³´³‘>±NŸ šr ÉrŽ(ž•–—–»Ç²z•–±.šÍ̜²³Ž0—–˜³“0jŽ^‘fžjŽ y Ö ²³Ž0X˜ X, . . . «,ª\Xîâ¥sïX§)ªIñ X Ö ²‘>—až^Ö
‘>˜³´
ÉT—–•–• ŽE—–˜ y) Ÿ¸xž y —–˜³“0jŽ^‘fžjŽLžj”È—–˜!ö³˜œ—–\±NÖGž)” · ŽE”f²³Ž0x›G”f—–˜zjžx“L”f̜•a´  Ž’jŽ0›œ•6‘f“LŽL´
X
ž— · —–•6‘>•–±NXŸ Ž,~.ö³. ˜³. ,Ž’X‘,»Ç)‘>›œ² ÉT²³”ÇžjUŽ(k;
°NŽ0—a“LŽLžx‘>jŽ²³Žx›G”f—–˜zjž”>š S ֜‘>˜³´¨\Ér”°NŽ0—a“LŽLž^Ö X ‘>˜³´ Y ‘>jŽ“L”f˜œ˜³ŽL“0jŽL´
— X jŽ0›œ•6‘f“LŽL´ Y ´!̜—–˜œ»¶²³Ž›œj”#“LŽL´!̜jŽÇŸ ю(»NŽ0 k ›‘>²³ž^Ö P , P , . . . , P Ÿ ¡‘f“)² )‘>˜³ž•6‘>j޼”>š U ²‘>
“L”f˜X)‘>—–˜³ž,‘>•aŽ^‘fž k ›G”f—–˜zjž’”>š S Ö
“L”f˜X)‘>—–˜³ž‘>•aŽ^‘fž’”f˜³Ži°NŽ0jŽ~Ë[”>š'Ž^‘f“)² ”>š P , P , . . . , P Ÿ,ʲ³Ž k¹ Ìœ›œ•aŽ
—až“^‘>•–•aŽL´²³Ž î!ïÇò ¦Xªj§j¤fóî!¤>ñ~¬®òͬº¤f­Ñ¤u£ S ¬®ò «jªIñj§LòTò¤ W ¡¤u£E¤f«¦Xª~« k Ÿ
(P , P , . . . , P )
é’æ>Ä Ä xÄ ¾#ø ø ê Šю“^‘>˜[‘fžjžÌ · Ž²‘> W ⊃ V ”f W ⊃ −V ‘>˜³´ W “L”f˜X)‘>—–˜³ž²³Ž›G”Çž—–—–°NŽ x¹ ‘cË!—až^Ÿ
‹Ž0 (P , P , . . . , P )  Ž²³Ž’›‘>²µ´œŽL“L” · ›G”Çž—–—a”f˜Ü”>š S ÉT—–²µjŽLž›‡ŽL“0Tj” W ֜”>š”fj´œŽ0 k Ÿ
žjŽ0°NŽ(²‘>‘>˜X±Ü)‘>˜³ž•6‘>jŽ(”>š
—–˜zjŽ0jžjŽL“0jž‘>˜z± P —–˜ ‘>˜Ñ—–˜XjŽ0°Ç‘>•¡”>š—–^Ÿ V˜³´œŽLŽL´ Ö
—sš X < X <
V
 ÖG²³Ž0˜
ÖÉT²œ—a“)²Ü—až‘ÈžÌ  žjŽ0”>š V (X , X ) ∩ −V (X , X ) Ÿ #ŽLŽ
X ∈P
X ∈ W (X , X ) ∩ −W (X , X )
Õ—–»Ç̜jŽEÓ!Ÿ
юž²³”^É ²‘>ÉrŽ“^‘>˜¶“L”f•a”f²³Ž›G”f—–˜zjž”>š S ÉT—–²¶jŽL´¶‘>˜³´  •–̳ŽEžÌ³“)².²‘>x‘>˜z±)‘>˜³ž•6‘>jŽE”>š W ÉT²œ—a“)²
ý
#
!
#
U
k
1
1
2
1
k
2
2
U
2
3
U
1
U
k
&% &%
W
i
1
k
W
1
k
0
1
U
1
k
0
1
1
0
1
2
U
2
U
k
U
1
U
2
U
k
W
2
W
k
2
W
i
1
3
1
3
1
1
3
2
y
1
(
3
y
X3
X2
V
Õ—–»Ç̜jŽEÓ W
X1
W (X1 , X3 ) ∩ −W (X1 , X3 ) ⊂ V (X1 , X3 ) ∩ −V (X1 , X3 )
Ÿ
“L”f˜X)‘>—–˜³žx‘>T•aŽ^‘fž ›‡”f—–˜Xjž^Ö³‘>˜³´µ‘>˜z±¼)‘>˜³ž•6‘>jŽE”>š V ÉT²œ—a“)².“L”f˜X)‘>—–˜³žx‘>T•aŽ^‘fž’Ð ›‡”f—–˜Xjž^֜“L”f˜z)‘>—–˜³ž›‡”f—–˜Xjž”>š
 Ÿ”fÑ²Ü“LŽR”f“L•a”f”fj•až^”fŸ ß'”f˜³ž—a‘>´œ˜³Ž0´  (P ,žP̳“)² ,P²‘>i, Ž0P°NŽ0)± ÖG²³Ž›‘>²Ü´œŽL“L” · “L›‡”f”ǘXž)—–‘>—a—–”f˜³˜[ž(”>‘ š S•–̳ÉTŽR—–›G²¶”f—–j˜zŽL,ž›‡”>ŽLš“0x²³jŽ ” W֑>Ö☳”>´ š¡Ž0”f°Nj´œŽ0Ž0± 
4
P
“LP”f˜z)‘>—–˜³ž›‡”f—–˜Xjž”>š  ”fW²µ“L”f∈•a”fjTž^Ÿ r z— (S)
—–•6‘>•–±NÖÉrŽ’“L”f•a”f P ‘>˜³´ P žj̳· “j²µ²‘>Ž0°NŽ0±
·
V ∈ T r (P ∪ P )
“L”f˜z)‘>—–˜³ž‘(jŽL´¶›G”f—–˜zx”>š²³Ž Ö ‘>˜³´ÜŽ0°NŽ0±
“L”f˜X)‘>—–˜³ž›‡”f—–˜Xjž”>š ”f²
W ∈ T r (S)
“L”f•a”fjž^Ÿ Õ—–˜‘>•–•–±NÖ'ÉrŽ.“L”f•a”f(²³ŽjŽLž(”>š²³Ž· ›‡”f—–˜Xjž R = SV\ (P∈ T r ∪(PP ∪∪ PP ) ∪ P ) žÌ³“)² ²‘>ÈŽ0 °NŽ0±
“L”f˜z)‘>—–˜³ž›‡”f—–˜Xjž”>š ”f²µ“L”f•a”fjž^Ÿ
V ∈ T r (R)
ŽL“^‘>•–•`²‘>šÍ”f‘>˜X± W ∈ T r  (S) Ö |W ∩ P | = |W ∩ P | = |W ∩ P | = |W ∩ P | = 1 ŸÕ³”f
‘>˜X± X ∈ P Ö Y ∈ P ֜—sš²³Ž0jŽ—až‘ W ∈ T r (S) ÉT—–² W ∩ P = {X} ‘>˜³´ W ∩ P = {Y } ֜²³Ž0˜
ÉrŽiž)‘^±.²‘> X ‘>˜³´ Y ‘>jŽ £I«~¬ª~­â¦>ñ Ÿx Vš X û®jŽLž›‹Ÿ Y ü²‘fž”f˜œ•–±.”f˜³Žš®—aŽ0˜³´ Y û®jŽLž›‹Ÿ X ü~և²³Ž0˜™ÉrŽi“^‘>•–•{—–‘
£L‘>ïfj­Ž˜³û͔>Ž0š —–Y²³Ž0Ö!TjŽLš‘>žj›‹˜³ž^ŸÖ!”>˜³š ”fXü~žjŸ)‘> Vjš žX‘>j”fŽ “^Y‘>•–•aŽL²´ ‘fž‘>•aŽ^‘fžŸ T”f˜³Žxšº‘>˜‹Ö#²³Ž0˜ÉrŽžI‘L±È²‘>T—–—až‘ ñ~òïÇ« ŸÊ²³”ÇžjŽ›‡”f—–˜Xjž²‘>
Õ³”f’‘>˜[Ž~˳‘ · ›œ•aŽÇ֋žjŽLŽ,Õ`—–»Ç̜jŽ(à!Ÿ «j˜Üªº¯ #²³¥sïǎ,« •aŽ~šÍö³»Ç̜jŽÇÖ Y —až‘¼žj)‘>^Ö◖jžxš‘>˜³ž‘>jŽ X ‘>˜³´ X և²³Ži”f²³Ž0
›‡”f—–˜Xjž‘>jŽjŽ0»Ç̜•6‘>^Ÿ ˜µ²³Ž—–»Ç²X^Ö Y —až‘ižj)‘>^֜—–jžš‘>˜—až X Ö!²³Ž’”f²³Ž0›‡”f—–˜Xjžx‘>jŽjŽ0»Ç̜•6‘>^Ÿ
z̜›œ›‡”ÇžjŽEö³jž²‘>’‘>•–•›G”f—–˜zjž”>š ‘>˜³´ ‘>jŽjŽ0»Ç̜•6‘>^Ÿ’ß'”f•a”fŽ0°NŽ0±.²œ—–j´¶›‡”f—–˜X”>š ֝jŽL´™‘>˜³´
²³Žx”f²³Ž0jž  •–̳ŽÇŸ¡ V˜ P Ö#“L”f•a”fr²³ŽšÍP—aŽ0˜³´œžr”>š
