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: G ^V>(     x70e0e?Rectangle 2 ) :2Carathodory, Helly and Radon Numbers For Sublattice Convexities Maurice Queyranne Sauder School of Business, University of British Columbia, Vancouver, B.C., Canada Joint work with Fabio Tardella Facolt di Economia, Sapienza Universit di Roma IASI  CNR, Rome, 31 May 2011 @ T4b@   f e       33y___PPT10Y+D=' = @B +">9T H%@%0(%(     p:0e0e ?Title 1(h#  ) MPrelude     <0e0e ?,Content Placeholder 2"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!?Јq drs/shapexml.xml]nH_iam_j#ew(T{Y8Ʃ}}}=3 N\8`31s'&qc-of|Ӻ0H )Q4,0_;N7sicsOT®xhK*Bv:öKmn\O?j˺{ bLcpSzq 'C-ks)tֳW4Kf~[Mup7t.]{ǡs2~9<^a8i{G3:Zͅ[kWț \;w::z ݥ#Ă>0-ËaآDY5$WZ= J=lo0T084.};yvD4y/kQ-UӐ浄=uSy'{E WR 'lɂL7OT'P G@NaEE&`m>.уO! {'|ahiyv 2c{:n[j.::]^;~<]N77p+;0az:|)PMGqڮ wm>O$jMb &z 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?,Content Placeholder 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Ąldrs/downrev.xmlDN0DHHܨӠVU[J@N|&ġ:M=8fFol/.CX| UyzyXYc\)vs{Bt9V$BPfn N]Řo8&eeKi`p| a|ҫ=ף\*Ui"Ǧ"YU/ Dv|"yI/&QPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Ąldrs/downrev.xmlPK x$@) )___PPT9x HGiven a set system (X, A) where |X| = d, and a subset S of X that can be expressed using unions and intersections of subsets from A then S can be expressed using at most max{d 2/4 , d } subsets from A Proof: If there exist either x in S but not in A (i.e., no AA contains x), or y in A (i.e., every AA contains y) but y is not in S , or x in S and y not in S, but no AA contains x and not y then S cannot be expressed using  s and  s of subsets from A (i.e., we have a non-membership certificate for S), else if S = X then "xX $ AxA s.t. xAx, so S = xX Ax S = then "yX $ AyA s.t. yAy, so S = yX Ay else "xS "yS $ Ax,yA s.t. xAx,y and yAx,y , so S = xX yX Ax,y (i.e., we have a membership certificate for S) QED     } i  8 H@   cccc ccccccccEccc% ccc ccgccc c ccc   c     c   c   c  c  c   c  g  c  c  c  g  c cccccccccccgccccc  @  f@X?6Slide Number Placeholder 3,  j(*    33___PPT10.01+ED' = @B Di' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Dx%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+CT --P-(  ,   p80e0e ?Title 1(h  ) h"How About Helly& ?     0e0e ?,Content Placeholder 2" ã PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!:Udrs/shapexml.xml\n8_iᷔ+1*dY۴Ы}}}=BV\@'9>ϏI#*<Տ(0rw]ja4CR?]yV)\N G4!H)HȄh(r*\ /JXV4 9G~ړ*yI|R1$v H!Yז@"|PY2JC-tq1U ʶKy8tμ4A'ngIFlDʎ'L69i(BjuVMܪVZ-ҦB>6k H3:SG;0׳'Wb85 wb]04p_g$^DR\h@״-4NNkP5>N:y Cxju~P2IJtRɟD\B#͍HJ-y*mDVNf 4*g`[{5M9A4U[W?$kupG`۞Gzc}u~ɦѫ{] . pwYV xh>#ަ=Xi2 O8!

4Ԗ7oh9|}"SH<}U7&}h{u%͖;oQCmB?POmٝ]yxW^ㇵ/:֓Kŝ'gwvOz}3:Hrf<1Bu lbI6̓|0p줰_~-9 œv"}_@zث @q,呧YĽE[Ru{o{Z<ԥne?UM0+ PTuiY Vj8b-~}N`ať=>Xx^AYc˔9C4(yߣZ@#D2gLJ:RA=Ta,R19Ydݔ[ųrT~Ļث(QעƓN izYJs-h؋xc+9җh7-FPo&'l.23c/2v pr9qLx2$DBwrGEGpV=G59jqOsϸ;+hY\_vңJư6 ^GišG3zL'r`>K@WVw(x?