DOMAINS and REDUCED DOMAINS 
                  of WELL-ROUNDED MINIMAL CLASSES 
                      in DIMENSIONS 2,3 and 4 

The notation of classes and corresponding matrices is that of my home page 
IV A "catalogue of perfect lattices", 4, file 
"Minimal classes in dimensions 2 to 4" above. 
The notation is S{n}x{r} for the classes (via their minimal vectors) 
and M{n}x{r} for the corresponding Gram matrices, where 
-- n is the dimension (n=2,3,or 4);
-- r is the parfection rank (n<=r<=n(n+1)/2); 
-- the letter x (a,b,c,or d) is used to distinguish classes with same n,r. 
-- the matrices depend on k=n(n+1)/2-r parameters (the perfection co-rank), 
   denoted by x1,\dots,xk (for x_1,\dots,x_k); we have 0<=k<=6. 

We disregard perfect classes (those with k=0). Thus among the 2+5+18 
well-rounded classes, we are left with 1+4+16 non-perfect classes. 

Domains are denoted by D{n}x{r} and reduced domains by R{n}x{r} 
Various properties can be read in XXXXXX 

               ************ DIMENSION 2 ************ 

D2a2 : -1<x1<1 ;
R2a2 : 0<=x1<1 . 

               ************ DIMENSION 3 ************ 

D3a5 : 0<x1<1 ;
R3a5 : 1/2<=x1<1 . 

D3a4 : x1,x2>-1, x1+x2<-1 ; (===> -1<x1,x2<0) 
R3a4 : -1<x1<=x2, x1+x2<-1 . (===> x1<-1/2,x2<0) 

D3b4 : -1<x1,x2,x1+x2<1 ; 
R3b4 : -1<x1<=x2<=0<-x1-x2<1 . 

D3a3 : -1<x1,x2,x3<1,x1+x2+x3>-2,-x1+x2+x3<2 and permutations ; 
R3a3 : -1<x1<=x2<=0,-1<x3<1,  x3>-2-x1-x2 . 

               ************ DIMENSION 4 ************ 

D4a9 : 0<x1<1 ; 
R4a9 : 0<x1<=1/2 . 

D4b9 : 0<x1<1 ; 
R4b9 =  D4b9 . 

D4a8 : x1,x2<1,x1+x2>1 ; (===> x1,x2>0) 
R4a8 : x1<=x2<1,x1+x2>1 . 

D4b8 : 0<x1,x2<1 ; 
R4b8 : 0<x1<=1/2,x1<=x2<1 . 

D4a7 : -<1x1<1,x2,x3,x1+x3<1,x2+x3>1,x1+x2+x3>1 ; (===> x1<x2,1-x3) 
R4a7 : x3<1,1-x3<x2<1,0<=x1<=2-2*x3-x2 

D4b7 : 0<x1,x2,x3<1 ; 
R4b7 : 0<x1<=x2<=1/2, x2<=x3<=1-x2 . 

D4c7 : 0<x1,x2<1,-1<x3<1,x1+x3,x2+x3<1,x1+x2+x3>0 ; 
R4c7 : 0<x1,x2<=1/2,-x1-x2<x3<1-x2 . 

D4d7 : -1<x1,x2,x1+x2,x1+x3,x2-x3<1 ; 
R4d7 : -1<x1<=0<=x2<=x3<=-x1 . 

D4a6 : x1+x2,x1+x3,x1-x4<-1,x1,x2,x3>-1 ; (===> x1,x2,x3<0<1+x1<x4<1) 
R4a6 : x1>-1,-1<x2<=x3<=-1-x1/2,2+x2+x3<=x4<=-x3 . 

D4b6 : x1,x2>-1,x3,x4<1,x1+x2<-1,x1+x3,x2+x4>-1,x3+x4>0,x1+x2+x3+x4<0 ; 
       (===> x1,x2<0,x3,x4>-1,x3 or x4>0) 
R4b6 : -1<x1<=x2<=-1-x1/2,-1-x1<x3<=-x1/2, 
       max(-x3,-1-x2)<x4<min(1,-(x1+x2+x3)) 
       (===> x1<=-2/3,x2<-1/2, -1<x3<1/2, x4>-1/2) 

D4c6 : -1<x1,x3,x4<1,0<x2<1,-1<x1+x3,x1+x4<1,-x2+-(x3-x4)>-2,x2+-(x1+x3+x4)>-1; 
R4c6 : -1<x1<=0,-1-x1<x3<=-x1/2,x3<=x4<=-x1-x3,x2<=1/2,-1-x1-x3-x4<x2<2+x3-x4 . 

D4d6 : -1<x1,x2,x3,x4,x1+x2,x1+x3,x2+x3,x2+x4<1, -1<x1+x2+x3+x4<1 ; 
R4d6 : -1<x1<=0<=x2<=x3<=-x1/2,-x2-x3<=x4<=-x1-x3. 

D4a5 : x1,x2,x3,x4,x5>-1,x1+x2+x4,x1+x3+x5>-2, x1+x2+x3,x1+x4+x5<-1, 
       x1+x2+x3+x4+x5<-2 ; 
R4a5 : -1<x1<=x2<=x3<=-2-x1-x2-x3,x4>=-1-i2,x5>=x2,x5>-2-x1-x3,
       x5<-1-i1-i4,x5<-2-2*i1-i4,x5<-3-2*x1-x2-x3-x4. 

D4b5 : x1,x2>-1,x1+x2<-1,-1<x3,x4,x5<1,x1+-(x3+x4)>-2,x2+-(x3+x5)>-2, 
       x3+x4+x5>-1,-(x1+x2)+-(x3+x4)>0 (===> x1,x2<0); 
R4b5 : -1<x1<=x2<=-1-x1/2,-1<x3<=0, x3<=x4<=-x3, -x4<=x5<=-x3, 
       x4>-2-x1-x3, x5>-2-x2-x3, x5<-x1-x2-x4. 

D4c5 : (a) -1<xi<1,-1<x1+x3,x2+x4<1, 
       (b) +-x1+-x2+-x5,+-x3+-x4+-x5>-2 (even nb of - signs), 
       (c)  +-x1+-...+-x5>-2 (+ signs at {1,3},{2,4},{5} or everywhere), 
       (d) +-x1+-...+-x5>-4 (+ signs at {1,4},{2,3},{1,2,5},{3,4,5} ; 
R4c5 : -1<x1<=0<=x2<=-x1,x1<=x3<=-x1 & x3>-1-x1,x1<=x4<=-x2/2,-1<x1<1; 
       set y4=|x1+x2+x3+x4| and z4=x1+x3-x2-x4; 
       x5>-2-x3-x4,x5>-2+y4,-2+z4<x5<2+z4,x5<4+x1-x2-x3+x4. 
       
D4a4 : (a) -1<xi<1 (i=1...,6), 
       (b) -2<xi+-xj+-xk<2, i<j<k,(i,j,k)=(1,2,4),(1,3,5),(2,3,6),(4,5,6), 
       (c) x1+-...+-x6>-3 with sets of + signs one of 
       {1,6},{2,5},{3,4},{1,2,4},{1,3,5},{2,3,6},{4,5,6},{1,2,3,4,5,6} ; 
R4a4 : -1<x1<=x2<=x3<=0, -1<x4,x5,x6<1 plus conditions (b),(c) above . 

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