P²³ŽjŽL´¼›G”f—–˜zjž  •–̳ŽÇ֜‘>˜³´¨“L”f•a”f'²³ŽjŽLž”>š
²³Ž›GP”f—–˜zjž”>š P
û͎0°NŽ0±µ²œ—–j´üjŽL´ ŸÕ³”f’‘>˜X±
Ö
‘>˜³´
‘>jŽ’šÍ—aŽ0˜³´œž^ÖG²³Ž0jŽ~šÍ”fjŽÇÖ ‘>•aŽ^‘fž”f˜³Ž”>š
²³Ž · —až  •–̳ŽÇŸ ˜.²³Ž,”f²³Ž0W²‘>˜³∈´ ևT‘>r˜X± VW∈∩TPr (P ∪WP ∩)P“L”f˜z)‘>—–˜³žx²œjŽLŽ“L”f˜³žjŽL“0̜—–°NŽ›‡”f—–˜Xjžx”>š P
”f P Ö³‘>˜³´µ²³Ž0±¼²‘^°NŽ  ”f²µ“L”f•a”fjž^Ÿ
z̜›œ›‡”ÇžjŽ˜³”^É ²‘>r˜³”fT‘>•–•G²³Ž›‡”f—–˜XjžT”>š ‘>˜³´ ‘>jŽxjŽ0»Ç̜•6‘>^Ÿ¡ß'”f•a”f‘>•–•‡ž)‘>jž •–̳ŽÇŸÊ²³Žö³jž‘>˜³´
•6‘fžš®—aŽ0˜³´™”>š'‘Rž)‘>^ÖG—–˜™²³Ž y¹ ”fj´œŽ0—–˜œ»³Ö‡—ažŽ0P—–²³Ž0E‘¼žP)‘>”f‘¼jŽ0»Ç̜•6‘>°NŽ0jŽ~ˇև²³Ži”f²³Ž0 jž’‘>jŽš‘>˜³ž^Ÿ’ß'”f•a”f
²³Žš®—aŽ0˜³´œž”>š'Ž^‘f“j²Ñž)‘>‘>•–jŽ0˜‘>—–˜œ»Ç•–±NÖ`‘f“L“L”fj´!—–˜œ»j”R²³Ž ”fj´œŽ0—–˜œ»³Ö‹žj)‘>—–˜œ»RÉT—–² •–̳ŽÇÖ Ž~˜“LŽ0›œ²³Ž,•6‘fž
VÉ'”¼š®—aŽ0˜³´œž G“L”f•a”f²³Ž’•6‘fž”f˜³Ž  •–̳ŽÇ֏²³Ž›œjŽ0°z—a”f̳ž”f˜³ŽjŽLy´ ¹ ŸÊ²³Ž,žj”š‘>Ìœ˜³“L”f•a”fjŽL´Üj Ž0»Ç̜•6‘>›‡”f—–˜Xjžx”>š P
‘>˜³´ P šÍ”f · ›‘>—–jžE”>šT—–˜XjŽ0°Ç‘>•až^Ÿ ю(“L”f•a”f,Ž^‘f“j² žÌ³“j² ›‘>—–”>šT—–˜XjŽ0°Ç‘>•r²³ŽÈž)‘ · Ž(ɑL± ‘fž’É'ŽÈ´!—a´ —–˜Ñ²³Ž
‘>•–• ¹ jŽ0»Ç̜•6‘>“^‘fž)ŽÇ֜“L”f•a”f—–˜œ»È²³Žö³jžT›‡”f—–˜X”>š{Ž^‘f“)²µ›‘>—–”>š{—–˜XjŽ0°Ç‘>•ažjŽL´ Ÿ #ŽLŽ’Õ—–»Ç̜jŽEÒ!Ÿ
ßr•aŽ^‘>•–±NÖz—sš
²³Ž0˜(—–¡“L”f˜z)‘>—–˜³ž'‘>¡•aŽ^‘fž¡”f˜³Ž •–̳ŽT›‡”f—–˜Xr”>š
ŸÂ Vš
Ö
²³Ž0˜ —–“L”f˜X)‘>—–W˜³žšº∈”f̜Tr“L”f˜³žjŽL“0̜—–°N޼›‡”f—–˜Xjži”>š P ”f P Ö{ž)‘L±NÖ X , XP , X∪ P, X Ö—–˜ VP ∈ ŸT r š X(P < ∪YP <)
‘>˜³´ Y —až’‘¼ž)‘>^և²³Ž0˜ V · ̳ž’“L”f˜X)‘>—–˜ ‘>•–•š‘>˜³žx”>š Y ‘fžxÉrŽ0•–•Ÿ’ \˜³´œŽLŽL´ և²³Ž,šº‘>˜³ž”>š Y ‘>jŽ
Z ∈ P ∩V
0
V
W
7
1
W
4
0
0
W
1
W
1
W
2
W
2
W
2
W
3
W
4
0
W
4
0
V
7
W
1
V
2
0
W
1
W
4
W
2
0
0
W
1
W
4
0
0
W
2
W
3
W
4
W
3
W
3
W
2
0
W
3
W
1
1
2
W
2
W
4
0
0
W
2
2
W
4
3
W
2
0
W
1
W
1
V
7
0
W
2
0
W
1
W
1
W
2
W
2
W
2
W
1
W
2
W
1
W
2
0
W
1
W
4
2
W
1
0
W
4
0
W
4
0
W
1
W
2
W
1
1
2
W
2
3
4
0
W
1
V
7
W
1
W
2
y
y
–— ˜ W (X, Z) \ (W (X) ∪ W (Z)) Öâ‘>˜³´  ±µ”f̜Ž^‘>•–—aŽ0”  žjŽ0°Ç‘>—a”f˜³ž^Ö³²œ—až—až—–˜ V (X, Z) ⊂ V Ÿ #”œÖâ—sšŽ0—–²³Ž0
”f X —až‘Rž)‘>^ÖG²³Ž0˜ V “L”f˜X)‘>—–˜³žE‘¼jŽL´Ü›‡”f—–˜X^Ö ž—–˜³“LŽiŽ0°NŽ0±¶žj)‘>x²‘fž‘¼jŽL´¶šº‘>˜‹Ÿ z—–˜³“LŽ²³Žiž)‘>—–jžjŽ0•sš
X
—až  •–̳ŽÇÖ!É'ŽE‘>jŽ´œ”f˜³Ž—–˜²œ—až“^‘fžjŽÇŸ Vš X , X , X , X “L”f˜z)‘>—–˜³ž²œjŽLŽ’“L”f˜³ž)ŽL“0̜—–°NŽjŽ0»Ç̜•6‘>°NŽ0—a“LŽLž²³Ž0˜É'Ž
‘>jŽ´œ”f˜³Ž‘>»X‘>—–˜‹Ö  ±²³Žx“L”f•a”f—–˜œ»Ìœ•aŽšº”f¡²³ŽjŽ0»Ç̜•6‘>'—–˜XjŽ0°Ç‘>•až^Ÿ #”EÉrŽ‘>jŽ•aŽ~šÍ¡ÉT—–²R²³Ž“^‘fž)ŽÉT²³Ž0˜ X ‘>˜³´
‘>jŽž)‘>jž^Ö X ‘>˜³´ X ‘>jŽjŽ0»Ç̜•6‘>^ŸÙr̜r—–˜¼²œ—až'“^‘fžjŽ V ‘>•až)”E“L”f˜X)‘>—–˜³ž'²³Žx“L” ·· ”f˜ÈšÍ—aŽ0˜³´ Y ”>š X ‘>˜³´
X
—–˜ ֜ÉT²œ—a“j²Ü—až‘>•až)”È‘ijŽ0»Ç̜•6‘>x°NŽ0jŽ~ˇŸÙ'±¼²³Ž“L”f•a”f—–˜œ»¼Ìœ•aŽšÍ”f²³ŽjŽ0»Ç̜•6‘>—–˜zjŽ0°f‘>•ažO֔f˜³ŽE”>š Y Ö X
X
‘>˜³´ XP —ažTjŽL´ Ö!²³ŽE”f²³Ž0VÉ'”¼‘>jŽ  •–̳ŽÇÖ³ž)”iÉrŽE‘>jŽ´œ”f˜³ŽÇŸ
0 "
2
0
3
1
2
3
4
1
4
2
0
3
2
W
2
3
2
3
b
P1W
r
P2W
b
b
b
P1W
r
P2W
b
b
r
b
b
Õ—–»Ç̜jŽEÒ Ê‹É'”ÈŽ~˳‘ · ›œ•aŽLž”>š“L”f•a”f—–˜œ»È”>š
b
b
r
r
r
b
b
r
b
P1W ∪ P2W
b
b
Ÿ¡Õœ—aŽ0˜³´œž‘>jŽ“L”f˜œ˜³ŽL“0jŽL´  ± ŽL´!»NŽLž^Ÿ
Õ³”f
É'ŽÌ³ž)Ž²³Žž)‘ Ž“L”f•a”f—–˜œ»iÌœ•aŽ‘fž¡šº”f
̜'É'ŽžÉT—–j“j²¨²³Žxj”f•aŽLžT”>š
²³Žx“L”f•a”fjž^Ÿ
#”i‘>˜X± WP ∈∪TPr “L”f˜z)‘>—–˜³ž‘>r•aŽ^· ‘fžj'”f˜³ŽjŽL´R›G”f—–˜z'”>š P P ∪ P∪ P ‘>˜³´R‘>˜z± V ∈ T r (P ∪ P ) “L”f˜X)‘>—–˜³ž
 ”f²µÕ—–“L˜”f‘>•a•–”f•–j±Nž^ևŸ ÉrŽ,²‘^°NŽj”R“L”f•a”f²³Ž,jŽLžx”>šr²³ŽE›‡”f—–˜Xjž
žÌ³“)²Ü²‘>Ž0°NŽ0±
R = S \ (P ∪ P ∪ P ∪ P )
L
“
f
”
z
˜
)