>-AL aZa{:B#"} \J_d@:!FHpp4I`Ł-U߇`OG,T2+jUNۛD)r]?TdDrtޣD(EAPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!:U*drs/shapexml.xmlPK-!vJv drs/downrev.xmlPK  w. H$@) ) 2Helly s Theorem: If a family A of (standard) convex sets in Rd is such that B for every subfamily BA with |B| = d+1 then A there exists a concise infeasibility certificate for an infeasible system of linear (or convex) inequalities A Helly-type theorem for integers: [Doignon 1973; also Bell 1977, and Scarf 1977] V i   m # p/  & cccAkcccccc"<#&/c*=      fD?6Slide Number Placeholder 3,  j(*     NX?Rectangle 3X hX( s           JA?TextBox 7= Eduard Helly (1884-1943) 2 "    A??$Eduard_HellyPicture 8Eduard_Helly.jpeg+T  \@?,Content Placeholder 2"^ X PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!$ndrs/shapexml.xml\R8oվ-&0Ԙ)Cs[&^d%+pgG'O8 qbB&#'oݭVKL#NLXNC!Lx[v}~sH)sK/ǿ9 Ae+ bQH2BGoͣFDC)q?LR?yIBcPY,gF^\$jzOQQ "|tt>`KC 3blC˾\7ս>J'*6#2ڴ{=8>6 uaѳ}6U^ޣaQhо.TGEtC#LE.tw">9ߔf\l ypOC3ld(T$F_3Qw"<{-0!׼yΟَlI&>R5T3jiеd83tڽ \!sڄ;g5 @ WReuN, 'ʈ BK5p&C's1L<=4"Ͷ(%NYPUiVրbx{\f\v49zjc4>1 f \D$HBirg !S2$7o׮d\(,29ּG s7(+\oChmmH>@2"J]w!wL4ۣy3_ y45L<x 80~(4PV.A yC_Ckd0P4EM[Z*ł BDfg5%_K%{ WqD$ZX؞"ho8`4KK/y 5uL4cq-DmG0,*pbsZYI]lֳ9וS-^Z>C&+{S^`9d5dXv[g׾blC~Osxr+*M=+T2~s:`5oSZ^BLX@mdNHوZla¤\vlVh+?lhhlEmLm"E}5 U4!ָqCV}~mpjSR߽lE{{ Vk զvLh8 sK,ܰzR?;[*h| DٲUY)㳬84#dUK.BEB:>zLh<@uKPC| Vz$JV̪LTJ[ ?oaAW:W:ZA.uI-*IOK^yS;`ԞC< &GCR*.3q@UVzd:ETXxj!E4 N" \ $ vSn`='**BF;z@6cZpĺSuiW RRj.+<!f Df™#Öf&kL殡/㒕}_\d1%R~ܧZG[z؉׸^X}d 1òjCBTSMmẅa8xJE\8ɼ"muv;/H^x}qt/b0?t{on?M>H'!0j gHe{6ᦍ xdb&PK! drs/downrev.xmlDJ1@!ج.Ⱥ6-bUDa~^&Ynzq g8d{\27k|l/ 138xG a8=ciUS+Xb Rʨ;g~$Ǭb1<:nGZwwUmbWzjbٴhJ>g-<?Co*p*#PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!$n*drs/shapexml.xmlPK-! I drs/downrev.xmlPKM  $D )___PPT9tl :A Zd is convex iff A = C Zd for some (std) convex set C in Rd If a family A of convex sets in Zd is such that B for every subfamily BA with |B| = 2d then A an infeasible system of linear inequalities in integers has an infeasible subsystem of at most 2d inequalities B j q  cCC kk C  C  k cccC kcccC  k  c   c   c   c   c          c e    ~    "  #   s 33___PPT10.01+KΖ D' = @B DY' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I{%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*{%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+GT O/G/`//(      p 0e0e ?Title 1(h  ) \& and Radon?       @80e0e ?,Content Placeholder 2" PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!<}gRdrs/shapexml.xml\n8_iᷴEMG3S^V&dӁ^g;iSF%%„ c;؇P; moP¥y/W#J䘉Hr.xB?IKb2jğ%KM"2NJ'\jP[F@SJ 2)nH0AզDO=Fc2H"-ZO Gmҫb0'"2MaLMR;ΙStqNs=lu]4zQbGz E [6dH:>h7ۦ(>l6Gv۵؁==|Ґ~)d5cS"GmspBFKF9 "a LGu<ȡ1}9}Sь%VTLL h8e1]?|]ub07bł>!S}gfrVR…@AֲL܀܂1\>]6'uk<>&#@Zɔpf0>-M1X|,3 \O|\s*XD/nzihv&Jy4?$14Az@|; 08be :F#+1WeQG[- m[ i-3WqI-){Z[6PQ4q4ȌIgBoB+I$h%ht381I2&L3 2*M9A"d քNu{R k?\!OF.IJ@sl5e*cO5>_!1 !q7{M830wkzl <#_³EJl|BZB>0ptF8Z7 6Wflǰ.`2]sv[YWkp|]2 ųW\咛7ᄌdՒ⨵XuMʚ,ER T[D*:៞;H.4*G2$MH¦j_T33^ _-/+T Pwhvxp꣌=Tf=ۣ7[[iHw(jqg~0f xtPv_PZ RZ-[g3&XeʻyS5C8]9Ӣ55EeyRTem]Wתr努UmVcpgLgki2N%MD4ZGGFᆅ6.EX(+DlPF'* ]l\K?|x+By?_n:b,iQ AO۝]Dd  "͊gC_ xK 8ꋷbnh}c̶Ph="+-nڏ6Hld^{dtUG!7MBEHJCxpX/{XMu98w~4շ`UQo0+_'дTB{@Uy$xgvZ$@*uEgwnl o Tw!{Xfa_MpC\-'G:fe9_̥pY< \Z4uݫpj//.G?6s'+9G:ɃM8; Yl8 J, %)=$.uxRRXFJuWvQЂ5;TޠDӈ* iWwiz[AɊ*s+YрYTTg!^?'"nRaO]f,ҍmc ,;j~9߇`gYQZ]V;x*A4HL4'0 n/}i.Oktрa)Έn[(7Dt]bp/bc dܣn Ŋ>,ATtɋN8 ]'HƑM̜MIF\Ifv Sh7PK!հ*drs/downrev.xmlDN0EH5H@*tZA#6Hlx٦I,չ:ͤ{qb;kʪ4d0™=l֗+ʕ͞OЈ1>'6!W-k3;]mkr4Fe$ 3񡥁-W÷Fc.