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On the Dimension of the Hull

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(1)On the Dimension of the Hull Nicolas Sendrier. To cite this version: Nicolas Sendrier. On the Dimension of the Hull. [Research Report] RR-2682, INRIA. 1995. �inria00074009�. HAL Id: inria-00074009 https://hal.inria.fr/inria-00074009 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(7) . . On the dimension of the hull Nicolas Sendrier. N˚ 2682 Octobre 1995. PROGRAMME 2. ISSN 0249-6399. apport de recherche.

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(97) g   ] _ ‘v:g ]  y ?„ W /W ?U^Z \   ?y cd ] x

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(99)  

(100) g ]  y   ?y cdo   ] x

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(102) s4‡. †(

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(104) g ]  Ÿ   y + } L 7 + } 7+ s D + } L 7 6s D + } 7 +  =s t š g(ve SŠ RgPj6AMŽ)[NTUZQTU:=[:9ZQlNAqOQAV'LNA\[)JMA\O + } 7 + >[)j + } L 7 + X  Ael)>4”=A D w  + } L 7 + } 7+ s D   1 6s D w F s r s A &– % 0 s r s A &– % 6s % w  F % t   + + } L 7 ‘s D + } 7 + q  1 +8s D w F˜s r s A – q  13F s r s A – + s $W:=8>iIUI  XNs  ŸQX6>[j  l>iZQA\”'A\8TUOZQlWAHY>8;T_Z‘Pq: 9X  Al>4”'As r s A &– % ‘s %  w % s r s A – + s  w F X   lNTKJ;l<=TU”=AxOH LNOHZQ lNAq8QA\O;LNIUZŠ g+›  “N M^Z \  q   A B Ty v

(105) g ]  Ÿ   y + } L 7 O + } 7 +8s % D + } L 7 6s D + } 7 O J =s t š g(ve Š6’“A`l>4”=A'X6R0Pj6AMŽ[WT_Z;TU:=[˜X Dw  + } L 7 O + } 7 +8s % D  ‘s D s r s A &– % 0 s r s A – % ‘s % D w %  t + } L 7 6s D + } 7 O q  % DNs r s A –  q , s r s A – +8s %   +8s $W:=8>IUI  . ‡WX.>[j  l>Z;A\”=Ax8eTUO`ZQlWAY>i8QTUZ‘P :i 9X  Awl>4”=As r s A &– % ‘s % D w %  Jms r s A – + s % D w  Š ƒ`~?Ÿ „Ÿ.{ W»/} W5?Ÿ UYX?X[Z ( +87 - 7 9 q – 9 ' 9 w F ) % r  +87 - 7 9 h  :g ]  ™ y¬‡ ™ ™y cd ] x

(106)    –r 9 ' 9 s F ) - % - – – 9 ' 9 s F ) - % t – + % s  + 7 7 9  – r s – + % s š g(ve Š .AxZ  q  A B 0X  Ael)>4”=A  +87 - 7 9 q – 9 ' 9 w F ) % + + } } L L 7  + -v} w 7 + s+ }% -/w % 9 -/+ w} 7 + + s } -  +87 - s™9 ! 7 6s 9 7 +8s % 9 7 + LNO;TU[N< .A\@#@>`ŒNX ±A\@@>]‹>[j#ZQlWAC‘>=JMZHZ;l>Z + } 7 + >[)j + } L 7 + >8;Aj6AJM8;A>O;T_[W<X  A`:'RNZ?>iT_[ +}  + } 7 + s 7 O % -vw 9  + 7 s@9  – 9 ' 9 + w 7 F ) 7 9 %   +87 s™9 t š (g ve  +87 - 7 9. . . . . . . . . .   

(107).

(108) s=s. 

(109)   

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(112) g ]  ™ y˜‡  ™  ?y cd ] x

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(114) [NA\YN<=8;IU:A”=JrZQAAjZ;l5>:=Z 8gG Ž 9*NAT_Owj>™ OQP6@#lNA\YN[ ZQ:'

(115) ZQTK>J\[)>IUj IUP#YN<=8Q8;:': Y:'Š8QZ;T_:'[>IZQ: 99.  + 7 79   M X X ˆ W5~={x}Ÿ)~= 3›6~}4›6!~  ’“‘p A[W:Ae <'A\ Z T_O;l ZQ:#OQ:'I_”'AZQlWAAV'L>iZQTU:=[ C D29 1=, )kD9 . G D q – 9 ' 9 w F ) %6Š RgP LNO;A:iZQlWATU[6”=A\8;OQTU:=[#5:'8Q@SLNIK>  o 9 ™ G D  . L ! w  - /L . t Mso 9 q I D21=, Ÿ—p  r s 9 s – g+›  “   :g ] Š 3™ J ‡ y cˆ ] x

(116)  o 9 ™ Mvs Q. 9 q ID 1=, Ÿ p  r s 9 s D –  . L ! w - /L. ! q  ‡G 9  ™™ k k/x{ = 8y h $. . .  . °d¡v¢ Õ\Ì ¢. .