>
‘
–
—
³
˜
ž
‡
›
f
”
–
—
X
˜
j

ž
>
”
š
f
”

µ
²
L
“
f
”
a
•
f
”
j

^
ž
Ÿ
Ê
œ
²
a
—
ž
^
“
>
‘
˜
’
Ž
f
‘
)
“
œ
²
a
—
0
Ž
N
°
L
Ž
´
¨
±

³
²
x
Ž
³
ö
j

ž

œ
›
j

z
”
>
”
š
>
”
{
š

Ž ·· ‘Ðǟ
V ∈ T r (R)



x”^ɑ>˜X±
“L”f˜z)‘>—–˜³ž‘>¡•aŽ^‘fž”f˜³Ž •–̳Žx‘>˜³´¨‘>r•aŽ^‘fžjr”f˜³ŽjŽL´È›‡”f—–˜X^Ÿ Vš
ÖX²³Ž0˜¨Ž0—–²³Ž0
—–¡“L”f˜z)‘>—–˜³ž'‘>W•aŽ^‘fžj∈TVÉ'r”’›‡”f—–˜Xjž¡”>š R = P \ (P  ∪ P ∪ P ∪ P ) ÖX”fr‘>¡•aŽ^‘fž¡žjŽ0°NŽ0V˜(›‡∈”f—–T˜Xjrž'”>š P ∪ P Ö
”f‘>r•aŽ^‘fžjž)Ž0°NŽ0˜¼›‡”f—–˜XjžT”>š P ∪ P Ö#‘>˜³´R—–˜µ‘>•–•‡“^‘fžjŽLž—–'“L”f˜z)‘>—–˜³žT›‡”f—–˜XjžT”>š  ”f²R“L”f•a”fjžOŸÊ²œ—ažT“L” · ›œ•aŽ0jŽLž
²³Ž›œj”#”>š{”>š‹Ž ·· ‘(à!Ÿ ½¾N¿ÀÂÁ6ÃOÁ®ÄGÀ Å z̜›œ›‡”ÇžjŽ²‘> W = {V, W } —až(‘[›‘>—–(”>šxÉ'ŽL´!»NŽLž^Ÿ W —ažÈž)‘>—a´ j”  Ž ïfñIÿfóió(ªLòÍ«I¬º§­â¤Ç­³¢
§j¤f­ “^r‘>˜ ¬º§Lòͬ®ŽE­z¯ “L”f”f•a'”fjžŽL— · ´.›œÉT•–±¼—–²¶¸ jŽLßx´¶ÖN‘>—sš‹˜³´²³Ž0j•–Ì³ŽŽ—ažTžÌ³‘’“j“L²¶”f˜³²ž)‘>‘>˜z‘> ˜zk±ÉT)—–‘>²¼˜³ž•6²³‘>ŽjŽEšº”f”>•–š •a”^ÉT—–˜œ²»i‘>›œxj“L”f”f›G˜XŽ0)‘>\—–±N˜³Ÿž¸‘>˜X±i•aŽ^ö³‘f˜œž—–jŽžj›‡Ž0”fr—–”>˜Xš‹jžx›‡”>”fš—–˜Xjž Ö
S
“L”f˜X)‘>—–˜³ ž‘jŽL´È›G”f—–˜z^Ö#‘>˜³´È‘>˜z±)‘>˜³ž•6‘>jŽ”>š W ²‘>r“L”f˜z)‘>—–˜³ž'‘>•aŽ^‘fVžj k ›‡”f—–˜Xjžr”>š S Öz“L”f˜X)‘>—–˜³žk ‘  •–Ì³Ž›‡”f—–˜XS^Ÿ
ʲ³Ž˜³Ž~Ë!TjŽL“j²œ˜œ—a“^‘>•jŽLžÌœ•–‘>•–•a”Oɞ̳žTj”(žj— · ›œ•–—s𮱑>•–• šº”f•–•a”^ÉT—–˜œ»(›œj”#”>šÍžOŸ
¾#ø ø ê Å £Eïîœïf¬8«,¤u£ ª¦^¯Xª~ñE¬6ñ­â¤Nòr¤u£’òÍÿI Çj§0¬ÍïÇ¥ 'ïÇ­â¦ í ¡ò ª0­¶¬Íò¬8ñEïf¥ ñL¤ õ
Ó
W
3
0
0
V
2
W
4
W
4
W
3
W
1
W
2
W
4
W
1
0
W
4
0
W
2
V
7
W
3
W
4
W
1
W
2
W
3
W
4
!
#
W
4
W
4
0
W
3
W
3
!$#
V
14
W
1
W
2
é’æ>Ä Ä `Ä ¾#ø øÑê Šю“^‘>˜(‘fžjžÌ · ŽrÉT—–²³”f̜{•a”Çžjž{”>š»NŽ0˜³Ž0)‘>•–—–V±,²‘> V “L”f˜X)‘>—–˜³ž²³Žr›‡”Çž—–—–°NŽ x¹ ‘cË!—až^ÖǑ>˜³´
“L”f˜z)‘>—–˜³žŽ0—–²³Ž0²³Ž'›G”Çž—–—–°NŽT”f{²³Ž'˜³Ž0»X‘>—–°NŽ x¹ ‘cË!—až^Ÿ z̜›œ›G”Çž)Žr²‘> {V, W } —až¸ ßxÖ>•aŽ0 k > 0 ‘>  —–)‘>±NÖ
W
‘>˜³´•aŽ0 S  Ž’‘ižjŽ0”>š{›G”f—–˜zjž^ŸÕ—–jžTÉrŽE“L”f•a”f Bd (S) Ÿ‹Ž0 U  Ž’‘,ÉrŽL´!»NŽ’²‘>‘>•až)”(“L”f˜X)‘>—–˜³ž²³Ž›G”Çžj—–—–°NŽ
‘cË!—až^Ö Ìœ²‘fž‘E°NŽ0±Rž ‘>•–•‹‘>˜œ»Ç•aŽÇŸÊ²³Ž0˜)‘>˜³ž•6‘>jŽLžT”>š ‘>˜³´R)‘>˜³ž•6‘>jŽLžT”>š ”f²R—–˜zjŽ0jžjŽL“0
—–˜
x¹
—–jžr—–˜XjŽ0 °Ç‘>•až^Ÿ¡ßr•aŽ^‘>•–±NÖ#²³· Ž›‘>—– {U, W } —ažr”>š‹\±z›‡Žxß'”f˜z)‘>—–V˜‹Öz²³Ž0jŽ~šº”fjŽÇÖ  ±(‹Ž ·U· ‘i à!ÖzÉrŽ“^‘>˜R“L”f•a”fBd Bd(S)(S)
žÌ³“)²²‘>‘>˜X±)‘>˜³ž•6‘>jŽ'”>š W Ö W ∈ T r (Bd (S)) ‘>˜³´,‘>˜z±’)‘>˜³ž•6‘>jŽ'”>š U Ö U ∈ T r (Bd (S)) “L”f˜X)‘>—–˜³ž
“L”f˜z)‘>—–˜³ž  ”f².“L”f•a”fjž‘fžTÉ'Ž0•–•Ÿ
 ”f²µx”^“Lɔf•a”fÉ'jž^ŽŸ²Ù'‘L̜°NŽ’j²³”(Ž0“L˜¶”f•a‘>”f˜z ±¼)‘>˜³ž•6‘>jŽ”>šŸ VÑÖ ŽV´!—–°z∈—a´œTŽ’r—–T(Bd
(S))
—–˜Xj”(²œjŽLŽ’›‘>jž‘fžšº”f•–•a”^ɞ^Ÿ
S \ Bd (S)
—až  •–Ì³Ž },
S = {X ∈ S \ Bd (S) | ∀Y ∈ V (X) ∩ Bd (S), Y
—ažTjŽL´ },