ŮZx}5=܃_Tyz ~=NvQ/FQPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!<}gR*drs/shapexml.xmlPK-!հ* drs/downrev.xmlPK w% $@) )~___PPT9`X TvRadon s Theorem: Every set A of at least d+2 points in Rd contains a subset B such that conv B and conv(A \ B) intersect Radon s Theorem has applications in statistics and computational geometry: a  Radon point (a point in that intersection of two convex hulls) is a  pseudocenter (a  generalized median )  i % ' 2 " B  &cc  CAkc cccccccc %c' c 2cBc*7     f` ?6Slide Number Placeholder 3,  j(*     N@(=?Rectangle 3X hX( s          \G?,Content Placeholder 2" PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!@}zBdrs/shapexml.xml]r6LANfli/;L^g-7E$=ld5ƆB!˲O|iqdu:a‹P<ηCREOy,X|I?M\,~:c~zcC0XFTGJ$KPT"g¹0H}}}Hh: .%N=6$:x @mv{Oi:Zu/B-Ӡmf\7w|}h(ش`hA@[ӓvc:=".<DwKh_DMd~cƭ""בS6UV-D|rNri5u.Ne(T0xkQ3*|SDѐcK.̌А빧?ӽ 7GjY-$3sg^׃Y\37C?] 3!?4YpOGWs!Rk, WM RBd1@Dxpq.&SB9E,˪Ԗ5`C1~q*PyΥru~x j\ɐ$74 tLeʠ'|_v*:̢};+yzѾ}d&Q(a6 z Ȃd'#@k.'h!OLh0J<ВwQC2sRA7ۆj/(CV F¢LrJ Vd>df,}|e/.XAm F.%GZ0tĺBEM]>,^<<=(0tfaEd-1Fv"mT} '53oU.LT0أ \2WI0P-EK{8F1 (sV>XY`V:Z=@38v=`6٭*㇭Ijyg Md*Sσ k/q0rthp"B2i nlx% O:aQUP~%8V Nk;KW=z8׌M|lVpNvuB/Wfbxr9]Y$lCN6 ) k77V}PѺ+#d.FPS 1U0{^:òE\9TXv>%הQd<[DΪz /eAn{B  \"Bw*o[n&vf~9^4V0VcHqF=4T~k걭(qw++3}xd"ThTѨ]壘F#;㰉pR6mfX-oѩb2,]JgA[QZ_\њFkzgtMbdX\3kGi%|IhYoղ^@iKbj||<,<|-Y&#<=WȻ%Ԯ`ٷS~ރ GIS*Lq x,h=tkRG r\AXJ n \Qdϳ7WLa-" Zn}sJEx@^2 'S_D٧<󤤠V!-WV(XKO!+1i=hxhLcLS]T]XMm}&d`|p,69sފ(J-ꂷ8+zإU #^Ji8k3̢<Au#u )zݣgΠvKAܯ<پ K?ptY*Ia, |] JZڵP%>+@sV _%Ois}Xo 1@C(]Cy]Gޥڰ#c||<ؓe܉xmQތǂIT&]!Z:ƌtm8w١ׁb#YvA.fnjPDz;oo7?@҈}m7`W$!P~79vBRLB5 WxA6{枲μg2pd]SDyp *6A7V33PK! drs/downrev.xmlDJ1@߅C/bUumZJŪc2{dIb|Ǚ3,VőBS0e io:(?] bBg)j9X`aɕtR#Xb ڔBʨ[g~ Ǭb14<&ڴ՗Uя}jv[X\~hL=V^ pJ| )1& xé@.PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!@}zB*drs/shapexml.xmlPK-!  drs/downrev.xmlPK  n$@ )vn___PPT9PH  statistical (or computational) learning (or classification) halfspaces cannot  shatter sets of d+2 or more points in Rd a set system (X, H) shatters a subset SX iff for every subset AS there is an HH such that A = SH the VC-dimension of halfspaces is (at most) d+1 the VC-dimension [Vapnik & Chervonenkis, 1971] of a set system (X, H) is the largest size of a subset it can shatterJ< > e 2` u <c!c cC  k    c  c  c  c  c                     c           c  c .ccccc0*v   s  A(??0Johann_Radon_ca1920Picture 10Johann_Radon_ca1920.png!7  P`I?TextBox 7"n) Johann Radon (1887-1956) 2   "   336.___PPT10.01+f\D' = @B D' = @BA?%,( < +O%,( < +DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E{%(Dd' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*{%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!;%(DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<z%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*z%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+T p (      p@B0e0e ?Title 1(h#  ) R Talk Outline    d   F0e0e ?,Content Placeholder 2xx )___PPT9 Sublattices, and certain types of linear inequalities Convexities, and some of their invariants Summary of our results Carathodory numbers for sublattice and convex sublattice convexities Carathodory numbers for Ln& convexities Some open problems " } E aia  D  fK?6Slide Number Placeholder 4,  j(*    33___PPT10i.P++D=' = @B +]9T /)')$)(  $! 