(117) s4B. †(

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(119) zv

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(121)  y 9 q  ™ kŒv T

(122) h ‘  – s 9 s 9?r š g(ve Š6’“A  TUI_I)YN8;:”=A`ZQlNAq8QA\O;LNIUZHR0P#TU[j6LJMZ;TU:=[˜ŠNFIUA>i8QIUP G , q s=SŠ $W8Q:'@ ?s4o H>i[j %BA \X  Al>4”=A o 9 G ™ r s – 9 s D H o ™ r s U  r s 9 s D – .  L ! w  - /L . 9 q I D 1=, T Ÿ r s p Ÿ p o F o 9 ™r s D D  .  L ! w -  L . H 9 s ™ r s D  . L ! w - /L . I D21=, Ÿ p  r s 9 s – q I D 13F Ÿ r (s p  r s 9 s – 9 s – F o F o L 9s ™ r s 9 s ' w Š ) w D F D F ™ r s  r s 9 s D s F –  .  L ! w -  L . .  /  ..  !  L s - L r –  s  – 9 9 s s r I D 1=, Ÿ p I D21=, Ÿ p q š g(ve S$.   . 1. . - !.

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(132)    ' F) % – – H 9  + 7 7 9  %™ s – r s + } 7 k9 % 9 s – +. % s - t š g(ve Š6ƒHlNATU[6”=Ax8QOQTU:'[S5:'8Q@SLNIK>]<=TU”=AxO o 9  +87 - 7 9 q ID21=, ™Ÿ—p  r s 9 s D –  - /L . +87 - 7 D ! o o 9 9 ™ –  + 7 - 7 9  ID 1=, Ÿ—p –  - /L . +87 - 7 D  ‘– r s ;– + % s - ID21=, Ÿ—™ p –  - /L . w L. ! >i[j#OQTU[JMA € 9 s % D ‚ H € %D ‚ q € %9 ‚ r Ÿ %™ r Ÿ \X  Al>4”'A o   LL9 – –  +87 - 7 9  – r s – + % s - ID21=, Ÿ™ p – D ' 9s s D ) q – r – s – + – % s - ID21=9 , + } 7 D+ + } } 7 97 9 s D t $3TU[>IUIUP6X  Awl>4”=A + } 7 9 + } 7 D >[j + } 7 9 + } 7 9 s D XZ;l6LNO + } 7 9 A  + } 7 D + } 7 9 s D @TU[ Ms A + } 7 D !4s A + } 7 9 s D    s A + } 7 k9 % XN>i[j  H  L +87 - 7 9  – r – s – + – % s - + ™} 7 k9 s% t `Ÿ~Ÿ ' '5N~œ M i3gm

(133) k( ] †(

(134) ( ]  y 

(135) 8 n T‰8_ =  T– y 8†( ] _v

(136) g ]  ™ y˜‡  ™  ™y  +8G 7 - 7 9 – %  -vw t + s 9 9  š g(ve SŠ $W8Q:=@ 798Q:'Y:'OQTUZQTU:=[“sx‹ >i[jF:=8Q:'IUI >i8QPŒWX  AJ\>i[A4>OQTUIUPwŽ[j#O;LJ;l >Jr:=[NO;Z?>[6Z4Š š g(ve S$. . . .   . . . . . . . . . . . . . .