S = {X ∈ S \ Bd (S) | ∀Y ∈ V (X) ∩ Bd (S), Y

S = S \ (Bd (S) ∪ S ∪ S ).
¸˜X±™)‘>˜³ž•6‘>jŽ
²‘>—–˜XjŽ0jžjŽL“0jž —–˜ ‘>•aŽ^‘fžE”f˜³Ž›‡”f—–˜X^Ö Ì³žj“L”f˜X)‘>—–˜ ‘>E•aŽ^‘fž”f˜³Ž •–̳Ž
›‡”f—–˜X^Öš͝j” · Bd (S)V Ö∈žj”.TÉrrŽR”f˜œ•–±[²‘L°N޼j” · 0‘ SN ŽRžÌœjŽÈ²‘>—–,“L”f˜X)‘>—–˜³ž(‘· jŽL´ ›‡”f—–˜X,j”z”œŸ z— · —–•6‘>•–±N֐ ‘>˜z±
²‘>x—–˜zjŽ0jžjŽL“0jž  —–˜Ñ‘>x•aŽ^‘fžjx”f˜³Ži›‡”f—–˜X^Ö Ì³ž“L”f˜X)‘>—–˜ ‘¼jŽL´Ü›G”f—–˜z^Ö
‘>˜³´[‘>˜X± V ∈ T r ²‘>
V ∈ Tr
—–˜XjŽ0jž)ŽL“0jž S · ̳ž“L”f˜X)‘>—–˜¶S›G”f—–˜zjž”>š  ”f².“L”f•a”fjžOŸ ·
ʲz̳ž^Ö{ÉrŽR“^‘>˜ žj— · ›œ•–± “L”f•a”f S žÌ³“j² ²‘>‘>˜X± W ∈ T r (S ) “L”f˜X)‘>—–˜³ž  ”f² “L”f•a”fjžOÖ{ÉT²œ—a“j² “^‘>˜  Ž
´œ”f˜³Ž  ±R‹Ž ·· ‘.Ðǟ
—–² S ֑>˜³´.ÉT—–² S ֜jŽLžj›GŽL“0—–°NŽ0•–±NÖÉrŽ’›œj”#“LŽLŽL´.Ž~˳‘f“0•–±²³Ž’ž)‘ · ŽEÉ'‘^±.‘fžÉrŽ´!—a´ÉT—–² S —–jžjŽ0•sšuÖ  ̜
˜³”^É É'Žx“)²‘>˜œ»NŽx²³Žj”f•aŽLž”>š V ‘>˜³´ W Ÿ ю»NŽ0'²³Žû͞—–•–•G̜˜³“L”f•a”fjŽL´ü'žÌ  žjŽ0jž S Ö  Ö S Ö  Ö S Ö S^Ö  Ö S^Ö  Ö
 ÉT—–²µ²³Žšº”f•–•a”^ÉT—–˜œ»È›œj”f›GŽ0—aŽLž^Ÿ
S
¸˜X±Ü)‘>˜³ž•6‘>jŽ V ∈ T r ”f W ∈ T r Ö ²‘>’—–˜zjŽ0jžjŽL“0jž S Ö  û®jŽLž›‹Ÿ S^Ö  ü—–˜ ‘>•aŽ^‘fžjE”f˜³Ži›G”f—–˜z^Ö
•
· ̳ž“L”f˜z)‘>—–˜Ü‘>•aŽ^‘fž”f˜³Ž  •–̳ŽÈû®jŽLž›‹Ÿ{jŽL´ü'›‡”f—–˜X^Ÿ
¸˜X±)‘>˜³ž•6‘>jŽ V ∈ T r ²‘>x—–˜XjŽ0jžjŽL“0jž S Ö  û®jŽLž›‹Ÿ S^Ö  üT“L”f˜z)‘>—–˜³ž‘  •–Ì³Ž¼û®jŽLžj›‹ŸTjŽL´ü›‡”f—–˜X^և‘>˜³´
•
‘>˜z±¨)‘>˜³ž•6‘>jŽ W ∈ T r ²‘>—–˜XjŽ0jžjŽL“0jž S Ö  û®jŽLž›‹Ÿ S^Ö  ü'“L”f˜z)‘>—–˜³žx‘,jŽL´ û®jŽLž›‹Ÿ  •–̳ŽOü'›‡”f—–˜X^Ÿ
¸˜X±µ)‘>˜³žj•6‘>jŽ V ∈ T r ²‘>—–˜zjŽ0jžjŽL“0jž S û®jŽLž›‹Ÿ S ü“L”f˜z)‘>—–˜³ž’‘  •–̳Žû®jŽLž›‹ŸxjŽL´ü›‡”f—–˜X^Ö ‘>˜³´
•
‘>˜z±¨)‘>˜³ž•6‘>jŽ W ∈ T r ²‘>—–˜XjŽ0jžjŽL“0jž S û®jŽLž›‹Ÿ S ü'“L”f˜z)‘>—–˜³ž›‡”f—–˜Xjž”>š  ”f²µ“L”f•a”fjž^Ÿ
ß'”f•a”f‘>•–•³›G”f—–˜zjž¡”>š S Ö  ‘>˜³´ S jŽL´ Öz“L”f•a”f'‘>•–•³›‡”f—–˜Xjžr”>š S^Ö  ‘>˜³´ S  •–̳ŽÇŸÕ—–˜‘>•–•–±NÖX“L”f•a”f S Ö  ̳ž—–˜œ»
²³ŽE¸ ß ›œj”f›‡Ž0V±.”>š²³Ž›‘>—– (V, W ) ÖG‘>˜³´Üž— · —–•6‘>•–±NÖG“L”f•a”f S^Ö  ‘>•až)”(̳ž—–˜œ»¼²³Ž¸ ß ›œj”f›GŽ0\±NÖ  ̜²³Ž
j”f•aŽLž”>šjŽL´Ü‘>˜³´  •–̳Ž,žÉT—–j“j²³ŽL´ Ÿ ”OÉ —–x—ažxŽ^‘fž±j”¼“)²³ŽLQ“ ¶²‘>—–˜¶²œ—ažx“L”f•a”f—–˜œ»µ‘>˜z±µ)‘>˜³ž•6‘>jŽ,”>š V ”f W
²‘>“L”f˜z)‘>—–˜³žžÌ ȓ0—aŽ0˜z•–± · ‘>˜X±¨›G”f—–˜zjž”>š S ֜“L”f˜X)‘>—–˜³ž‘,›G”f—–˜z”>š  ”f²µ“L”f•a”fjž^Ÿ R¾zøÑêœæ {ÅE \˜ ØÚÏrÔÇÓ>Þ—–x²‘fž  ŽLŽ0˜Ü›œj”^°NŽL´Ü²‘>—sš {V, W } —až‘ z›‡ŽL“0—6‘>•`›‘>—–^ÖG²³Ž0˜ {V, W } —až ­â¤Çò ¸ ßxŸ #”œÖ
²³Žšº”f•–•a”^ÉT—–˜œ»Èžj)‘>jŽ · Ž0˜X²³”f•a´œžx‘fžTÉrŽ0•–•Ÿ
¾#ø ø ê ÍÅ £Eïî!ïǬ8«¤£ ª¦^¯XªIñ¬6ñ í ¡ò ª~­Ü¬®ò¡¬6ñEïf¥yñ0¤ õ
Ò
&% &%
V
V
V
0
W
4
V
V
14
V
U
14
V
V
V
V
V
V
V
0
V
V
0
0
0
0
V
0
V
0
0
0
W
2
0
,0
,0
0
V
0
V
0
W
0
V
0
W
0
W
,0
,0
,0
,0
,0
,0
!$#
!
#
¾#ø ø ê‹Å ª0ò W = {V, W } ôjªiï(ñLª0ò¤u£’ò ¤ ª¦^¯XªIñ ¤u£’òÍÿI ¬s¯ Çõ ª~­ W ¬8ñ õ
é’æ>Ä Ä Ä ¡¾zø ø ê ‹ÅÙ'±.
Ž ‘ !Ö◖x—ažŽ0˜³”f̜»Ç²™j”¼ž²³”OÉ ²‘> W } —až¸ ßxŸG‹Ž0 W  Ž²³ŽÉrŽL´!»NŽ
ÉT²³”ÇžjŽR‘>˜œ»Ç•aŽ(—ažE‘>•aŽ^‘fž π Ÿ¨Ê²³·Ž0· ˜ W —až’²³ŽÈ̜˜œ—a”f˜ ”>šTVÉ'”µ²‘>•sšÍ›œ•6‘>{V,
˜³ŽLž^Ö{ž)‘L±NÖ H ‘>˜³´ H Ÿ¼Ê
)‘>˜³ž•6‘>jŽ  ”f²
²‘>•sšÍ›œ•6‘>˜³ŽLž,žj̳“j² ²‘>²³Ž0± “L”f˜X)‘>—–˜ Ž~˜‘f“0•–± ”f˜³ŽÈ›‡”f—–˜Xi”>š S Ö{´œŽ0˜³”fjŽR²³Ž ·  ± X ‘>˜³´ X Ö{jŽLž›‡ŽL“0—–°NŽ0•–±NŸ
x”fjŽE²‘> X · ‘L±¶“L”f—–˜³“0—a´œŽ,ÉT—–² X Ÿ’ß'”f•a”f X ‘>˜³´ X jŽL´ Ö ‘>˜³´[‘>•–•{²³Ž”f²³Ž0›‡”f—–˜Xjž  •–̳ŽÇŸÊ²³Ž0˜Ñ‘>˜X±
)‘>˜³ž•6‘>jŽ”>š W ²‘>(“L”f˜X)‘>—–˜³žR‘>•aŽ^‘fž”f˜³Ž¨›G”f—–˜z^Ör“L”f˜z)‘>—–˜³žÈ‘ܝjŽL´ ›‡”f—–˜X^Ör‘>˜³´ ‘>˜z± )‘>˜³žj•6‘>j޵”>š V ²‘>
“L”f˜X)‘>—–˜³ž‘>T•aŽ^‘fž²œjŽLŽ’›G”f—–˜zjž^֜“L”f˜X)‘>—–˜³ž‘  •–Ì³Ž›‡”f—–˜X^Ÿ ¾#ø ø ê ‡Å ª0ò W = {V, W } ôjªiï(ñLª0ò¤u£’ò ¤ ª¦^¯XªIñ ¤u£’òÍÿI Eïf¥ £Vî⥖ïf­Gª Nõ ª~­ W ¬6ñ õ
é’æ>Ä Ä ¼Ä ¡¾zø øÑê Å ¸»X‘>—–˜‹Ö—–—až(Ž0˜³”f̜»Ç² j”[ž²³”^ɏ²‘>(²³Ž0± ‘>jޏ ßxŸ z—–˜³“LŽ {V, W } —až”>šV±#›G޵à
û ‘>•sš®›œ•6‘>˜³ŽOü~Ö Bd (S) ‘>˜³´ Bd (S) ²‘^°N޼‘> · ”ÇžjE”f˜³Ž›‡”f—–˜XE—–˜ “L” ·· ”f˜‹ŸR Vš Bd (S) ‘>˜³´ Bd (S) ‘>jŽ
