Notes on polytope decomposition

The notes here are derived from these notes of Pace Nielsen.

Let [math]\displaystyle{ 1/4 \leq \varepsilon \leq 1/3 }[/math]. The problem here is to optimise the ratio J/I, where

[math]\displaystyle{ J := 3 \int\int_{x+y \leq 1-\varepsilon} (\int_0^{3/2-x-y} F(x,y,z)\ dz)^2\ dx dy }[/math]

[math]\displaystyle{ I := \int\int\int_{x+y+z \leq 3/2} F(x,y,z)^2\ dx dy dz }[/math]

for symmetric F supported on [math]\displaystyle{ R := \{ (x,y,z): x+y+z \leq 3/2 \} }[/math] subject to the vanishing marginal condition

[math]\displaystyle{ \int_0^{3/2-x-y} F(x,y,z)\ dz = 0 }[/math] when [math]\displaystyle{ x+y \geq 1+\varepsilon }[/math].

(Throughout these notes, [math]\displaystyle{ x,y,z }[/math] are understood to be non-negative.

We partition R six identical pieces, one of which is

[math]\displaystyle{ R_{xyz} := \{ (x,y,z) \in R: x+y \leq y+z \leq z+x \}. }[/math]

We then partition [math]\displaystyle{ R_{xyz} }[/math] further into 10 pieces:

Name	Inequalities	[math]\displaystyle{ x,y }[/math] inequalities	[math]\displaystyle{ z }[/math] inequalities
[math]\displaystyle{ R_{xyz} }[/math]	[math]\displaystyle{ x,y,z \geq 0; x+y+z \leq 3/2; x+y \leq y+z \leq z+x }[/math]	[math]\displaystyle{ x \geq y \geq 0; 2x+y \leq 3/2 }[/math]	[math]\displaystyle{ x \leq z \leq 3/2-x-y }[/math]
[math]\displaystyle{ A_{xyz} }[/math]	[math]\displaystyle{ z+x \leq 1-\varepsilon }[/math]	[math]\displaystyle{ 2x \leq 1-\varepsilon }[/math]	[math]\displaystyle{ z \leq 1-\varepsilon-x }[/math]
[math]\displaystyle{ B_{xyz} }[/math]	[math]\displaystyle{ y+z \leq 1-\varepsilon \leq z+x \leq 1+\varepsilon }[/math]	[math]\displaystyle{ 2x \leq 1+\varepsilon; x+y \leq 1-\varepsilon; y \leq 1/2+\varepsilon }[/math]	[math]\displaystyle{ 1-\varepsilon-x \leq z \leq \min(1+\varepsilon-x, 1-\varepsilon-y) }[/math]
[math]\displaystyle{ C_{xyz} }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon \leq y+z; z+x \leq 1+\varepsilon }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon; x \leq 1/2+\varepsilon }[/math]	[math]\displaystyle{ 1-\varepsilon-y \leq z \leq \min(1+\varepsilon-x }[/math]
[math]\displaystyle{ D_{xyz} }[/math]	[math]\displaystyle{ 1-\varepsilon \leq x+y; z+x \leq 1+\varepsilon }[/math]	[math]\displaystyle{ x+y \geq 1-\varepsilon; 2x \leq 1+\varepsilon }[/math]	[math]\displaystyle{ z \leq 1+\varepsilon-x }[/math]
[math]\displaystyle{ E_{xyz} }[/math]	[math]\displaystyle{ y+z \leq 1-\varepsilon; 1+\varepsilon \leq z+x }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon; y \leq 1/2-\varepsilon }[/math]	[math]\displaystyle{ 1+\varepsilon-x \leq z \leq 1-\varepsilon-y }[/math]
[math]\displaystyle{ S_{xyz} }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon \leq y+z \leq 1+\varepsilon \leq z+x }[/math] [math]\displaystyle{ z \leq 1/2+\varepsilon }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon; x \geq 1/2; y \geq 1/2-2\varepsilon }[/math]	[math]\displaystyle{ \max(1-\varepsilon-y,1+\varepsilon-x) \leq z \leq \min(1/2+\varepsilon,1+\varepsilon-y) |- | \lt math\gt T_{xyz} }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon \leq y+z \leq 1+\varepsilon \leq z+x }[/math] [math]\displaystyle{ z \geq 1/2+\varepsilon; x \geq 1/2-\varepsilon }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon; x \geq 1/2-\varepsilon }[/math]	[math]\displaystyle{ \max(1-\varepsilon-y,1+\varepsilon-x,1/2-\varepsilon) \leq z \leq 1+\varepsilon-y }[/math]
[math]\displaystyle{ U_{xyz} }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon \leq y+z \leq 1+\varepsilon \leq z+x }[/math] [math]\displaystyle{ x \leq 1/2-\varepsilon }[/math]	[math]\displaystyle{ x \leq 1/2-\varepsilon }[/math]	[math]\displaystyle{ \max(1-\varepsilon-y,1+\varepsilon-x) \leq z \leq 1+\varepsilon-y }[/math]
[math]\displaystyle{ G_{xyz} }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon; 1+\varepsilon \leq y+z }[/math]	[math]\displaystyle{ x+y \leq 1-\varepsilon; x \leq 1/2-\varepsilon }[/math]	[math]\displaystyle{ 1+\varepsilon-y \leq z }[/math]
[math]\displaystyle{ H_{xyz} }[/math]	[math]\displaystyle{ 1-\varepsilon \leq x+y; y+z \leq 1+\varepsilon \leq z+x }[/math]	[math]\displaystyle{ x+y \geq 1-\varepsilon }[/math]	[math]\displaystyle{ 1+\varepsilon-x \leq z \leq 1+\varepsilon-y }[/math]