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" ! " ' (  0e0e ?,Content Placeholder 2"3 - PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!YJkdrs/shapexml.xml\n6?ǯ86 i g倖XD%=$%[dұcK/d/^^R<8B22 bcJQ <蔒T11ba,G<, AfNJzZOxqƱ£|%\((5Oj =GQb6HR$ȣzI`z ̤2Op%iZeM?Yj$4T!"%T/*~i2Am5ImgimdZ(5-cXFo1z ,^bjꥉ1cz]1%?O >X&EcGum G6a:o t8(0yWy6|oG; q9 &Ѫ5FݸcZ}+ }gKlaR!aZ-.YxRKil3qKnt(KEcz;J MU)ۖu١RyAicS[fw n7Q:T>{n%ŕt8W ˽e|ul-'z嚅_~%:_n+t]nsEFfof/`KJݎVDRUZ Rsr132͡ j]iϞ2ovC@x1Rg*)]8uyv;wξܪ݂!ū7VoD\;y~7ۚö,zqgݳ]T9ʳ)xuNsw23 ^-yoWXBCr`Vb.6]}vkbT;Wn1~*]HmQ)z@ 95^plo,}>IK{BZEJx[c3$l->=ŸɽQ)"ے?RV<%"Ϸ22j aL(dYjm[QRtg+7k)8tbY襒e7,MS# ܴW8;"dz '|%,x|WO>.dS}ָdJjMrs AV.+K٬ ?OznYBDz 0l-ELeGqD;q!QjNMjCwKu '_ !#3>@\ffd(79AJ9%LW6%F#>8ƜfzCs`eY,R4TI͵9c_0QURĈԱf"KoXDa>(Gx3ao =uYأLu@ôF2z:[_z&?m;imGhFiZ?#pba Ru:SM|ߦn\^TJ C Uxwze˾tpr<\x+g5n ϡ5mz;PK!,vdrs/downrev.xmlDj0E@wf@[ZnZT.^lvnE7 g 6UΧ5(ja {m'wʔ́cDBBI鋚5ĮNSUR9"\2MԘPSFIT r;W"nc+<](._QBԋQPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!YJk*drs/shapexml.xmlPK-!,v drs/downrev.xmlPK"  H$D) ) Sublattices form  natural feasible sets for Submodular Optimization the characteristic (or penalty) function cS for a subset S of groundset X, defined by cS (x) = 0 if xS, + otherwise, is submodular iff S is a sublattice of X Submodular Optimization (and Lattice Fixed Point Theory) study conditions under which solutions exist solutions can be found efficiently the set of all solutions has a nice structure (e.g., is a lattice itself, or a sublattice of X) (the set of all) solutions varies  in the same direction as parameters in parametric problems: qualitative sensitivity analysis (a.k.a., monotone comparative statics)E y  < -)c cc ccccc cccccccc c! }  ( f`?6Slide Number Placeholder 3,  j(*    33___PPT10.01+YD' = @B DA' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(E%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(E%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(A%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(AQ%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(Qt%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(t%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(}%(+X=T %%,z%(  ,; ,  p0e0e ?Title 1(h  ) w%Sublattices also have nice properties & % & " ,  `0e0e ?,Content Placeholder 2" PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!G drs/shapexml.xml]]o8_iv7MtVJ*j:ˑCL ciO_Ri@r*5}|ϔGn,keљiDF-F?{8B.f(̹Ӊ9I-)sN}"?vBN#Ew'̼[I8e=,s#>|u\l=by(7f' n)JFEՌ,g/!c08A|p~r?&}pEhpס]|'q pZ/yzٕ# # N{ӡiax~={t⁨G*ȫ?$`j?8~_NM,7Ѕ: Op3S [ׁ'0ᮖ)浀#[6 mQ>K#!p]06b:AċDxt)QؤRxbc>>걡q3L̃f@'IGh4fy k2rEܛ[vş:@b)$R@,9 {B/ gؓ-3(zWyWo=.Q|__rvRMt1y]5oQVwTA1d3r? 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NM *6ƍj|#̠ݲcEjɰLlvNBfcfT,R-1kk*S+\m؋8# E-A"u~>~q1s8Λ#X_ۦܔ$ݿ7sr=Nȇֱ$A\9rt"1)8SbvtH*012d1L\Oy1EHqHp    N     0 f@ ?6Slide Number Placeholder 3,  j(*    33___PPT10.01+YDf' = @B D!' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0@%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0@k%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0k%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0!