DUAL-EXTREME LATTICES among PERFECT LATTICES in DIMENSIONS n <= 8 
                                            Put on line: Feb. 11th, 2017. 

Recall that a lattice L is dual-extreme if 
\gamma'(L):=(\min L \times \min L^*)^1/2 achieves a local maximum on L, 
and that this is equivalent to L to be both 
(1) dual-perfect and (2) dual-eutactic, i.e. : 
(1) The orthogonal projections p_x onto (\mathbb R)x, x\in S(L)\cup S(L^*) 
    span E; 
(2) There exists a relation 
         \sum_{x\in S(L)} a_x p_x = \sum_{y\in S(L^*)} b_y p_y 
     with stricly positive coefficients a_x, b_y.  

Here (1) is a consequence of perfection. 

n < = 5: all perfect lattices are extreme and have (strongly) eutactic duals, 
         hence are dual-extreme (1+1+1+2+3=8 lattices). 

Next: test of s(L^*)>=n, then of rank (S^*)>=n. 

n=6. Survivors P6[i], i=1,2,3,5,7; all have a (strongly) eutactic dual, 
     hence are dual-extreme (5 lattices). 

n=7. Survivors P7[i], 1 2 4 15 20 27 33 ; 1,2,4,33: E_7,E_7^*,D_7,A_7. 
     i=15. 4 orbits on S, 1 on S^*. Dual-eutaxy needs negative coefficients. 
     i=20. 4 orbits on S, 1 on S^*. Dual-eutaxy needs negative coefficients. 
     i=27. 2 orbits on S, 1 on S^*. Unique relation: (4,23,216). Dual-extreme. 
     Dual-extreme: P7[i], i=1,2,4,27,33 (5 lattices). 
      
n=8. Survivors. p8d7[i], i=1,4,8,245,274,1175;  p8d3[i], i=3296. 
                i=1,245,1175: E_8,D_8,A_8. 
     i=4.    L and L^* strongly eutatic. Dual-extreme. 
     i=8.    3 orbits on S, 2 on S^*. L eutactic e.g., (1,4,2;4);
             L^* strongly semi-eutactic.  Dual-extreme. 
     i=274.  L and L^* strongly eutatic. Dual-extreme. 
     i=3296. 5 orbits on S, 1 on S^*. M=x1*M1+...+x5*M5-y*Md; 
             M[1,1]=0 <===> y=52*x1+45*x2+42*x3+4*x4+10*x5; 
             Giving y this value makes M=0. Dual-extreme, 
             with L eutactic, L^* NOT eutactic (7 lattices). 

Remark. All perfect, dual-extreme lattices up to n = 8 are extreme; moreover 
        their duals have a single orbit of minimal vectors except L=p8d7[8], 
        the dual of which is strongly semi-eutactic with two orbits.