Code

eps:=1/4;

AA1  := 1;
AA10 := 1045142938208869499191181851782774507/ 1250583557010555646667061395704693241;
AA2  := -(316190143410376180877743423449314379/ 245464443795540077818629001202004919);
AA3  := -(1891769024614429265911454167348900079/ 1447939476712835671464363817808062688);
AA4  := -(6645228840627884543963048543908513027/ 5289876174681743792060084416381005195);
AA5  := 339382475899332801872344573142712540/ 382315024741410006634197807400450109;
AA6  := 800533017763591087028889327164205386/ 1244356905396458217205005409940313975;
AA7  := 610227319942583521343573321987866901/ 1364882503730947175994072119119099314;
AA8  := 889358636648145697618963860699826673/ 1089059167005007839552742646367084883;
AA9  := 1129697701772130766129061623987627555/ 1057677648341105966172642014701620358;
BB1  := 369411148833963466416948486507827637/ 662523432040271503868922466512394931;
BB10 := 70036650704097962432270211642055587/ 764012880419651000309703978698201099;
BB2  := -(1560220310742393302871638104929376767/ 2533466855538622711494072790849524188);
BB3  := -(675291956550121180518604625359119746/ 628900966577543384705927335762397661);
BB4  := -(2529381695462699016828722243204952689/ 10158934785795031265990553206322976713);
BB5  := 301473130543012653915144253204794977/ 671648655006425274620020662289591135;
BB6  := -(72843967588099728726005521157785775/ 1201374210126871248100329027582696868);
BB7  := 143825707895572733498873732281838861/ 642792029730168691317286168208033538;
BB8  := 649120491616113171860687738354432093/ 1372868684761191246523483175495609470;
BB9  := 923204127144474133750324575031071970/ 894483138835109750336516244948229253;
CC1  := 504054606061804606346982706190041169/ 1482009696315178162834773520112425988;
CC10 := 6393539409569434525570984945744351/ 1536858786446366256984445263822713064;
CC2  := -(790669890434879790226924633515000611/ 1202927660436616925000852487643859754);
CC3  := -(583951140942993837488005058061406582/ 904691553133220160159156823559029957);
CC4  := -(22155783335807716384604672631876769/ 2382741655599610617339893240473178207);
CC5  := 861834403869460514116581559993754859/ 1483110658557996492977375659547372425;
CC6  := 1929146726486743959255980765537648/ 583834071458801022674364752868975235;
CC7  := 21780572077072279546180172035622479/ 1486260981903228631659801660491854709;
CC8  := 184535211848461345418175814435319135/ 407301027098512230853641880451245568;
CC9  := 232080145073024378344663228214923163/ 453834445204584035710168200740458909;
EE4  := 5472538654920722429742789691613843676/ 974479232954197975340589995394266455;
EE6  := 927222828149050158923800771583485723/ 2427346765941624985384346616239603038;
EE7  := 250517508328910949894678249652185722/ 927265216411134102504733935734045705;
EE9  := 1240108547818177956234040673419197568/ 2268677602592314368751160380697174029;
HH9  := 9405819102419059244834716575963924175/ 811372333735457494738136754843943653;
SS10 := -(1432088304215654669021352650864060725/ 1173831359938549301293098635807566287);
SS3  := -(11612115950284073971150222762767416620/ 1189758425639212572034398845328452639);
SS4  := 103937977558232440508940501348834518/ 968720338025380091790884725502334715;
SS6  := 214224468539246347682035898225657185/ 578340038314441580717115396074382778;
SS7  := 16011015326505338708151388841891253041/ 1481970237969351533821283886719874867;
SS8  := -(2152329004242745879844194442185193763/ 1112441007795743584614598777366749922);
SS9  := 8652296470217515560135397830496043201/1393032063270399805786566196554189095;
TT4  := 2724262739331687731618153452386489771/7777163575787859138926611143971853064;
TT7  := -(3533612330801100162685856352893683/318742786972619370533192728831094789);
TT8  := -(8634152014937802133186289083541939/326132136552676474614615351931787892);
TT9  := 639258673765815764176758921945001459/1023652233362766137377498777586011863;
UU16 := 0;
UU18 := -(1036177365908403576222625465669165224191/1396118752614173365815190752515315262);
UU2  := 683163754447358515357759652655917657307/899770442732486890915132953316493212;
UU20 := 3977430161340892036145546371180618658240/942423718900792196615855892814714641;
UU21 := 4369967465028547394227392689218793956/1118925727402664937850908867587807095;
UU22 := -(6493264317462327139909085594207259201/2656583931468774802084917498666106861);
UU23 := 208533806588523947204108028241851658601/1593749576650909843879153107097169586;
UU24 := -(2586867413257547105405021630653379594931/1017361494191501491128242008483670506);
UU25 := -(46075356336908979453982139912989909769/2186736965136790998934152365709299888);
UU26 := 21583120699474906684881271067473540892/2327719037190114284507554697536132109;
UU27 := 0;
UU28 := 0;
UU29 := -(4493884950736564857307858442065834123411/2487469455179284947187381390582168137);
UU3  := -(912131634231853063945020626889013945/1117765710674185487041664983420906397);
UU30 := 385418773588866232850515087462213507/1797985734299399330339559894335414707;
UU31 := -(60846714782262228976526525072918278/1419255270556600115075698318707631525);
UU4  := 38086532340155117976291869658511569399/429342259347747976339594153498728242;
UU5  := 9900518239375432526911040812620204965/1202954557712616339652022357371972894;
UU6  := -(1363826261161475275558885344800775294047/1385770492279731028804032285608964254);
UU7  := 0;
UU8  := -(3986739775635725156458528972312106107630/1340165793339507490854268037177642059);
UU9  := -(2967616495233435231968510933124363413/610343298456544672866212720080088296);