%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0!g%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0g%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0V%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0V%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(+#8T  %%4$(  4n 4  p 0e0e ?Title 1(h  ) 8Ln&-Convex Sets and Polyhedra@ ai  "" 4  0e0e ?,Content Placeholder 2"d^PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!~d drs/shapexml.xml]]o:_iCۣ-mi9Aꞕ8*Suc:WO_CjOsܳCN|s>ܞ\N, rs_huȁy܍zLʨjތ$Fþi("aU<"Ac%CkOO;ϣh(Ԓw<?鹗gs$_\ΐB6is, S\&|[]3-}Iq8 .qs/77e4Vw;#TE/G S~}yqz hܫ橺ҥ xu!Sx8{p֞+ž58 Ej%|XQp]8yKq^5ݠNb9/Mʗ;V1з z=CA?|<ы=LGI5%E<鮝>C$VHSyDι7=Twk#oI(Q\>j_tzls 9V->OzLdxgav<.oV+w tIo҅zh5зO[{.'_G @:֟'*gzPT/O|!5dQϨq[1s+fK26TB?p5l+:RO34ɲ~)4vw> wȗ>=͈~5d-'ř۞a oQ[`=B>a={YC9qa]+i %!RNkږ8.mE'XErA!qgA~SWvvS>(44ܲ>k2KpNCHyE9'Οkfu{WYyJ^KYWg u)ze֍VEఽ%'*U ߬A#B+M}R@P@8oT+`M;&o!uOv2[`b_hUs^rJΙt$oΘfL\bA J R>uv6`g暿m/9P4ۘQQ$-;b~p-XYk:zm6bPXg\MV[13jlɽy1f4}&ܼG;YNeC2ʭo%hRh{q؜f)9r~q霻N@Ľ.XaϬ炏 瞤-{=yOІy2)Ifm'D> @7D~&y~:ތBSxއ-zs_U3ʓ$A,Nh(^dSk|9"N]'yÑ^r'*8hh*&0W$pxs u/`ii425tEmsJejx)Rqa9Jd-EKw}󆝤Xv%-\&*Uhd2Im(}S[)AS~^̲9)/yiIq-3H*=B>a$@u:$C>]A\n JŜJO2؋%v!|J +~BΉÚn* ' |uRjЂϲU+7n[<&cR/gGJ2o;ӎZRV&Ѫ~,N|۩Rךڸ6.\  kyOJ`ԽԽG1Z2_Aj >[pi&lb5:TAźLdHA.?Owj5ͨXvQν2yWnjlCU U!/R՚uie7eu(s " Z~R)1HY0Hά"7?D*c'6!W-k ;]mkr4Fe$k3񡥁-WӷF8c6kF*jU3p"l(U(2@/ԑ|`lPK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!~d *drs/shapexml.xmlPK-!0KopNdrs/downrev.xmlPKSxB$@) )___PPT9 >Let Ln& be the family of all convex polyhedra in Rd defined by a system of bimonotone linear inequalities with 0-1 coefficients and integer right hand sides: P = {xRd : xi  xj bij (i,j)I; xi bi0 iIu ; - xj b0j jIl} where all bij , bi0 and b0j are integers dual network flow constraints (plus simple bounds on variables) These are integral Ln&-convex polyhedra [Murota & Shioura 2000] defined by a totally unimodular system, with integer RHS s all extreme points (if any) are integer Ln& |Zd is the family of all Ln&-convex sets [Fujishige & Murota 2000] Remark: Ln& |Bd = L|Bd by Birkhoff s (1937) Representation Theorem for finite distributive lattices JH +` @ " ? ; " , E c ccgo*cAkk A k  @ c gogc;, $g$$o$$$(E(,o,,,,c,,g,,o, ,g,,c,,g,,,,,,c,,k,,,0A04k4444c4448A8+B#style.visibility<*4%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4R%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4R%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4=%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4=%(+9T ))8)(  8( 8  p0e0e ?Title 1(h  ) dAbstract Convexity    ' 8   0e0e ?,Content Placeholder 2"~PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!sL :drs/shapexml.xml]nH_iq$N:>$RvWFqFsy 4,s\Lu7ؐ11M EU}U_}=Gp+owW{K&&  ۿ~_8d¡=2Ό,%c ~?x̅d~N} ]JI?rH;Q8{DSpC[l'C&(t@,a˻YZjN F3r==^R8;ViPb9 WM_JO~,Ϣkش]ox4A4MG{5;6='Zќ]AOX\2fqH)3&Qn¡˛t{C֑#STͼ4|{HWX?tdzUvi\rt<'=yyEܹ!H66fQBxl`>}Ig\6"!HwاfJFFI*m=;QwՌ{mS=q#IKƾ8WTw WWG1āNԗz84Ofi4Q|:Lgg~-ٛ$m"SCylg¶BW:+o3cQ Yb3#8su镈 7,7͞boz?4ڻn(}DV&ݽ R.ZcfWRYBnuY6B%F~F8p+o+nՔ@b"j=S +,2br"F}XruЭ]M_M_4w{#Z9kw7|:fn: m⺱qqșL;򇬺Рʺ]nic` @[hNrNX6)gRiYT ~|$k.&1S`<'d=z^n^B.w`ӛ 5I^s;ےѓEORTBڵZ0'׮Bbu: !1L喿VJgvҌe`tuDPQ<ss7`cjjx.