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(141)  ‡ef:gh  :g ]    A &] _ ]  ?y 88 ` g6e  "!    †(/ =\c"{vŠ? ]  :  | : —(˜ ] @‡ o + H  UU t 5 + 7 - 7 9 q p ' w F ) + } L '- } 7 ) + } 9 9 % 7 s™9 % 7 9 T T + % s@9 G- H - š g(ve R. 5 + 7 - 7 - s™9 -*'-/w F ) % q ) + - .u) .  +87 - 7 9  + 7 7 s ™ 9 @s 9 q 9 - +87 7 s@9 o o (  ' v w ) F  U U t 5 + 7- 79. % q p  mp G - s™9 H T T + % s@9. . 9›6Ÿ)~›. ™. B. ™. „ – 'ˆ. s. s – – Š gP jkAMŽ[NTUZQTU:'[˜X ASl>4”=A – Š.’“Awl>4”=A  >[j#ZQl6LNO»8Q:=@  F:=8;:=IUIK>8;P‹)X  A`<=AxZ – s – ™ $3TU[>IUIUPw58;:=@ F:'8Q:=IUIK>8;P ŒS>i[j %"Œ  A`:=RWZ?>TU[#ZQlNA`8QAxOQLNIUZŠ R)ƒHAlNAxZ;ƒH:=lN8QlWAAxT_@O]J\>i8QOQA\B A#O;LNT´IUTUZIUI˜ <=>TUr YN”'A\YWBOI_P6>Š [“TUO>=IKJ\>JM8;LW<=8?A=>@Š Z;2»AS A\ OQZ;r TU@"B > Z;ATUO>O;O`@IU:=>i[WI_I<>i>[Oj“™g™TUOeTUO[WJM:=IUZ:=O;JMASIU:=ZQO;: AZ;.: X3ZQ lWA A\BN[

(142) X)ZQ5:=lW8;T_O@S LNIKTU>IUI¬M>is=I s]:>4P6iO  ZQTU:=[5 - ),+- . 8QA\YW8QA\O;A\[6ZQOHZQlWACYN8;:=Y):=8QZ;TU:=[S:„Á– "! 0ˆWJr:kjkA\O  lW:=OQA]l6LNIUINl>O>`<=TU”=Ax[ H ƒ W l  A 5  ? 8 ' > M J j[66LNTU@#@SA\R)[NA\OQ8QTUOH:'[<'8Q™‘: ŠƒHŠlNAH[W+8A 7*NZ¬7 9 AZ;lNA\:'8QA\@ OQZM>Z;A\OZ;l>Z9Z;lNTUO¬8M>Z;T_:`TUO9TU[j6A\Y)A\[j6Ax[0Zg58;:=@ >i[j   lNA\[wZQlNA\O;A  9›6Ÿ)~›  M Z \ 5 +87 - 7 9 T_{e8 ` gŒ „Á– "! 'ˆ †m/ =\Œc"{v‘? ]  :   : ™ h  :g ]  ™ y 8Œ‰?gm\‰:g( 

(143)  5 + 7 - 7 9 A ),+-/. e †\Y†m/ =\& k&†(

(144) 8x

(145) gef:Œc"8  ] _  f=v\e; lz8 k n h P cV @ =_/‡  kŒB  khP1 ] x{ˆ/

(146) g ]  ™@J ‡y . . . 9.  ' ,w. . ,. c"gm , q + + } L 7 O t } 7O. – %– r s ‘– %ˆr s @tvt/t %– 9 r s š g(ve Š2‘@@#Aj6TK>Z;AC>YNYWI_TKJ\>iZQTU:=[S:i¬ƒHlNAx:=8QAx@ ŒNŠ `Ÿ(˜ ~] Ÿ @' ‡'5 N~œ  i3 ] x

(147) g ] f=

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(149) [ž6e ] –  ] g n B ? ] gj†m/ = s I EGF ™  9 q I DE3F – D H s t 9 E š g(ve Š .AxZ $ Oe>iYNYNIUP %"u  T_Z;l q s A –q>[j G q E X  Ae:'RNZ?>iTU[  H w 0 s D– F@U q I %– r s ‘– %ˆr +s @t/tvt ‘– r s ! 9 DE=, T E-, + 58;:=@  lNTKJ;l  Ae:'RNZ?>iTU[Z;lNA`OQA\8;TUA\O    q I E-,  9  9 q  , D0 EGF T s H – D(U t 9 9 q + } 79 9 9 F ) % q. — ]  n ‘‰8‡/ † ]  n.   