´!—ažÍהf—–˜X^ÖG²³Ž0˜™“L”f•a”f (S)  •–̳ŽÇÖ Bd (S) jŽL´ Ö ‘>˜³´.²³Ž,”f²³Ž0›‡”f—–˜Xjž‘>  —–)‘>—–•–±NŸÊ²³Ž0˜[‘>˜X±µ˜³”f˜³Ž · ›œ\±
)‘>˜³ž•6‘>jŽ’”>š V û®jŽLž›‹Bd
Ÿ W ü'“L”f˜X)‘>—–˜³ž‘  •–̳ŽÈû®jŽLž›‹Ÿ{jŽL´ü'›‡”f—–˜X^Ÿ
²³Ž0ÉT—ažjŽÇÖ•aŽ0
Ž(²³Ž0—–,“L” ·· ”f˜ ›‡”f—–˜X^Ÿ‹Ž0 P = Bd (S) ∪ Bd (S) \ X ֑>˜³´ “L”f˜³ž—a´œŽ0E—–jž V ¹
X 
ÖX‘>˜³´ ”f̜˜³´³‘>±NÖ (P ) Ÿßr•aŽ^‘>•–±NÖNŽ^‘f“)²›G”f—–˜z—–˜ Bd (S)\X  Ž0•a”f˜œ»Nž{j” Bd (P ) Ö
‘>˜³”f´µÌœ˜³Ž^´³‘f‘>“)²µ±N֛GBd
”f—–˜z(P—–˜ )Bd (S)W ¹\ X  Ž0•a”f˜œ»NBd
žj” Bd (P ) Ÿ
Vš (P ) ‘>˜³´ Bd (P ) ‘>jŽ,´!—ažÍהf—–˜X^Ö⏲³Ž0˜Ü“L”f•a”f Bd (S)  •–̳ŽÇÖ Bd (P ) ‘>˜³´.²³Ž,”f²³Ž0x›G”f—–˜zjžjŽL´ Ÿ
ʲ³Ž0˜µBd
‘>˜X±(˜³”f˜³Ž ›œ\±¼)‘>˜³žj•6‘>jŽ”>š “L”f˜z)‘>—–˜³ž‘ •–̳Žx›‡”f—–˜X^Ÿ z̜›œ›‡”ÇžjŽ²‘>'É'޲‘^°NŽ’‘’)‘>˜³žj•6‘>jŽ”>š ÉT—–²
VÉ'”µ›‡”f—–˜Xjž^Ö  ”f· ²  •–̳ŽÇŸÈʲ³Ž0˜Ñ—–EžV²³”f̜•a´Ñ“L”f˜z)‘>—–˜  X Ö‘>˜³´Ñ‘¨›‡”f—–˜XE”>š Bd (P ) Ÿ(Ù'̜’²œ—až’“L”f˜z)‘f´!—aW“0jž”f̜
‘fžjžÌ ›œ—a”f˜ ²‘> (P ) ‘>˜³´ Bd (P ) ‘>jŽ(´!—ažÍהf—–˜X^Ÿ #”œÖ{‘>˜X±™)‘>˜³ž•6‘>jŽÈ”>š W ÉT²œ—a“)² “L”f˜X)‘>—–˜³ži‘>E•aŽ^‘fžj
VÉ'”(·›‡”f—–˜Xjž”>š S Ö³“LBd
”f˜z)‘>—–˜³ž‘ijŽL´¨›‡”f—–˜X^Ÿ
Vš
‘>˜³´
‘>jލ˜³”f¼´!—ažÍ×u”f—–˜z^Ö¡²³Ž0˜ ²³Ž0± ²‘^°NŽ.”f˜³Ž¨›‡”f—–˜X(—–˜ “L” ”f˜‹Ö'•aŽ0 ލ²³Ž0—–
“L” ·· ”fBd˜.›‡”f(P—–˜X)^Ÿ Vš Y Bd Ž0•a”f(P˜œ»Nž) j” Bd (S) Ö³²³Ž0˜¶“L”f•a”f Bd (S)  •–̳ŽÇÖ Bd (P ) ‘>˜³´·· ²³Ž”f²³Ž0Y›‡”f—– ˜XjžjŽL´ Ÿ
ʲ³Ž0˜‹Ö  ±¶²³Žiž)‘ · Ž‘>»ÇÌ · Ž0˜X‘fž  Ž~šÍ”fjŽÇ֋‘>˜z±.˜³”f˜³Ž · ›œV±Ü)‘>˜³ž•6‘>jŽ”>š V “L”f˜X)‘>—–˜³žE‘  •–̳Ž,›G”f—–˜z^֋‘>˜³´[‘>˜X±
)‘>˜³ž•6‘>jŽ,”>š W ÉT²œ—a“j²™“L”f˜X)‘>—–˜³ž’‘>•aŽ^‘fž\Ér”R›G”f—–˜zjž”>š S ֓L”f˜z)‘>—–˜³ž’‘jŽL´¶›G”f—–˜z^ŸÕ—–˜‘>•–•–±N֗sš Y  Ž0•a”f˜œ»Nžxj”
֜²³Ž0˜.ÉrŽ›œj”#“LŽLŽL´.‘>˜‘>•a”f»N”f̳ž•–±NÖ  ̜²³Žj”f•aŽLž”>š V ‘>˜³´ W Ö³‘>˜³´²³Ž’“L”f•a”fjžOÖ³‘>jŽ’žÉT—–j“j²³ŽL´ Ÿ Bd (P )
¾#ø ø ê Å ª0ò W = {V, W } ôjªiï(ñLª0ò¤u£’ò ¤ ª¦^¯XªIñ ¤u£’òÍÿIîœ
ª Eïf«u¦ Çõ ª~­ W ¬8ñ õ
é’æ>Ä Ä Ä ¾zø øÑê Å ¸xžE̳žÌ‘>•ÖÉrލ”f˜œ•–±Ñ›œj”^°NŽR²‘> {V, W } —až,¸ ßxŸ`¸xžjžÌ · ޼²‘> W “L”f˜X)‘>—–˜³ži²³Ž
›‡”Çž—–—–°NŽ x¹ ‘cË!—až^ŸµÎN̳ž•–— NŽÈ—–˜ ²³ŽR´œŽ~ö³˜œ—–—a”f˜ ”>šV±#›GŽ û ‘>j´ü~֎~Ë#jŽ0˜³´ ²³Ž  ”f̜˜³´³‘>±[²‘>• ³—–˜³ŽLži”>š W j”
•–—–˜³ŽLž^֜²³Ž0±´!—–°#—a´œŽ²³Ž›œ•6‘>˜³Ž—–˜zj”šº”f̜›‘>jž^֛œ›‡Ž0^Ö³‹”^É'Ž0^Öâ
Ž~š®^֑>˜³´ —–»Ç²X^ÖÉT²œ—a“j²¶—až W —–jž)Ž0•sšuŸ юE“^‘>˜
‘fžjžÌ · ŽEÉT—–²³”f̜•a”Çž)ž”>š»NŽ0˜³Ž0)‘>•–—–V±µ²‘> V “L”f˜X)‘>—–˜³žx²³Ž’˜³Ž0»X‘>—–°NŽ x¹ ‘cË!—až^֔f˜³Žž—a´œŽE”>š V —až—–˜ ›œ›‡Ž0^Öâ‘>˜³´
”f˜³Ž’ž—a´œŽ—až—–˜µ‹Ž~š®^Ÿ
žjŽ0°NŽ²‘>{—sšG‘)‘>˜³ž•6‘>jŽ”>š
‘>˜³´i‘)‘>˜³ž•6‘>jŽT”>š W —–˜XjŽ0jž)ŽL“0Ž^‘f“j²(”f²³Ž0^ÖǏ²³Ž0˜”f˜³Ž”>š²³Ž · “L”f˜X)‘>—–˜³ž
V
²³Ž’”f ²³Ž0”f˜³Ž ڞ‘>›GŽ~Ë Ÿ
á ®ê³ÁÍø żճ”fE‘>˜X±Ü›G”f—–˜z’žjŽ0 P ‘>˜³´ X ∈ P Ö Ž0—–²³Ž0 Bd (P \ X) \ Bd (P ) = ∅ ”f Bd (P \ X) \
Ÿ
Bd (P ) = ∅
é’æ>Ä Ä Ä á ®ê³ÁÍ
ø âÅ z̜›œ›G”Çž)Ž’”f˜µ²³ŽE“L”f˜z)‘>±²‘> Y ∈ Bd (P \ X) \ Bd (P ) ‘>˜³´ Z ∈ Bd (P \
Ÿxʲ³Ž0˜ X ∈ V (Y ) ‘>˜³´ X ∈ W (Z) Öâžj” V (Y ) ‘>˜³´ W (Z) —–˜XjŽ0jžjŽL“0’Ž^‘f“)²[”f²³Ž0^և²³Ž0jŽ~šÍ”fjŽÇÖ
X) \ Bd (P )
&% &%
!$#
1
2
1
1
2
1
&%
&%
V
!
W
#
V
V
W
W
V
V
W
W
V
W
V
2
2
V
W
W
V
W
V
V
V
W
W
W
V
W
V
&%
%
!$#
V
V
W
W
&% &%
V
W
Ð^Ô
V
W
”f˜³Ž”>š²³Ž · “L”f˜X)‘>—–˜³žE²³Ž”f²³Ž0E”f˜³ŽÚžE‘>›‡Ž~ˇ֋žI‘L±NÖ Z ∈ V (Y ) ŸiÙ'̜’²œ—až—až’‘¨“L”f˜z)‘f´!—a“0—a”f˜‹Öžj—–˜³“LŽ Y —ažE‘
 ”f̜˜³´³Ž0‘>̜±R˜Ü›‡j”f”R—–˜X²³”>Ž,š ›œPj”#\”>Xšr”>Ÿ šr‹Ž ‘ý!ŸEß'”f•a”f
jŽL´ ֋‘>˜³´ Bd (S) \ Bd (S)  •–̳ŽÇև²³Ž
Bd (S) \ Bd (S)
·
·
—–˜XjŽ0—a”fi›‡”f—–˜Xjž‘>  —–)‘>—–•–±NŸ ”OɓL”f˜³ž—a´œŽ0i²³Ž¼›G”f—–˜zjž”>š Bd (S) ∩ Bd (S) Ÿ¶Õ³”f‘>˜X± X ∈ Bd (S) ∩
Öâ—sš Bd (S \ X) \ Bd (S) 6= ∅ Ö³²³Ž0˜™“L”f•a”f—–jŽL´ Öâ—sš Bd (S \ X) \ Bd (S) 6= ∅ ֏²³Ž0˜Ü“L”f•a”f—–
Bd (S)
Ÿ
ß'•–”f̳•aŽÇ”fŸµÕ³Ž^”f‘f“j² Ž^‘f“j”>²š”>²³š‹ŽLžj²³ŽÜŽ›‡jŽ ”f· —–˜X‘>j—–žµ˜œ—–ž˜œÌ³»“)²›G”f—–²˜z‘>jžY²³É'Ž0± ޲²‘^‘^°N°NŽ Ž[Bd²³Ž (S \ Y ) \ Bd
(S) = Bd (S \ Y ) \ Bd (S) = ∅
L
“
f
”
a
•
f
”