( BR"H eQeXj?zY@w˝)/rxhG(t=t.4ٺ-V_&tΧ҄9 ykiNݝGSRW,u?wn[<C' Us;AQA-U;2w ykXQs%) *?"],Ǚ&ʪ1I"r=Ϙ;bQ87FcmS&n$zo+㌈RP\4zGjU5Voj][-t?}Z>%K39<;z'vƢZ F=ܶzT7. ;`3^Գ]6RnY읾kp/iu D,.*[L9NwQJ0Z̅;Ln1$.5154yW49dvTJC@m۴=¢@Fydnk)e+ `цUEI[ ArQۥXt9mBpp'U60{`r<;yO\?t36Iሚ26d\WP˜jL(v:-Gcɩ Vz%JhTdP~y,6c|E+=6ZQ^p|J71fQ]R\ɃI1\=3mJݧ@_tfp}N;Ue*x}ZUv :e'@׭ˎ2t[|]A/(ڲPK!9drs/downrev.xmlDn0xk+V҂Pʂh9>6. o_8ffԝ85iMiUkjdu0…=l֓ecEBBIˆ5ĮNSR9"\w2MԚPyW#~zRbK E zG|/ 69͟wU(,}Q}\\;e(PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!sL :*drs/shapexml.xmlPK-!9ndrs/downrev.xmlPKsw$@) )jb___PPT9D<   These 7 set systems, (Rd, L), (Zd, L|Zd), (Rd, LC), (Zd, LC|Zd), (Rd, Ln&), (Zd, Ln&|Zd) and (Bd, L), are instances of abstract convexity A closure system (or Moore family) is a set system (X, F) containing the empty and full sets (F and XF), and stable for (arbitrary) intersection (i.e., GF for all GF ) In a closure system every subset SX has a convex hull coF S coF S is the smallest set in F that contains S (i.e., the intersection of all FF that contain S) A convex structure (or aligned space) is a closure system (X, F) s.t. stable for nested unions (i.e., if GF is totally ordered by inclusion then GF) (3) implies that for all SX and all xcoF S there is a finite subset AS such that xcoF A (the closure operator of F is domain finite)  : w " =@ c@ F S " @ cAkccc A kccAkcA k  c  c   c  c $A$(k((c((c((((c(((,A,0k00c04A48k88c88c88k88c8+B#style.visibility<*8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8>%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8>{%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8{%(DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8$w%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8w%(+T +#BB<~(  <F <  @ABSCpDE^FQ>R"UVWX@@Ԕ? 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Our-ResultsPicture 6Our-Results.png 33___PPT10i.ʣ +D=' = @B + TS f ^ @LF (  L$ L # A? ?,$Known-ResultsContent Placeholder 4Known-Results.tif ) L  pV|0e0e ?Title 1(h#  ) 8Prelude Result  L f W|?6Slide Number Placeholder 3,  j(*    L A?? Our-ResultsPicture 6Our-Results.png L lY|PRRV?Rectangle 4"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!]Pdrs/downrev.xmlDj1ECxnfڅS 5y N!y\^匧kőbjW8,@76R+B#style.visibility<*L%(+8+0+L) +T   PP (  P$ P # A? ?,$Known-ResultsContent Placeholder 4Known-Results.tif )7 P  pZ|0e0e ?Title 1(h#  ) sMore Carathodory Numbers(   P f\|?6Slide Number Placeholder 3,  j(*    P A?? Our-ResultsPicture 6Our-Results.png P l]|PRRV?Rectangle 4"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDJ1EB!i3dڴB]*g:38INB{93_g ^AZf"1z]J +.^yϵ*T040MCO>wr6%]'[!v ngeK4kS󺔬xX@0 ?>Qgq^?ck5& el rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK ~` ,$D) R    33___PPT10.ʣ +Do' = @B D*' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(+8+0+P) + 9T /('(`T((  TZ T  p^|0e0e ?Title 1  ) DDetermining Carathodory Numbers for (Convex) Sublattice Convexities E D E  T f`|?6Slide Number Placeholder 3,  j(*   \% T  @0e0e ?,Content Placeholder 2"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! w 1drs/shapexml.xml]n6_ H8N1)2] 82($g\Yh}=8SɢlYuP(Ywyg߳xǿN0q|}г(fb¼@#׋qޏ],|`x?<YK0Mg1G\,١\a_Эh> e˹};g%O?ys,&<:a;c>+H}=%Bd&g|9}蜟}tu"Z< vԢvT/n'sl9t;Iԛ%6ϟɋzLin>4b |!