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(154) I. ”=ZQTU>iIUlWI9I_ALWRA\LNAOCO;>L:ig>”=Ig–6AŠWT_Jr[6ZQ’“”=:=>i8qA8QOQTK@>Y[6>'>4ZQJMPSOA Z;»lN @A\lN[

(155) TU[N:'OQTULN@SASOQAwLNj6@mTUZQ@lWAj6A\[WTUl6OQOQZMLNTU>:=IU[I˜[ JrZQA=TU:#O`X :=OQRN@A\ZMTU>><=IUTUl6I˜[ZRW[NLNj6:=Z`TU[OQZQ[N8;ZQ:=TU8QRNZ`TU”6LN[NTKZQ>ATU:'I3JrA\[YT_[kOQOM5tv>:=t/8Q8;t TU@`>IUP >8;Z;\ATUA\:=8Q[Sjk:)TBX¬:='[

(156) >iJrZ`LNJMI_:NIUA4ZCj6>ZQA\OQ: OZ JM:'@ H YNZgLWZQI >iA=OQŠ Z4X=T¢ AC8QAx@>8 0ZQl)>Z   ’ ’ ’?>  ’ H  ’?> > X6TUZ>iYNYA4>8QOZQl)>ZZQlNA]l0LWI_I)J\>[#RA JM:'@YNLWZQAj#R6P>%>LWOQOQTK>i[A\IUTU@TU[>iZQTU:=[wq :=[ >  q @>Z;8QT *˜X  lNA\8;A T_OHZ;lNA`IUA\[N<=Z;l:¬Z;lNAJM:Nj6A'Š  5 `

(157)  Gq ƒHY>ilN8QA“Z;T Jr>LNLWIKZQ>lN8Q:'IUP#8 ZQ lT_>iO;8[lN0'A\O5LNZQI˜:“Z;:Z;ldH>Š [$0 ŠN ]7|OQl˜O;@SŠ $˜LNIKO>\X ar=:=8HIUA\»Zw:=8C5:'lW8ST_OHlN”=TUO>iIUTUL[>j6RNTKIUJ\A>iZQ>=TUjk:=[N”0TKOSJMAw:i>[<=j8QA45>:'Z8CTUlN@#TUOCY:'TU[68QZQZAxŠ 8QA\O;A ZeTUTUO[>iZQlWI_O;T_: O  :=80 Š `= q ]e „ e†`t'‡4ˆ dHp=Œ8Š $›Us4Š4oN ]s .OQOQs@SpTQNLWXNOs4X t='t=8|‡W>iŠ [j ŠÁcwŠÊ†]A\P6ŠN“ '[NAH>[jqYN8Q:4aMAJMZ;T_”'AHYNIK>[NAxOŠ —†gm\‡ ]  ]  †( X . H. . . . . . . .

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(161)  ] _  g 7 / =\ ŠFH>@SRN8;TKj6<=A< [NTU”=A\8;OQTUZ‘P. ŠNF:=@SZ;A\ZŠ 9

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(165) R`A\ Š8g :] ¬‡

(166) O;gLN RN†(O; YX9>=YJM>iAx<=OCA\:O¬Q=>qu ”=NA4p=JMŒWZQŠ9:'8H eOQJ\Y>'>'j6JMA\A=@Š"TK2‘J[79’ 8;A\Š OQƒO4XŠ st=o'tNŠ „,I f8QTQ4ˆ $HŠ:=)8;Š ZQl6I— H>=JM:'’ IUI >iTU[IUIUj.TK>X)@stTOSQ=>QN[)Š j qŠ )Š ŠHf6IU:0>i[NA=Š i3*i38(

(167) g n dgg(

(168) g 7 :ggm(†\ Šf 7 v T( Š „Á7) QWsMˆ %SŠ˜] 7S8:'^I_ P6> ] > [)Xj B'o8%S›Us Š ‡'BŠ) Cs4IU‡=A tW*X0>s4[)t=j6QWA\s=8QO;Š :=[˜Š %>LWOQOQTK>i[“RNTU[N:=@#T >iI¬JM:6(A 'JrT_Ax[0Z;OŠ d  |‡ =g „Á79IUAo'u4ˆŠ.] 79 hˆIUA\ OQO4Œ]Š`ƒHh lNy A9 [6†(†LN] @S :hR)A\]8C :i h T_Z O; :=_Z;†8Q:=  YWy T JE™OQgLNh RNO; Y<!>=<<JrXA\OŒ't8TU[ › ‹s4>wp Ž)6[N‹0BNTUZQs'AX<'cCA\A4:=JM@#A\@SA\ZQR)8;P0A\8Š s4 t=o=_uW :Š h#@†h  kh „Á79IUAo'p4ˆu8›ÁŠNBN79s4u IUkA\O;B'OB=ŠpWb]XNs[t'Z;o=lNpNA`Š LN[NTKV'LNA\[NA\O;OC:i˜ZQlNŒA %`:'I >4PSJM:Nj6A\O4 Š

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