FA := AA1 + AA2*x + AA8*x^2 + AA3*y + AA5*x*y + AA9*y^2 + AA4*z + AA6*x*z + AA7*y*z + AA10*z^2;
FB := BB1 + BB2*x + BB8*x^2 + BB3*y + BB5*x*y + BB9*y^2 + BB4*z + BB6*x*z + BB7*y*z + BB10*z^2;
FC := CC1 + CC2*x + CC8*x^2 + CC3*y + CC5*x*y + CC9*y^2 + CC4*z + CC6*x*z + CC7*y*z + CC10*z^2;
FD := 0;
FE := -((3*EE4)/4) - (9*EE7)/32 + (3*EE9)/16 - (9*SS10)/16 - (3*SS3)/8 - ( 3*SS9)/8 + (-((3*EE6)/4) + SS3/2 + (3*SS7)/8 - (3*SS8)/4 + SS9/4)*x + SS8*x^2 + (2*EE4 + (3*EE7)/4 - EE9 + SS3 - 2*SS4 - (3*SS7)/4 + SS9)*y + (2*EE6 - 2*SS6 + 2*SS8 + (2*SS9)/3)*x*y + EE9*y^2 + EE4*z + EE6*x*z + EE7*y*z + (EE7/2 - EE9/3 + SS10 - SS7/2 + SS9/3)*z^2;
FS := -((9*SS10)/16) - (3*SS3)/8 - (3*SS4)/4 - (9*SS7)/32 - (3*SS9)/16 + (SS3/2 - (3*SS6)/4 + (3*SS7)/8 - (3*SS8)/4 + SS9/4)*x + SS8*x^2 + SS3*y + (2*SS8 + (2*SS9)/3)*x*y + SS9*y^2 + SS4*z + SS6*x*z + SS7*y*z + SS10*z^2;
FT := -((3*TT4)/2) - (9*TT7)/8 + (3*TT9)/4 + (TT4 + (3*TT7)/4 - (3*TT8)/2 - TT9/2)*x + TT8*x^2 + (2*TT4 + (3*TT7)/2 - 2*TT9)*y + (2*TT8 + (2*TT9)/3)*x *y + TT9*y^2 + TT4*z + (TT7/2 + TT8 - TT9/3)*x*z + TT7*y*z + (TT7/2 - TT9/3)*z^2;
FU := -((125*UU16)/256) - UU2/16 + UU20/512 + (125*UU22)/128 + (5*UU23)/8192 + (5*UU24)/4096 + (125*UU26)/2048 - (625*UU28)/1024 + UU29/1280 - (5*UU3)/16 + (625*UU31)/256 - (5*UU4)/8 - (5*UU5)/256 - (5*UU6)/128 - (25*UU7)/64 + UU2*x +UU8*x^2 + UU20*x^3 + UU29*x^4 + UU3*y + UU5*x*y + (-((225*UU16)/26) + (5*UU18)/2 - (24*UU2)/5 + (333*UU20)/ 260 + (225*UU21)/26 + (225*UU22)/13 - (423*UU23)/832 + (1089*UU24)/416 + (45*UU25)/16 - (105*UU26)/16 - (1125*UU28)/104 - (3*UU29)/25 + (72*UU3)/13 + (225*UU30)/26 + (1125*UU31)/26 - (144*UU4)/13 - (3*UU5)/2 - 3*UU6 - (90*UU7)/13 - (8*UU8)/5 + (120*UU9)/13)*x^2*y + UU23*x^3*y + UU9*y^2 + ((165*UU16)/26 - (27*UU20)/52 - (165*UU21)/26 - (165*UU22)/13 - (75*UU23)/416 - (15*UU24)/52 - (69*UU25)/80 - (87*UU26)/40 + (825*UU28)/104 - (27*UU29)/25 - (264*UU3)/65 - (165*UU30)/26 - (825*UU31)/26 + (528*UU4)/65 - (3*UU5)/5 + (6*UU6)/5 + (66*UU7)/13 - (88*UU9)/13)*x*y^2 + (-3*UU18 + (81*UU23)/80 - (81*UU24)/40 - (147*UU25)/40 + (147*UU26)/10 + (108*UU29)/25)*x^2*y^2 + UU21*y^3 + UU25*x*y^3 + UU30*y^4 + UU4*z + UU6*x*z + ((225*UU16)/52 - (5*UU18)/4 - (12*UU2)/5 - (567*UU20)/520 - (225*UU21)/52 - (225*UU22)/26 + (189*UU23)/1664 - (1323*UU24)/832 - (45*UU25)/32 + (255*UU26)/32 + (1125*UU28)/208 - (3*UU29)/50 - (36*UU3)/13 - (225*UU30)/52 - (1125*UU31)/52 + (72*UU4)/13 - (3*UU5)/4 - (3*UU6)/2 + (45*UU7)/13 - (4*UU8)/5 - (60*UU9)/13)*x^2*z + UU24*x^3*z + UU7*y*z + ((3*UU16)/2 + (3*UU21)/4 - 3*UU22 + (3*UU25)/16 - (3*UU26)/4 - (15*UU27)/16 + (15*UU28)/8 + (3*UU30)/2 - (15*UU31)/2)*y^2*z + UU27*y^3*z + ((5*UU16)/13 - (3*UU20)/208 - (15*UU21)/208 - (105*UU22)/52 + (5*UU23)/3328 - (35*UU24)/1664 - (3*UU25)/128 + (3*UU26)/32 + (25*UU28)/52 + (2*UU3)/13 - (15*UU30)/208 - (725*UU31)/208 - (4*UU4)/13 + (4*UU7)/13 - UU9/13)*z^2 + (-((15*UU16)/13) + (3*UU20)/13 + (15*UU21)/13 + (30*UU22)/13 - (63*UU23)/2080 + (363*UU24)/1040 + (3*UU25)/8 - (27*UU26)/8 - (75*UU28)/52 + (48*UU3)/65 + (15*UU30)/13 + (75*UU31)/13 - (96*UU4)/65 + UU5/5 - (2*UU6)/5 - (12*UU7)/13 + (16*UU9)/13)*x*z^2 + UU18*x^2*z^2 + UU16*y*z^2 + ((3*UU27)/4 + (3*UU28)/2 - (3*UU30)/5 - 3*UU31)*y^2*z^2 + UU22*z^3 + UU26*x*z^3 + UU28*y*z^3 + UU31*z^4;
FG:= -((9*UU2)/4) + (27*UU20)/64 + (513*UU23)/2048 + (27*UU24)/1024 - (243*UU25)/512 + (2097*UU26)/512 + (153*UU29)/160 - (45*UU5)/64 - (45*UU6)/32 + (6*UU2 - (99*UU23)/256 + (99*UU24)/128 + (99*UU25)/64 - (771*UU26)/64 - (33*UU29)/20 + (15*UU5)/8 + (15*UU6)/4)*x + (-3*UU2 - (27*UU20)/16 - (117*UU23)/512 - (423*UU24)/256 - (153*UU25)/128 + (987*UU26)/128 + (33*UU29)/40 - (15*UU5)/16 - (15*UU6)/8)*x^2 + (UU20 + UU23/4 + (3*UU24)/4 + UU25/4 - UU26 - (8*UU29)/5)*x^3 + UU29*x^4 + (-((675*UU16)/104) + 6*UU2 - (81*UU20)/208 + (675*UU21)/104 + (675*UU22)/52 - (8919*UU23)/16640 + (4869*UU24)/8320 + (369*UU25)/160 - (4827*UU26)/320 - (3375*UU28)/416 - (123*UU29)/50 + (54*UU3)/13 + (675*UU30)/104 + (3375*UU31)/104 - (108*UU4)/13 + (15*UU5)/8 + (15*UU6)/4 - (135*UU7)/26 + (90*UU9)/13)*y + ((225*UU16)/13 - 8*UU2 - (45*UU20)/13 - (225*UU21)/13 - (450*UU22)/13 + (1929*UU23)/4160 - (10929*UU24)/2080 - (453*UU25)/80 + (2437*UU26)/80 + (1125*UU28)/52 + UU29/25 - (144*UU3)/13 - (225*UU30)/13 - (1125*UU31)/13 + (288*UU4)/13 - (5*UU5)/2 - 5*UU6 + (180*UU7)/13 - (240*UU9)/13)*x*y + (-((225*UU16)/26) + (207*UU20)/52 + (225*UU21)/26 + (225*UU22)/13 - (441*UU23)/1040 + (6057*UU24)/1040 + (243*UU25)/80 - (243*UU26)/20 - (1125*UU28)/104 + (9*UU29)/25 + (72*UU3)/13 + (225*UU30)/26 + (1125*UU31)/26 - (144*UU4)/13 - (90*UU7)/13 + (120*UU9)/13)*x^2*y + ((3*UU23)/5 - (6*UU24)/5 - (2*UU25)/5 + (8*UU26)/5 + (64*UU29)/25)*x^3*y + ((225*UU16)/26 - 3*UU2 - (243*UU20)/208 - (225*UU21)/26 - (225*UU22)/13 - (981*UU23)/33280 - (23319*UU24)/16640 - (1773*UU25)/640 + (8967*UU26)/640 + (1125*UU28)/104 + (381*UU29)/200 - (72*UU3)/13 - (225*UU30)/26 - (1125*UU31)/26 + (144*UU4)/13 - (15*UU5)/16 - (15*UU6)/8 + (90*UU7)/13 - (120*UU9)/13)*y^2 + (-((375*UU16)/26) + (189*UU20)/52 + (375*UU21)/26 + (375*UU22)/13 - (579*UU23)/1040 + (5883*UU24)/1040 + (417*UU25)/80 - (417*UU26)/20 - (1875*UU28)/104 - (9*UU29)/25 + (120*UU3)/13 + (375*UU30)/26 + (1875*UU31)/26 - (240*UU4)/13 - (150*UU7)/13 + (200*UU9)/13)*x*y^2 + ((27*UU23)/20 - (27*UU24)/10 - (33*UU25)/20 + (33*UU26)/5 + (54*UU29)/25)*x^2*y^2 + (-((25*UU16)/13) + (23*UU20)/26 + (25*UU21)/13 + (50*UU22)/13 + (107*UU23)/520 + (361*UU24)/520 + (39*UU25)/40 - (39*UU26)/10 - (125*UU28)/52 - (46*UU29)/25 + (16*UU3)/13 + (25*UU30)/13 + (125*UU31)/13 - (32*UU4)/13 - (20*UU7)/13 + (80*UU9)/39)*y^3 + ((3*UU23)/5 - (6*UU24)/5 - (7*UU25)/5 + (28*UU26)/5 + (64*UU29)/25)*x*y^3 +UU29*y^4 + ((675*UU16)/208 + 3*UU2 - (621*UU20)/416 - (675*UU21)/208 - (675*UU22)/104 - (20799*UU23)/33280 - (10251*UU24)/16640 + (99*UU25)/320 - (2667*UU26)/640 + (3375*UU28)/832 - (129*UU29)/50 - (27*UU3)/13 - (675*UU30)/208 - (3375*UU31)/208 + (54*UU4)/13 + (15*UU5)/16 + (15*UU6)/8 + (135*UU7)/52 - (45*UU9)/13)*z + (-((225*UU16)/26) - 4*UU2 + (45*UU20)/26 + (225*UU21)/26 + (225*UU22)/13 + (5169*UU23)/8320 + (3831*UU24)/4160 - (93*UU25)/160 + (997*UU26)/160 - (1125*UU28)/104 + (181*UU29)/50 + (72*UU3)/13 + (225*UU30)/26 + (1125*UU31)/26 - (144*UU4)/13 - (5*UU5)/4 - (5*UU6)/2 - (90*UU7)/13 + (120*UU9)/13)*x*z + ((225*UU16)/52 + (27*UU20)/104 - (225*UU21)/52 - (225*UU22)/26 + (909*UU23)/2080 - (1143*UU24)/2080 + (63*UU25)/160 - (63*UU26)/40 + (1125*UU28)/208 - (81*UU29)/50 - (36*UU3)/13 - (225*UU30)/52 - (1125*UU31)/52 + (72*UU4)/13 + (45*UU7)/13 - (60*UU9)/13)*x^2*z + (-(UU23/5) + (2*UU24)/5 - UU25/5 + (4*UU26)/5 + (32*UU29)/25)*x^3*z + (-4*UU2 + (9*UU20)/4 + (105*UU23)/128 + (75*UU24)/64 - (15*UU25)/32 + (185*UU26)/32 + (47*UU29)/10 - (5*UU5)/4 - (5*UU6)/2)*y*z + (-((75*UU16)/52) - (9*UU20)/104 + (75*UU21)/52 + (75*UU22)/26 + (633*UU23)/2080 - (1491*UU24)/2080 + (51*UU25)/160 - (51*UU26)/40 - (375*UU28)/208 - (117*UU29)/50 + (12*UU3)/13 + (75*UU30)/52 + (375*UU31)/52 - (24*UU4)/13 - (15*UU7)/13 + (20*UU9)/13)*y^2*z + (-(UU23/5) + (2*UU24)/5 - UU25/5 + (4*UU26)/5 + (32*UU29)/25)*y^3*z + (-((225*UU16)/52) - UU2 + (297*UU20)/208 + (225*UU21)/52 + (225*UU22)/26 + (21297*UU23)/33280 + (8403*UU24)/16640 - (39*UU25)/640 + (781*UU26)/640 - (1125*UU28)/208 + (523*UU29)/200 + (36*UU3)/13 + (225*UU30)/52 + (1125*UU31)/52 - (72*UU4)/13 - (5*UU5)/16 - (5*UU6)/8 - (45*UU7)/13 + (60*UU9)/13)*z^2 + ((75*UU16)/13 - (15*UU20)/13 - (75*UU21)/13 - (150*UU22)/13 - (243*UU23)/520 - (33*UU24)/65 + (3*UU25)/5 - (12*UU26)/5 + (375*UU28)/52 - (66*UU29)/25 - (48*UU3)/13 - (75*UU30)/13 - (375*UU31)/13 + (96*UU4)/13 + (60*UU7)/13 - (80*UU9)/13)*x*z^2 + (-((9*UU23)/80) + (9*UU24)/40 - (27*UU25)/40 + (27*UU26)/10 + (18*UU29)/25)*x^2*z^2 + ((75*UU16)/26 - (69*UU20)/52 - (75*UU21)/26 - (75*UU22)/13 - (111*UU23)/208 - (123*UU24)/208 - (9*UU25)/16 + (9*UU26)/4 + (375*UU28)/104 - 3*UU29 - (24*UU3)/13 - (75*UU30)/26 - (375*UU31)/26 + (48*UU4)/13 + (30*UU7)/13 - (40*UU9)/13)*y*z^2 + (-((9*UU23)/80) + (9*UU24)/40 + (3*UU25)/40 - (3*UU26)/10 + (18*UU29)/25)*y^2*z^2 + ((75*UU16)/52 - (43*UU20)/104 - (75*UU21)/52 - (75*UU22)/26 - (659*UU23)/2080 + (243*UU24)/2080 + (37*UU25)/160 - (37*UU26)/40 + (375*UU28)/208 - (59*UU29)/50 - (12*UU3)/13 - (75*UU30)/52 - (375*UU31)/52 + (24*UU4)/13 + (15*UU7)/13 - (20*UU9)/13)*z^3 + ((3*UU23)/20 - (3*UU24)/10 - (3*UU25)/5 + (12*UU26)/5 + (16*UU29)/25)*x*z^3 + ((3*UU23)/20 - (3*UU24)/10 - UU25/10 + (2*UU26)/5 + (16*UU29)/25)*y*z^3 + (UU23/16 - UU24/8 - UU25/8 + UU26/2 + UU29/5)*z^4;
FH:= (3*HH9)/8 - (9*SS10)/8 - (3*SS4)/2 - (9*SS7)/16 + (-((3*HH9)/4) + (3*SS10)/4 + SS4 - (3*SS6)/2 + (9*SS7)/8)*x + (HH9/3 + SS6 - SS7/2)*x^2 + (-((3*HH9)/2) + (3*SS10)/2 + 2*SS4 + (3*SS7)/4)*y + ((4*HH9)/3 + 2*SS6 - SS7)*x*y + HH9*y^2 + (-(HH9/4) - (3*SS10)/4 + SS4 + (3*SS7)/8)*z + (HH9/3 + SS10 + SS6 - SS7/2)*x*z + ((2*HH9)/3 + 2*SS10)*y*z + SS10*z^2;