`!-?".=Z:tgOb¡8k^t$2,#;wQ|_-٤hlnь͹nXXa7xx(dhx`k}|ިצƝjxO4@4PTyg϶&|zƣtXIonħQ 0v.HRp*tɌBp!}2žnL:$;Rxz|վ޺==.j/=;^\ybּWy;%lЛiDaJz iO>hlɰ[l.> vu5NZ}.<2tY`ZCw$^rWB2F}忩C9+٫:7 `7 mWg=!?Epz^*7QAɎ9SB9jD2Ng-mҶV2Sbyd]3a[s7vfw=ɓHh83F\MFEδg r"DzÄ'[pyNȓYOٖLY4qL *Z䰚4c TY8YW^g/+ ;Pn֥]Ų4|gBA*"vgP Eh c"m⡭uK_xUЏ#G%A¥[L2*x+xx5x+x+x7jZ؞= i~uXÞ _3g`ȵ #Pթ=L[^d۬190s`|cE%b7_ȍiu)Xa}ap"T)MS1rs4.99ԖemW9z4MY?2$)iH V/CQa\ŕ.ac Uȡo?JB8?czupP"Vb8 i2yk$o٪DTjGZF9]~u]NKDdHno٦B[;sJp8<  .\. l{edg$f8!UT&Ƃr-R` ?T/.Bv=hşVCW[ !^v>mxnO\L,dS]Ҥԫm^D4">$e\-hhIm&SS5JIfpСvAX'SC+cP5C;q[h,0CVHφuCoDnۂ`5BU`XUWRaU2}mhP|6hn?oSEu`kU?N\s{DbRw+Z`_3 k$E0 `:=qØkƑeD'& Shߍ3c;PK+J%Z>,},BakoG)b(Y%ov3iʨ{>2Jyb4wR麔,;XoҖ.qzѦnm j-T6yfXcȀ);0͔`Bi]8mXCذ۰(!2{Z̬vy/܀21rӹ@ =ع|EsaB-[nj# \gh\030V R"3MF+c`R &v@0KsD=8kw/o`homoĬZGeUlhm2Z1Er_vm_ il@$ G 2)S׮%ɇ7B#g}~AHԍe>n5U2vAtҪ6 (MkKbWh ,M_ަ}Ҫؒ-t / Ar_o՘EsЅ !Ngޒr A=~'8~TP#Gdj'L*# i+pF`wBo<;rժJ2ՌC?JfJ ”i)kj8 : ʲf9nq+ت vry;@#HMKlk:C#FA)!Iej, C /Mj`Ȟنɰ0lP  (Z4$,PX0#X0Y|DZGrF|wPwwD-X gxųW(H 8q^¦!V50E^UU@ll}~_O^i2VBЅԅ`o`Sq c%a@.9Ffo :fU2P͢幘 Y]?#IxJu9T)U͎0~׊XmQz@+L\e;GtRPjyvX8XQC)rJcO@k(a-A<рY TqVnyrޏ%j 6nf^+c6;2_nXdh6YYFl4H)1 n eܘ'+TKj9WU^C:`4S m9NݎNYm0LRH^P^;3l7T٢6l\=r1/J9{Lp{JI~h[E:3:'`xA>S>"4󯟿=6! ږj9d-$[V =80_Tws4HH:*8 G%س-vQ(>i$KIprGpi\IOipff&jӮxdIܲ^EN4s8hbn邎_߭WR%,;AJO^@:H / FwA_V]յՏffa?5ՁQK7PK!^mdrs/downrev.xmlDQK0Co.u`uwcԀ6Mg$ݿ7sr=^9;Y]uZDL41…#WWKݞχԊ q"PIkÖr`)Zn{9/RZ\~04puX>Bz'E/7D) (J3tzO1q@n4[\PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-! w 1*drs/shapexml.xmlPK-!^m drs/downrev.xmlPK  xH$@) ) $We determine the Carathodory Numbers for the convexities (Rd, L), (Zd, L|Zd), (Rd, LC), (Zd, LC|Zd) and (Bd, L) using upper bounds from algorithms for the convex hull membership problem: given finite subset A X and point xX, does xcoF A? that provide a concise membership certificate, i.e., a small subset B A and a short proof that xcoF B (or else a non-membership certificate) and lower bounds, from  bad instances We get matching lower and upper bounds max{d 2/4 , d } for (Bd, L) d 2/4 + d for (Rd, L) and (Zd, L|Zd) and small gaps (of lower order) between 2 d 2/4 and 2 d 2/4 + d +1, for (Rd, L C) and (Zd, L C |Zd) D ; @ |` @ " ' ( G   L  <cAkccc A kccAkcA k  c  c   c  c $A$(k((c((c((((c(((,A,0k00c04A48k88c88c88c8 +B#style.visibility<*T{%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T>%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T>%(+UT   pX (  X$ X # A? ?,$Known-ResultsContent Placeholder 4Known-Results.tif ) X  p0e0e ?Title 1#  ) NCarathodory Numbers for Ln&-ConvexitiesH( ai (  X fU?6Slide Number Placeholder 3,  j(*    X A?? Our-ResultsPicture 6Our-Results.png X lPRRV?Rectangle 4"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDJ1EB!i3dڴB]*g:38INB{93_g ^AZf"1z]J +.^yϵ*T040MCO>wr6%]'[!v ngeK4kS󺔬xX@0 ?>Qgq^?ck5& el rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK  ~` ,$D) R    33___PPT10.ʣ +Do' = @B D*' = @BA?%,( < +O%,( < +Da' =%(%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(+8+0+X) +;+T \(  \ \  p ?0e0e ?Title 17  ) VCarathodory Numbers for Ln&-Convexities (2)T, "ck",  \ f`??6Slide Number Placeholder 3,  j(*    \  ?0e0e ?,Content Placeholder 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!LYdrs/downrev.xmlDN0HHܨJD%ԭJ$JN<o~xn=z}h:q%[ qiu˵;ˤ`" 3mj!3TЄgR!