²
>
‘
˜

³
²
™
Ž

³
²
Ü
Ž
œ
›
j

0
Ž
z
°
a
—
f
”
³
Ì
ž
‡
›
f
”
–
—
X
˜
µ

>
”
š
¤IîNî!¤fñI¬ÍòVª
֜—–˜²³Ž y¹ ”fj´œŽ0—–˜œ»³Ÿ
Bd (S) ∩ Bd (S)
Ê`”›œj”^°NŽr²‘>
²œ—až`“L”f•a”f—–˜œ»—až
»N”z”#´ Öc•aŽ0 T r Ö V ∩S = {X, Y } ŸÂ Vš#—–`—–˜XjŽ0jžjŽL“0jž Bd (S)\Bd (S) Ö
Ÿ‹Ž0 X ∈ V ∩ Bd (S) Ÿ Vš X —ažjŽL´ Ö
ÉrŽ,‘>jŽE´œ”f˜³ŽÇŸ #”¼‘fž)žÌ · Ž²‘> V ∩ Bd (S)V ⊂∈Bd
∩ Bd (S)
²³Ž0˜  ± ²³ŽR“L”f•a”f—–˜œ»™Ìœ•aŽÇÖ Bd (S \ X) \ Bd (S) (S)
[
Ÿ
'
Ù
œ
Ì
i


³
²
0
Ž
˜ Y —až‘>•ažj”[‘ V ¹ ”f̜˜³´³‘>± ›G”f—–˜z^֞j”
= ∅
ÉrŽ,²‘^°NŽ Y ∈ Bd (S) ∩ Bd (S) Ÿx¸»X‘>—–˜¶É'Ž,“^‘>˜[‘fžjžÌ · Ž²‘> Y —ažjŽL´ ևžj” Bd (S \ Y ) \ Bd (S) = ∅ Ÿ
z̜›œ›‡”ÇžjŽx²‘> X < Y Ÿ z—–˜³“LŽ V ∩ S = {X, Y } Ö X ‘>˜³´ Y ‘>jŽx“L”f˜³ž)ŽL“0̜—–°NŽ›G”f—–˜zjžT”>š Bd (S) ∩ Bd (S) Ÿ
x”^É —–—až˜³”fx²‘>j´µj”¼žjŽLŽ’²‘> (S \ Y ) \ Bd (S) = ∅ ŸÊ²³Ž0jŽ~šº”fjŽÇÖ  ±µ²³ŽE“L”f•a”f—–˜œ»RÌœ•aŽÇÖ X ‘>˜³´ Y
²‘L°NŽE´!— ‡Ž0jŽ0˜z“L”f•a”fjž^ŸÕ”f)‘>˜³ž•6Bd
‘>jŽLž”>š W ²³Ž‘>»ÇÌ · Ž0˜X—až‘>˜‘>•a”f»N”f̳ž^Ö³ÉT—–²µ²³Ž“L”f•a”fjžžÉT—–j“j²³ŽL´ Ÿ x”^É É'ŽÌœ˜µj”i²³Ž’“^‘fž)ŽÉT²³Ž0˜µÉrŽ’²‘L°NŽ · ”fjŽ²‘>˜.VÉr”ÈÉrŽL´!»NŽLž^Ÿ
¾#ø ø ê âÅ ¤f«EïÇ­³ÿ s, t > 0 ¬®­òVªº¯zª~«)ñ rò ª~«jª¬8ñ’ïÈ-­ #óÈôjª~« f (s, t) ¬Íò ò ª¡£0¤f¥®¥–¤ ¬8­#¯xî⫤I~«0òÍÿXõ
ª0ò W = {W, W , . . . , W } ôjªÈï¼ñLª0òT¤u£ t ª¦O¯XªIñ ñ!§ ò ïÇòTïf­³ÿEî!ïf¬®« {W , W } ¬6ñ 'ïf­â¦R¥aª0ò S ôjª
ï(ñLª0ò¤u£'î!¤f¬®­ò®ñ^õ ª~­ S §jïf­ ôjªi¦zª§j¤fóî!¤>ñ^ª¦¨¬®­ò¤ t î!ïf«0ò®ñ S , S , . . . , S ñœ§ ò ïÇòâ£L¤f« i = 1, 2, . . . , t³£L¤f«
ïf­³ÿ¼òÍ«uïf­³ñI¥–ïÇòVª W ¤£ W ¬ £ |W ∩ S| ≥ f (s, t) ò ª~­ |W ∩ S | ≥ s õ
é’æ>Ä Ä Ä ¡¾zø ø ê âÅʲ³ŽŽ~Ë!—ažjŽ0˜³“LŽ”>š f (1, 2) —až¡Ž z̜—–°Ç‘>•aŽ0˜X'j”²³Ž›œj”f›‡Ž0V±²‘>r²³Ž“L”fjŽLž›‡”f˜³´!—–˜œ»E\Ér”
ÉrŽL´!»NŽLžx‘>jޏ ßxŸ ”OÉÉrŽž²³”Oɏ²‘> f (s, 2) Ž~Ë#—ažjjž'šÍ”fŽ0°NŽ0± s Ÿ‹Ž0 V ‘>˜³´ W  ŽxVÉr”(ÉrŽL´!»NŽLž²‘>šº”f ·
‘ ß ›‘>—–^Ÿ‹Ž0
 Ž²³Ž›‘>²™´œŽL“L” · ›‡”Çž—–—a”f˜[”>š S ”>šr”fj´œŽ0 s f (1, 2) ևÉT—–²™jŽLž›GŽL“0
j” V ŸÕ³”f i = 1,P2, . ,. P. , s,Ö³.•aŽ0. . , P
V
W
W
V
W
V
W
V
V
W
V
V
V
W
V
W
W
W
0
0
V
2
V
V
V
V
0
V
W
0
V
W
V
V
0
y
W
V
V
W
W
W
1
2
t
i
1
&%
0
i
%
0
i
2
t
i
V
1
Õ³”fŽ^‘f“)²
0
i
i
V
2
2
V
s2 f (1,2)
isf (1,2)
Hi = ∪j=(i−1)sf (1,2)+1 PjV .
Hi
(!$#
j
Ö!)‘0NŽ²³Ž W ¹ ›‘>²µ´œŽL“L” · ›G”Çžj—–—a”f˜‹Ö P
W
W
1 (Hi ), . . . , Psf (1,2) (Hi )
Ö³‘>˜³´¨šº”f
j = 1, 2, . . . , s
Ö!•aŽ0
jf (1,2)
Hij = ∪k=(j−1)f (1,2)+1 PkW (Hi ).
Õ³”fEŽ0°NŽ0±
֋“L”f•a”f ֋žÌ³“j²[²‘>,‘>˜X±Ü)‘>˜³ž•6‘>jŽ(”>š û®jŽLž›‹Ÿ ü²‘>’—–˜XjŽ0jžjŽL“0jžE—–—–˜ ‘>
•aŽ^‘fž f (1, 2)i, j›‡=”f—–˜X1,jžO2,Ö “L.”f.˜X.),‘>s—–˜³ž’‘>•aŽ^H‘fž”f˜³Ž,jŽL´ û®jŽLžj›‹Ÿ  •–̳ŽOü›G”f—–˜z”>Vš'—–^ŸÊ²œ—ažW—ažx›G”Çžjžj—  •aŽÇÖGžj—–˜³“LŽ²³Ž,›‘>—–
—až¸ ßxŸ
{V, W }
ß'”f˜³ž—a´œŽ0x‘,)‘>˜³ž•6‘>jŽ V ”>š V ²‘>“L”f˜X)‘>—–˜³žx‘>T•aŽ^‘fž s f (1, 2) ›‡”f—–˜Xjž”>š S ŸÕ³”fŽ0°NŽ0± i Ö V —–˜XjŽ0jž)ŽL“0jž
—–˜ sf (1, 2) ›‡”f—–˜XjžO֜žj”,²³Ž0jŽ—až‘ j žÌ³“j²²‘>T—–—–˜zjŽ0jžjŽL“0jž H —–˜.‘>'•aŽ^‘fžj f (1, 2) ›‡”f—–˜Xjž^ŸÊ²³Ž0jŽ~šº”fjŽÇÖ V
H
“L”f˜X)‘>—–˜³ž‘>T•aŽ^‘fž”f˜³ŽjŽL´µ›G”f—–˜z”>š H Ö³ž)”(‘>T•aŽ^‘fž s jŽL´›‡”f—–˜Xjž”>š S Ÿ
ß'”f˜³ž—a´œŽ0`˜³”^É ‘)‘>˜³ž•6‘>jŽ W ”>š W ²‘>`“L”f˜z)‘>—–˜³žÂ‘>`•aŽ^‘fž s f (1, 2) ›G”f—–˜zjž`”>š S Ÿ
ʲ³Ž0jŽ¡—až`‘>˜ i žÌ³“)²E²‘>
—–˜XjŽ0jžjŽL“0jž H —–˜™‘>•aŽ^‘fžj sf (1, 2) ›‡”f—–˜XjžOŸÊ²³Ž0jŽ~šº”fjŽÇÖ◖—–˜XjŽ0jžjŽL“0jžŽ^‘f“)²Ü”>š P (H ), . . . , P (H ) Ö
W
ÐÇÐ
j
i
2
0
0
j
i
i
0
0
i
j
i
0
2
W
1
i
W
sf (1,2)
i
–— ˜[‘>•aŽ^‘fžx”f˜³ŽE›‡”f—–˜X^Ö ž)”šº”f j = 1, 2, . . . , s Ö W —–˜XjŽ0jž)ŽL“0jž H —–˜[‘>•aŽ^‘fž f (1, 2) ›‡”f—–˜Xjž^Ÿß'”f˜³žjŽ z̳Ž0˜X•–±NÖ
—–“L”f˜X)‘>—–˜³ž‘>T•aŽ^‘fž”f˜³Ž  •–Ì³Ž›G”f—–˜z”>šŽ^‘f“j² H Ö³žj”(‘>T•aŽ^‘fžj s  •–Ì³Ž›‡”f—–˜Xjž”>š S Ÿ
x”^É •aŽ0
öœËœŽL´ ‘>˜³´ÑžÌœ›œ›‡”ÇžjŽi²‘>
Ž~Ë#—ažjjžšÍ”f’Ž0°NŽ0± Ÿ(‹Ž0 , W , . . . , W }  Ž
”f̜žjŽ0”>šrÉrŽLs,´!»Nt ŽL>žOÖ 2žÌ³“j²™²‘>E‘>˜X±¶›‘>—–”>š¡²³Ž · f (s—až ,t ßx−Ÿ‡1)‹Ž0 s = f (s, 2) Ÿ’Ï{s‘>—–—a”f˜ {W
”f̜›‡”f—–˜XžjŽ0 S —–˜Xj”
j
ž
³
Ì
j
“
i
²