Axyz:=Heaviside((1-eps)-(z+x));
Dxyz:=Heaviside(1+eps-(z+x))*Heaviside((x+y)-(1-eps));
ABCDxyz:=Heaviside(1+eps-(z+x));
ABExyz:=Heaviside((1-eps)-(y+z));
Exyz:=Heaviside((z+x)-(1+eps))*Heaviside((1-eps)-(y+z));
BExyz := ABExyz - Axyz;
Bxyz:=ABExyz-Axyz-Exyz;
Gxyz:=Heaviside(y+z-(1+eps));
DHxyz:=Heaviside(x+y-(1-eps));
CFxyz:=Heaviside((y+z)-(1-eps))-DHxyz-Gxyz;
Hxyz:=DHxyz-Dxyz;
Fxyz:=CFxyz-Cxyz;

IA := int(int(int(FA*FA*Axyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2);
IB := int(int(int(FB*FB*BExyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2) - int(int(int(FB*FB,y=0..1-eps-z),x=1+eps-z..z),z=1/2+eps/2..1-eps);
IC := int(int(int(FC*FC*ABCDxyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2) - int(int(int(FC*FC*Axyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2) - int(int(int(FC*FC*BExyz,z=x..(3/2-y- x)),x=y..(3/2-y)/2),y=0..1/2) - int(int(int(FC*FC*Dxyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2) + int(int(int(FC*FC,y=0..1-eps-z),x=1+eps-z..z),z=1/2+eps/2..1-eps);
ID := int(int(int(FD*FD*Dxyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2);
IE := int(int(int(FE*FE,y=0..1-eps-z),x=1+eps-z..z),z=1/2+eps/2..1-eps);
IS := int(int(int(FS*FS,x=1+eps-z..(1-eps-y)),z=y+2*eps..1/2+eps),y=1/2-3*eps/2..1/2-eps)+int(int(int(FS*FS,x=1+eps-z..(1-eps-y)),z=1-eps-y..1/2+eps),y=0..1/2-3*eps/2);
IT := int(int(int(FT*FT,y=0..3/2-x-z),x=1/2-eps..3/2-z),z=1/2+2*eps..1+eps)+int(int(int(FT*FT,y=0..3/2-x-z),x=1+eps-z..3/2-z),z=1/2+eps..1/2+2*eps);
IU := int(int(int(FU*FU,z=1+eps-x..1+eps-y),y=0..x),x=0..1/2-eps);
IG := int(int(int(FG*FG*Gxyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2);
IH := int(int(int(FH*FH*Hxyz,z=x..(3/2-y-x)),x=y..(3/2-y)/2),y=0..1/2);