~f{U 1ZjCN.$[ 7T Q/υsQS;+@cK=CZA ~\9-FK_PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!LYdrs/downrev.xmlPK IxH$@) ) Recall that an integral Ln&-convex polyhedron is defined as P = {xRd : xi  xj bij (i,j)I; xi bi0 iIu ; - xj b0j jIl} where all bij , bi0 and b0j are integers. Thus the convex hull coLn& A of a finite subset ARd (or Zd) is of the same form with at most d(d+1) inequalities and right-hand sides bij = maxaA(xi  xj) (ij), bi0 = maxaA xi and b0j = maxaA (- xi) Let V = {0, 1, & , d} and E = {(i,j) : ijV} A permutation p = (p(1),& , p(d+1)) of V covers E iff for every (i,j)E, i appears before j in p (i.e., i = p(k) and j = p(l) with k < l) A set P of permutations of V is a cover of E iff for every (i,j)E there is a permutation in P that covers it; it is a minimal cover of E iff it is a cover of E but for every pP, P \{p} is not a cover of E F; J ,`  I` -   ck!cAk %kkc A kkAk#'    bB      "  33  ___PPT10 .01+AGD ' = @B D= ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\9%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\9%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\<%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\< %(+ȩT ƙ`(  ` `  p@B0e0e ?Title 1n  ) VCarathodory Numbers for Ln&-Convexities (3)T, "ck",  ` fB?6Slide Number Placeholder 3,  j(*   + `  `B0e0e ?,Content Placeholder 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!A-j8drs/downrev.xmlDN0HH\uDTx6MXq4o͗ĉ|0zemK|Uf|~i{_J Q; L%/PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!A-j8drs/downrev.xmlPK xH$@) ) 4<Let V = {0, 1, & , d} and E = {(i,j) : ijV} A permutation p = (p(1),& , p(d+1)) of V covers E iff for every (i,j)E, i appears before j in p (i.e., i = p(k) and j = p(l) with k < l) A set P of permutations of V is a cover of E iff for every (i,j)E there is a permutation in P that covers it; it is a minimal cover of E iff it is a cover of E but for every pP, P \{p} is not a cover of E A problem in extremal combinatorics: Let g(d) = max{ |P| : P is a minimal cover of E} We prove: c(Rd, Ln&) = c(Zd, Ln&|Zd) = g(d) Open question: is there a simple formula for g(d)? 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Talk Outline Sublattices"Why be interested in sublattices?&Sublattices also have nice propertiesConvex SublatticesL♮-Convex Sets and PolyhedraAbstract Convexity Convex HullsIndependence and Invariants&Known Results for Related ConvexitiesOur (Current) ResultsPrelude ResultMore Carathéodory NumbersFDetermining Carathéodory Numbers for (Convex) Sublattice Convexities+Carathéodory Numbers for L♮-Convexities/Carathéodory Numbers for L♮-Convexities (2)/Carathéodory Numbers for L♮-Convexities (3)Some Open Problems…Some Open ProblemsSome Basic References  Fonts UsedThemeEmbedded OLE Servers Slide Titles$_D )Windows UserWindows User  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ACDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesyCurrent UserSummaryInformation(BPowerPoint Document(=hDocumentSummaryInformation8Root EntrydO)P< PicturesyCurrent UserJSummaryInformation(B&_DFabio TardellaFabio Tardella