²
>
‘
{

Í
š
f
”

>
Ö
º
š
f
”

>
‘
X
˜
,
±

)

>
‘
³
˜
ž
6
•
>
‘
j

Ž
>
”
š
f
Ö
s
—
š
²³Ž0˜
S ,S ,...,S
i = 1, 2, . . . , t−1
W
W
|W ∩S| ≥ f (s , t−1)
ŸrÕ³”fŽ^‘f“j²
Ö³›‘>—–—a”f˜ —–˜zj”(VÉ'”(›‘>jž^Ö ‘>˜³´ Ö³žÌ³“)²µ²‘>
|W ∩ S | ≥ s = f (s, 2)
šº”f'‘>˜X±È)‘>˜³ž•6‘>jŽ W ”>š W ÖX—sš |W ∩i S= |1,≥2,f.(s,. . ,2)t −²³1Ž0˜ |W ∩ S |S≥ s Ö#‘>˜³´(šº”f'‘>˜z±S)‘>˜³ž•6‘>jŽ SW ”>š W ÖN—sš
²³Ž0˜
Ÿ{Õ`—–˜‘>•–•–±N֜šÍ”f
Ö!•aŽ0 S ‘>˜³´¨•aŽ0 S = ∪ S Ÿ
|W ∩ S | ≥ f (s, 2)
Õ³”f i = 1, 2, . . . , t − 1 ֑>˜X|W±¨)∩‘>˜³Sž•6‘>| j≥Ž sW ”>š W ֜—sš |Wi =∩1,S|2,≥. . f. ,(st −, t1− 1) S²³Ž0=˜ |W
Ö
∩ S | ≥ s = f (s, 2)
žj” |W ∩ S | ≥ s Ö
¸˜³´ šÍ”fµ‘>˜X± )‘>˜³ž•6‘>jŽ W ”>š W ֗sš |W ∩ S| ≥ f (s , t − 1) ֏²³Ž0˜ šÍ”fžj” · Ž i = 1, 2, . . . t − 1 Ö
Ö⏲³Ž0jŽ~šÍ”fjŽÇÖ |W ∩ S | ≥ s ֞j” |W ∩ S | ≥ s ŸÊ²œ—až“L”f˜³“0•–̳´œŽLžx²³ŽE›œj”z”>š”>š
≥ f (s, 2)
|W ∩ S | ≥
‹Ž ·· ‘ #Ÿ R¾zøÑêœæ {Åʲ³Ž›œj”#”>šÍž'”>š`‹Ž ·· ‘fžà!Ö ³Ö!Ý!֜‘>˜³´RýE— · ›œ•–±(²‘> f (1, 2) ≤ 8 Ÿ¡ß'” · —–˜œ—–˜œ»(—–'ÉT—–²¼²³Ž›œj”z”>šÂ”>š
‹Ž ·· ‘ ÉrŽ»NŽ0²³Ž  ”f̜˜³´ f (s, t) ≤ (8s) Ÿ
¾#ø ø ê Åí ñLª0ò¤u£ ª¦^¯zªIñ W = {W , W , . . . , W } ¬6ñ ¬ £ïǭ⦨¤f­³¥nÿ(¬ £’ïf­ÿî!ïf¬®« {W , W } ¬8ñ õ
é’æ>Ä Ä Ä ¾#ø ø ê Åßr•aŽ^‘>•–±NÖz—sš‹žj” · ŽT›‘>—– {W , W } —až ­â¤Nò ßxÖǏ²³Ž0˜¼²³ŽTÉT²³”f•aŽž)Ž0 W —až¡˜³”f ß Ž0—–²³Ž0^Ÿ
—až ßxŸ ŽL“L” ›‡”ÇžjŽ —–˜zj” ›‘>jž
ÉT—–²Ñ²³Ž›œj”f›‡Ž0V±
z̜›œ›‡”ÇžjŽ²‘>EŽ0°NŽ0±Ü›‘>—–
..,S
²‘>šº”f i = 1, 2, . . . , t ֜šº”fx{W‘>˜X±, W)‘>}˜³žj•6‘>jŽ W ”>š W· Ö³—sš |WS ∩ S|t≥ f (3, t)S ,S²³Ž0˜, . |W
ŸTʲ³Ž0˜‹Ö  ±
∩S| ≥3
‹Ž ·· ‘¨ÐÇÖ#Ž^‘f“j² S “^‘>˜  Ž“L”f•a”fjŽL´¨ÉT—–²¨jŽL´¨‘>˜³´  •–̳Žxžj̳“j²¨²‘>—sš |W ∩ S | ≥ 3 ²³Ž0˜ W “L”f˜z)‘>—–˜³ž›G”f—–˜zjž
”>š  ”f²Ü“L”f•a”fjžOŸ #”ȏ²œ—až“L”f•a”f—–˜œ»”>š S ²‘fžx²³Ž›œj”f›‡Ž0V±¶²‘>xšÍ”f i = 1, 2, . . . , t Ö⚺”f‘>˜z±.)‘>˜³ž•6‘>jŽ W ”>š
Ö!—sš |W ∩ S| ≥ f (3, t) ²³Ž0˜—–“L”f˜X)‘>—–˜³ž›‡”f—–˜Xjž”>š  ”f².“L”f•a”fjž^Ÿ W
…‡†† ܆ †T…  z̜›œ›‡”ÇžjŽR²‘> P —až(‘>˜ ”f›‡Ž0˜ “L”f˜z°NŽ~Ë ›‡”f•–±#»N”f˜ ”>š n °NŽ0—a“LŽLž(‘>˜³´ P = { P | i ∈ I } —až(‘Ü“L”f•–•aŽL“0—a”f˜ ”>š
)‘>˜³ž•6‘>jŽLž”>š P ÉT²œ—a“j²ÑšÍ”f · žE‘>˜ M ¹ šº”f•a´Ñ“L”^°NŽ0—–˜œ»Ü”>šr²³Ž›œ•6‘>˜³ŽÇŸ юiÉT—–•–•¡ž)Ž0²³Ži°f‘>•–Ì³Ž”>š M •6‘>jŽ0^Ÿ‹Ž0
Ž²³Ž —–˜œ— Ì ´!—až)‘>˜³“LŽ Ž0\ÉrŽLŽ0˜[‘>˜X±°NŽ0jŽ~˶‘>˜³´.˜³”f˜ ‘f´>בf“LŽ0˜zxž—a´œŽ”>š P ŸrÊÂ0‘ NŽ,‘(ž X̑>jŽE»Ç—a´ G ”>š
m
‘ fž—a“´!—ažj)‘>· ˜³“LŽ ·m/2· Ÿ  °z—a”f̳žj•–±NÖ#‘>˜z±,)‘>˜³žj•6‘>jŽ”>š P —–˜XjŽ0jž)ŽL“0¹ jž'‘> · ”Çžj K = 4π(diam(P
‘fž—a“
) + m) /m 
ž z̑>jŽLž^Ÿ
Քf`Ž^‘f“j²û͓0•a”ÇžjŽL´ü ‘fž—a“ž z̑>jŽ ÖO̳ž—–˜œ»—–jž
“L” ›‘f“0˜³ŽLžjžOÖcÉ'Ž¡“^‘>˜Eö³˜³´‘ žÌ “L”f•–•aŽL“0—a”f˜,”>šœ²³Ž
)‘>˜³ž•6‘>jŽLž'žÌ³“j²È²‘>¡²³Ž0±ž —–•–•œšº”f · ‘>˜ MB ¹ šÍ”f•a´¼“L”^°NŽ0—–˜œ· »”>š B ŸÂÊ`0‘ NŽ²³ŽT̜˜œ—a”f˜¼”>š‹‘>­•–¬®•òVª ²³ŽLžj ŽžÌ  “L”f•–•aŽL“0—a”f˜³žOŸ
юR²‘L°NŽ‘
šº”f•a´ “L”^°NŽ0—–˜œ»Ñ”>š²³ŽR›œ•6‘>˜³ŽÇŸÑʲ‘>i—až^֎0°NŽ0± “L” ›‘f“0(ž)Ž0—až,—–˜XjŽ0jž)ŽL“0jŽL´ ±
ö³˜œ—–jŽ0•–± · ‘>˜z±¼¥s¤O§j”>ïfš‹¥®¥yÿ ²³Žx­³)¬Í‘>òVª˜³žM•6‘>j¹ ŽLž^Ÿ \¡—ažžÌ ȓ0—aŽ0˜z'j”,´œŽL“L” · ›‡”ÇžjŽ²œ—až“L”O°NŽ0—–˜œ»³Ÿ·Õ³”fTž— · ›œ•–—a“0—–V±NÖ!̳ž)Ž²³Žž)‘ ·  Ž
˜³”f)‘>—a”f˜ P = { P | i ∈ I } šÍ”f²œ—ažžÌ  “L”f•–•aŽL“0—a”f˜‹Ÿ
юšÍ”f ̜•6‘>jŽ‘>˜³´Èž)”f•–°NŽT²³Ž›œj” •aŽ —–˜È—–jž¡´!̑>•³šÍ”f Ÿ‹Ž0 ŽT²³Ž“LŽ0˜zjŽ0'”>š »Ç)‘^°z—–\±(”>š Ÿ z—–˜³“LŽ
—až¡‘>˜ M ¹ šÍ”f· •a´È“L”^°NŽ0—–˜œ»”>š ²³ŽT›œ•6‘>˜³ŽÇÖzŽ0°NŽ0· ±i)‘>˜³ž•6‘>jŽ”>š P̄· ÖǏ²³ŽTjŽ OŽL“0 —a”f˜R”>š P ²œj”f̜»Ç²È²³Ž”f—–»ÇP—–˜‹Ö#“L”f˜X)‘>—–˜³Pž
‘>T•aŽ^‘fž M ›G”f—–˜zjž”>š{²³Ž•a”#“^‘>•–•–±Èö³˜œ—–jŽ’ž)Ž0 O = { O | i ∈ I } Ÿ
j
i
0
0
0
1
0
i
j
i
0
0
2
0
t
0
t−1
0
0
i
0
i
0
i
0
i
0
t
0
i
t
i
i
0
t
1
0
i
0
i
0
i
0
i
00
i
0
i
i
00
i
i
0
i
2
t
0
0
i
i
0
0
00
i
t
i
0
t
t
0
i
0
i
0
t
t−1 t
j=1 j
i
0
t
f (s0 ,t−1)
t−1
0
i
0
i
0
t
t
0
t
i
0
t
t
2t−1
&%
1
%
2
t
i
i
i
j
!$#
j
1
j
0
i
!$#
0
i
i
0
i
i
2
t
0
i
i
i
0
i
0
i
0
i
i
i
2
2
i
i
i
Ð^à
i
Ê ²³Ž“L”f•–•aŽL“0—a”f˜ P = { P | i ∈ I } “^‘>˜  ŽR´œŽL“L” · ›G”Çž)ŽL´ —–˜Xj”™VÉ'”[“L”^°NŽ0—–˜œ»Nž—sš‘>˜³´ ”f˜œ•–± —sš²³Ž¨žjŽ0
“^‘>˜  Ž“L”f•a”fjŽL´.ÉT—–²ÜVÉ'”R“L”f•a”fjžOÖâžÌ³“)²¶²‘>Ž0°NŽ0±)‘>˜³ž•6‘>jŽ,”>š P̄ “L”f˜X)‘>—–˜³ž’‘(›G”f—–˜zx”>š
O={O |i∈I }
 ”f²µ‹Ž0“L ”f•a”fjž^Ÿ
޼²³ŽRž)Ž0i”>šÉrŽL´!»NŽLži²‘>(“L”fjŽLž›‡”f˜³´ j”¶²³ŽR°NŽ0—a“LŽLž”>š P̄ Ÿ[Ùr± ²³Ž
..,W } 
“L”f˜X°NŽ~Ë!—–VW±¶”>=š P̄{W
֘³”, ›W‘>—– , .{W
a
—
ž”>šV±#›‡ŽEÝ.û z›‡ŽL“0—6‘>•8ü~Ö³²³Ž0jŽ~šº”fjŽÇÖ  ±²³Ž›œjŽ0°z—a”f̳žx‹Ž ·· ‘fž^ÖâŽ^‘f“j²¶›‘>—–
,W }
—až ßxŸß'”f˜³žjŽ z̳Ž0˜X•–±NÖ  ±‹Ž ·· ‘(Ó!Ö W —až ß ‘fžTÉ'Ž0•–•Ÿ #”i²³Ž0jŽ—až‘ k ÉT—–²µ²³Žšº”f•–•a”^ÉT—–˜œ»(›œj”f›‡Ž0V±NŸ
¸˜z±µžjŽ0”>𛇔f—–˜Xjž “^‘>˜ Ž“L”f•a”fjŽL´.ÉT—–²ÜVÉ'”¨“L”f•a”fjžžÌ³“)²¶²‘>‘>˜X±µ)‘>˜³žj•6‘>jŽ”>š W , W , . . . , W ²‘>
“L”f˜X)‘>—–˜³ž‘>T•aŽ^‘fž k ›‡S”f—–˜Xjž”> š S ֜“L”f˜z)‘>—–˜³ž›‡”f—–˜Xjž”>š  ”f²µ“L”f•a”fjž^Ÿ
ßr²³”#”ÇžjŽ M žÌ³“j² ²‘> M ≥ kK ֑>˜³´ “L”f•a”f²³ŽR›‡”f—–˜Xjži”>š O —–˜ Ž^‘f“)²  ‘fž—a“¼ž X̑>jŽRžjŽ0›‘>)‘>jŽ0•–±NÖÉT—–²
›œj”f›‡Ž0V± Ÿ
z—–˜³“LŽi‘>˜X±µ)‘>˜³žj•6‘>jŽ P ”>š P̄ —–˜zjŽ0jžjŽL“0jž’‘> · ”Çžj K  ‘fž—a“,ž z̑>jŽLž”>šr²³ŽE»Ç—a´ G Ö P “L”f˜X)‘>—–˜³ž’‘>x•aŽ^‘fž
›G”f—–˜zjž”>š —–˜.²³Ž,ž)‘ Ž ‘fžj—a“ž z̑>jŽ ŸÙ'±²³Ž,“j²³”f—a“LŽ,”>š²³Ž’»Ç—a´ Ö “L”f˜X)‘>—–˜³ž‘> ”Çž
M/K ≥ k
”f˜³Ž’°NŽ0jŽ~Ëܔ>š P Ö²³Ž0˜³O“LŽ B ∩ P · = B ∩ W ÖÉT²³Ž0jŽBW —až‘()‘>˜³ž•6‘>jŽ,”>š¡žj” · Ž W G∈ BW Ÿ #”œÖ  ±›œj”f›‡· Ž0V±
Ö P “L”f˜z)‘>—–˜³ž›‡”f—–˜Xjž”>š O ∩ B ”>š  ”f²µ“L”f•a”fjž^ŸÊ²œ—až“L”f˜³“0•–̳´œŽLž²³Ž›œj”#”>š{”>šÊ²³ŽL”fjŽ · Ðǟ †ƒŠ j‰,ˆŒj
ƒ T
… ʲœj”f̜»Ç²³”f̜²œ—až›‘>›‡Ž0xÉrŽ · ‘f´œŽE˜³”¨‘>jŽ · ›œj”R”f›œ— · — ^Ž²³Ž“L”f˜³ž)‘>˜zjž^Ÿ x”^É'Ž0°NŽ0^և—– · ‘^±  Ž,‘>˜¶—–˜zjŽ0 ¹
ŽLž—–˜œ»(›œj”  •aŽ · j”(´œŽ0jŽ0 · —–˜³Ž¼ûº‘fž± · ›œj”f—a“^‘>•–•–±³ü'²³Ž’ž · ‘>•–•aŽLž k —–˜²³Ž’›œj”z”>š”>š{ʲ³ŽL”fjŽ · Ðǟ
¸˜³”f²³Ž0—–˜XjŽ0jŽLž—–˜œ» X̳ŽLž—a”f˜ —ažij”[´œŽL“0—a´œŽ¨ÉT²³Ž0²³Ž0²œ—až“L”f˜³ž)‘>˜zÈ´œŽ0›‡Ž0˜³´œž”f˜œ•–± ”f˜ ²³ŽR˜XÌ Ž0È”>š
°NŽ0—a“LŽLž”>š{²³Ž›‡”f•–±z»N”f˜µ”f”f˜²³Ž’ž²‘>›‡Ž’‘fžTÉrŽ0•–•Ÿ \˜›‘>—a“0̜•6‘>^Ö³É'ŽE“^‘>˜œ˜³”f°NŽ0—sšÍ±R²³ŽšÍ”f•–•a”OÉT—–˜œ»³Ÿ ·
á ÄâÀœã0¾zäÇÃOå`æc¾Å ª~«ªµ¬8ñRïѧj¤f­³ñ~òïf­ò k ñ !§ ò ïÇò’ïÇ­³ÿ k ¢ £L¤f¥s¦ §j¤f©>ª~«~¬®­z¯ ¤£Rò ªîâ¥sïÇ­Gª ¬Íò òÍ«uïÇ­œñI¥–ïÇòVªIñµ¤u£ï
§j¤f­©>ªð FœïN¦f«~¬8¥–ïÇòVª~«ïf¥§jïf­ ôjª,¦Xª§)¤fóî!¤>ñLªj¦¬®­ò¤(ò ¤¨§j¤f©fª~«I¬®­z¯>ñ^õ
˜³Ž(“^‘>˜ ‘>•ažj”—–˜X°NŽLžj—–»X‘>jŽ(ÉT²³Ž0²³Ž0,‘R»Ç—–°NŽ0˜ “L”^°NŽ0—–˜œ»™“^‘>˜
Ž´œŽL“L” · ›‡”ÇžjŽL´[—–˜zj” s “L”^°NŽ0—–˜œ»Nž^Ö
šÍ”fžj” · Ž