Isum := 6*(IA+IB+IC+ID+IE+IS+IT+IU+IG+IH);
#                                  0.03062470572

FAxyz := FA;
FAyzx := subs( {x=y,y=z,z=x}, FA );
FAzyx := subs( {x=z,z=x}, FA );
FBxyz := FB;
FByzx := subs( {x=y,y=z,z=x}, FB );
FBzyx := subs( {x=z,z=x}, FB );
FCxyz := FC;
FCyzx := subs( {x=y,y=z,z=x}, FC );
FCzyx := subs( {x=z,z=x}, FC );
FExyz := FE;
FEyzx := subs( {x=y,y=z,z=x}, FE );
FEzyx := subs( {x=z,z=x}, FE );
FSxyz := FS;
FSyzx := subs( {x=y,y=z,z=x}, FS );
FSzyx := subs( {x=z,z=x}, FS );
FTxyz := FT;
FTyzx := subs( {x=y,y=z,z=x}, FT );
FTzyx := subs( {x=z,z=x}, FT );
FUxyz := FU;
FUyzx := subs( {x=y,y=z,z=x}, FU );
FUzyx := subs( {x=z,z=x}, FU );
FGxyz := FG;
FGyzx := subs( {x=y,y=z,z=x}, FG );
FGzyx := subs( {x=z,z=x}, FG );
FHxyz := FH;
FHyzx := subs( {x=y,y=z,z=x}, FH );
FHzyx := subs( {x=z,z=x}, FH ); 

J1func := int(FAyzx, z=0..y) + int(FAzyx, z=y..x) + int(FAxyz, z=x..1-eps-x) + int(FBxyz, z=1-eps-x..1-eps-y) + int(FCxyz, z=1-eps-y..1+eps-x) + int(FUxyz, z=1+eps-x..1+eps-y) + int(FGxyz, z=1+eps-y..3/2-x-y);
J1 := int(int(J1func*J1func, y=0..x), x=0..1/2-eps);

J2func := int(FAyzx, z=0..y) + int(FAzyx, z=y..x) + int(FAxyz, z=x..1-eps-x) + int(FBxyz, z=1-eps-x..1-eps-y) + int(FCxyz, z=1-eps-y..3/2-x-y);
J2 := int(int(J2func*J2func, y=1/2-eps..x),x=1/2-eps..1/2-eps/2);

J3func := int(FAyzx, z=0..y) + int(FAzyx, z=y..x) + int(FAxyz, z=x..1-eps-x) + int(FBxyz, z=1-eps-x..1-eps-y) + int(FCxyz, z=1-eps-y..1+eps-x) + int(FTxyz, z=1+eps-x..3/2-x-y);
J3 := int(int(J3func*J3func, y=0..1/2-eps),x=1/2-eps..1/2-eps/2);

J4func := int(FAyzx, z=0..y) + int(FAzyx, z=y..1-eps-x) + int(FBzyx, z=1-eps-x..x) + int(FBxyz, z=x..1-eps-y) + int(FCxyz, z=1-eps-y..3/2-x-y);
J4 := int(int(J4func*J4func, y=1/2-eps..1-eps-x), x=1/2-eps/2..1/2);

J5func := int(FAyzx, z=0..y) + int(FAzyx, z=y..1-eps-x) + int(FBzyx, z=1-eps-x..x) + int(FBxyz, z=x..1-eps-y) + int(FCxyz, z=1-eps-y..1+eps-x) + int(FTxyz, z=1+eps-x..3/2-x-y);
J5 := int(int(J5func*J5func,y=0..1/2-eps),x=1/2-eps/2..1/2);

J6func := int(FAyzx, z=0..y) + int(FAzyx, z=y..1-eps-x) + int(FBzyx, z=1-eps-x..x) + int(FBxyz, z=x..1-eps-y) + int(FCxyz, z=1-eps-y..1+eps-x) + int(FSxyz, z=1+eps-x..1/2+eps) + int(FTxyz, z=1/2+eps..3/2-x-y);
J6 := int(int(J6func*J6func,y=x-2*eps..1-eps-x),x=2*eps..1/2+eps/2) + int(int(J6func*J6func,y=0..1-eps-x),x=1/2..2*eps);

J7func := int(FAyzx, z=0..y) + int(FAzyx, z=y..1-eps-x) + int(FBzyx, z=1-eps-x..x) + int(FBxyz, z=x..1+eps-x) + int(FExyz, z=1+eps-x..1-eps-y) + int(FSxyz, z=1-eps-y..1/2+eps) + int(FTxyz, z=1/2+eps..3/2-x-y);
J7 := int(int(J7func*J7func,y=0..x-2*eps),x=2*eps..1/2+eps/2);

J8func := int(FAyzx, z=0..y) + int(FAzyx, z=y..1-eps-x) + int(FBzyx, z=1-eps-x..1+eps-x) + int(FEzyx, z=1+eps-x..x) + int(FExyz, z=x..1-eps-y) + int(FSxyz, z=1-eps-y..1/2+eps) + int(FTxyz, z=1/2+eps..3/2-x-y);
J8 := int(int(J8func*J8func,y=0..1-eps-x),x=1/2+eps/2..1-eps);

Jsum := 6*(J1+J2+J3+J4+J5+J6+J7+J8);
#                                  0.06125062172