öœËœŽL´ s ŸrÕ³”f‘>˜X±›œ•6‘>˜‘>xžjŽ0 P ‘>˜³´ s ≥ 1 ֗sšÂ—–Ž~Ë#—ažjžO֜•aŽ0 k = k(s, P )  Ž²³Ž’ž · ‘>•–•aŽLž˜XÌ ·E Ž0žÌ³“j²µ²‘>
‘>˜X± k¹ šº”f•a´¼“L”O°NŽ0—–˜œ»,”>š‹²³Ž›œ•6‘>˜³ŽÉT—–²R)‘>˜³ž•6‘>jŽLž'”>š P “^‘>˜  Ž´œŽL“L” · ›G”Çž)ŽL´È—–˜zj” s “L”^°NŽ0—–˜œ»Nž^Ÿ¡Ùr±Ê²³ŽL”fjŽ ·
ÐÇÖ P ) Ž~Ë!—ažjžxšÍ”f’‘>˜z±Ü”f›‡Ž0˜Ñ“L”f˜X°NŽ~Ë[›‡”f•–±z»N”f˜ P Ÿ —–² ‘Rž•–—–»Ç²z · ”#´!—sö“^‘>—a”f˜ ”>šr—–jž›œj”#”>šÖ‡ÉrŽi»NŽ0²³Ž
šº”f•–•ak(2,
”^ÉT—–˜œ» · ”fjŽ’»NŽ0˜³Ž0)‘>•‹jŽLžÌœ•–^Ÿ
÷ ë`¾XÄâæc¾#ø ®Å ¤f«xïf­³ÿ,¤~~­.§j¤f­©>ªðxf¥yÿ^¯N¤Ç­ P {ïf­â¦iïf­ÿ s `ò ª~«jª’ªðf¬8ñ~ò®ñï ñIó(ïf¥®¥–ª~ñ~ò ’­-#óÈôjª~« k = k(s, P )
ñ œ§ ò ïÇòïf­³ÿ k ¢y£0¤f¥–¦¨§j¤f©>ª0«I¬®­z¯¤u£’ò ªTî⥖ïf­Gª ¬Íò òÍ«ïf­œñ~¥sïNòVªIñ¤u£ P §jïf­ ôjªi¦Xª§)¤fóî!¤>ñLªj¦¨¬8­âò¤ s §j¤f©fª~«I¬®­z¯>ñ^õ
̜x›œj”z”>š»Ç—–°NŽLž
ÖâÉT²³Ž0jŽ —až²³Ž“L”f˜³ž)‘>˜z š®j” ²³ŽE›œj”#”>šr”>š¡Ê²³ŽL”fjŽ Ð
‘>˜³´ n —až²³Ž,˜XÌ · Ž0’k(s,”>š'P°NŽ0)<—a“LŽLKž”>(8s)
š P Ÿ’Õ”f §Iª~­òÍ«ïf¥8K¥nÿñ~ÿ>óióȪ0òÍ«~¬Í§ ”f›‡Ž0˜[“LK”f˜X°NŽ~˙· ›G”f•–±#»N”f˜³žÏ‘f“)² ‘>˜³´™Ê · f²
ØÚÏÊÔ cދŽLž)‘ •–—ažj²³ŽL´¨‘ ̳“j² Ž0jŽ0'̜›œ›‡Ž0 ”f̜˜³´ Ö#ÉT²œ—a“)²—až z̑f´!)‘>—a“—–˜ Ÿ ŽL“LŽ0˜z•–±N֜¸•a”f̜›œ—ažTŽ0‘>•ŸTØÚ¸ÔÇÓcÞ
— · ›œj”^°NŽL´µ—–' j”,•–—–˜³Ž^‘>^Ÿ· ʲ³Ž0±R ž²³”^É'ŽL´²‘>^Ö# šÍ”fT‘>˜X±R“LŽ0˜X)‘>•–•–±ž± ·· Ž0—a“s”f›GŽ0˜“L”f˜X°NŽ~˨›‡”f•–±z»N”f˜ P Ö#²³Ž0jŽx—až
‘>˜ α žÌ³“)²È²‘> k(s, P ) < α s ŸÂʲ³Ž  ŽLžj #˜³”^ÉT˜R•a”^ÉrŽ0  ”f̜˜³´È”f˜ k(s, P ) —až (s, P ) ≥ b4s/3c − 1 ØÚÏÊÔ fÞVŸ
̜x›œj”z”>šºžÌ³žjŽE²³Ži‘fžjžÌ ›œ—a”f˜™²‘>x²³Ž“L”O°NŽ0—–˜œ»¨—až•a”#“^‘>•–•–±¨ö³˜œ—–jŽÇ֋‘>˜³´.šÍ”fx”f›GŽ0˜Ü›G”f•–±#»N”f˜³žxÉrŽ,“L”f̜•a´
ö³˜³´(‘x•a”#“^‘>•–•–±’ö³˜œ—–jŽTžÌ  “L”f•–•aŽL“0· —a”f˜(ÉT²œ—a“j²È—ažž—–•–•³‘ k¹ šº”f•a´“L”^°NŽ0—–˜œ»³Ÿ z—–•–•ÖNÉrŽžj”f˜œ»Ç•–±  Ž0•–—aŽ0°NŽ²‘>Ê²³ŽL”fjŽ ·
Ð
i
i
1
2
n
i
j
"
1
2
0
0
0
0
0
0
0
0
"
0
i
0
P
2n−1
P
P
P
n
Ð ²³”f•a´œžTšº”f“0•a”ÇžjŽL´µ“L”f˜z°NŽ~˛G”f•–±#»N”f˜³ž‘fžTÉ'Ž0•–•Ÿ
ùiä‹À
Ä ®¾ ¾#øÑ¾zÀGÃÇŠю‘>jްNŽ0±R»Ç)‘>jŽ~šÍ̜•‹j”i²³ŽE‘>˜³”f˜z± · ”f̳žjŽ~šÍŽ0jŽLŽšÍ”fT²œ—ažjŽ · ‘>U#ž^Ÿ
{… ƒ,Š ØÚ¸ÔÇÓcÞ iŸ¸•a”f̜›œ—až^ÖγŸ¡ß‘>j´!—–˜‘>•Ö GŸrß'”f•–•aŽ0jŽÇÖ GŸ
‘>˜œ»NŽ0 · ‘>˜‹Ö iŸ j´œŽ0˜‹ÖTÏŸ ‘ · ”Çž xŽL“L” · ›‡”Çž—–—a”f˜ ”>š
۶̜•–—–›œ•aŽß'”^°NŽ0—–˜œ»Nž—–˜Xj”ÈÛ¶”fjŽ’Ï{‘>jž^֜—–˜.›œjŽ0›‘>)‘>—a”f˜‹Öâ‘>x—–° ÔÇÓÇÔ#Ÿ ÔÇÝÇÝÇà>°‡Ðǟ
ØÚÙÛ¶ÏrÔÇÝcÞÏ`ŸzÙ')‘fžjž^ÖXΟXÏ{‘f“j²‹Öœ‘>˜³´ ŸNÛ¶”Çž)Ž0 ª~ñLªïf«§ «u¤zô~¥–ª0óñx¬8­
’¬6ñL§0«jª0òVªª¤fóȪ0òÍ«~ÿ Ö z›œ—–˜œ»NŽ0^Ö#َ0•–—–˜‹Ö
àÇÔÇÔÇÝ!Ÿ
ØÚÛÜÏ¡ÓÇýcÞÏŸÛܑ>˜œ— ¹ 
Ž0°z—–jUž Ǒ!Ö'γŸÏ‘f“)² ŽL“L” · ›‡”Çž—–—a”f˜ ›œj”  •aŽ · žšº”f · ̜•–—–›œ•aŽ.“L”^°NŽ0—–˜œ»Nž¼ÉT—–² ̜˜œ—–  ‘>•–•až^Ö
· ‘>˜z̳žj“0—–›œ^Ö
Ð^ÒÇÓÇý!Ÿ
ØÚÏ¡ÓÇÔcÞiγŸ!Ï‘f“)² ŽL“L” · ›G”Çž—–—a”f˜µ”>š · ̜•–—–›œ•aŽ›‘f'“ z—–˜œ»(‘>˜³´“L”^°NŽ0—–˜œ»³Ö x—aUž #jŽ0jŽ ŽL” · Ž0—aŽÇÖ³à!Ÿ ”f•–•a” âŸ!ۙ‘>²‹Ÿ
\˜³ž^Ÿ ˜œ—–°GŸ !‘>•  ̜»³Ö`Ð^ýÇÒ ¹ Ð fÓ!֋Ð^ÒÇÓÇÔ!Ÿ
ØÚÏ¡ÓÇýcÞiγŸÏ‘f“)² ™ß'”O°NŽ0—–˜œ» ²³Ž¨›œ•6‘>˜³Ž.ÉT—–² “L”f˜X°NŽ~Ë ›‡”f•–±z»N”f˜³žOÖ ’¬6ñL§0«jª0òVªÜïf­â¦ ¤fó+î #òïNòͬͤf­Gïf¥ ª¤fóȪ0òÍ«~ÿ ûuÐ^ÒÇÓÇýNü~+Ö ¹ Ó!Ðǟ
ØÚÏÊTÊÔÇÝcÞiγŸzÏ{‘f“j²‹Ö iŸzÊ‘>j´œ”Çž^Ö iŸ#Ê f² V˜³´œŽL“L” · ›‡”Çž)‘  •aŽ“L”^°NŽ0—–˜œ»Nž^֜ \˜ {ʲ³Ž’ßr²œ—–˜‘ zÎX‘>›‘>˜µÎN”f—–˜Xß'”f˜!šºŽ0 ¹
Ž0˜³“LŽT”f˜ x—ažj“0jŽ0jŽŽL” Ž0±NÖ#ß'” —–˜‘>j”f—a“Lž‘>˜³´x)‘>›œ²(ʲ³ŽL”f±ûß'Îzß ’ßÊ àÇÔÇÔÇÝNü~Öz‹ŽL“0ÌœjŽ x”fjŽLž
—–˜Üß'” · ›œÌœjŽ0 #“0—aŽ0˜³“LŽ · GÖ z›œ·E—–˜œ »NŽ0^Öَ0•–—–˜‹ÖàÇÔÇÔ #Ÿ
ØÚÏÊÔ OÞiγŸÏ{‘f“j²‹Ö iŸâÊ f² ŽL“L” · ›G”Çž—–—a”f˜[”>š · ̜•–—–›œ•aŽi“L”O°NŽ0—–˜œ»Nž—–˜Xj” · ‘>˜X±µ›‘>jž^Ö◖˜ «¤^§Oõ Ç«¦ í ÿfóî!¤>ñ~¬ #í ó ¤f­ ¤fóî+Ö!‘fòïǓLò͓L¬ºŽ0¤f›œ­âjïfŽL¥´ Ÿ ª¤fóȪ0òÍ«~ÿ G¤ àÇÔÇÔ #ևР¹ Ð #Ÿ!¸•až)”E—–˜ ¤fóî+!òVõªj¤fóÈõ ª0¢
¤f«~ÿ¼ïf­G¦ îNîâ¥n¬Í§jïNòͬͤf­³ñ
ØÚÏ¡ÔÇÓcÞ ,ŸœÏ >•–!° f•–»Ç±#— ¡ \˜³´œŽL“L” · ›G”Çž)‘  •aŽE“L”^°NŽ0—–˜œ»NžÉT—–².“L”f˜³“^‘L°NŽE›G”f•–±#»N”f˜³ž^Ö!—–˜µ›œjŽ0›‘>)‘>—a”f˜‹Ÿ
ØÚÊTÊÔ O"Þ ŸOÊ`‘>j´œ”ÇžO#Ö iŸcÊ f² `Û.̜•–—–›œ•aŽ'“L”^°NŽ0—–˜œ»Nž”>š³²³Žr›œ•6‘>˜³Ž'ÉT—–²—6‘>˜œ»Ç•aŽLžOÖ ¬8ñ0§0«jª0òVªïf­â¦ ¤fó+î !òïÇòͬº¤f­âïf¥
ûºàÇÔÇÔ Çü~Ö ¹ XÝÇÔ!Ÿ
ª¤Çó(ªLòÍ«Iÿ
'
"
'
&
#
(
&
'
#
#
#
#
#
Ð