Page 68 - Textos de Matemática Vol. 39

58 CARLOS M. DA FONSECA
Finally, in 2004, the author presented a conjecture for the case of 3 × 3 block decompositions, after a careful analysis of all previous results.
Conjecture 4.2 ([14]). Under the conditions of Theorem 4.1, (I) (and, there- fore, (II)-(V)) is equivalent to the following system of linear inequalities:
π ≥max{π1,π2,π3,
r12 −ν1,r13 −ν1,r12 −ν2,r23 −ν2,r13 −ν3,r23 −ν3, π1 −ν2 +r23 −R12,π1 −ν3 +r23 −R13,
π2 −ν1 +r13 −R12,π2 −ν3 +r13 −R23,
π3 −ν1 +r12 −R13,π3 −ν2 +r12 −R23,
π1 +π2 −R12,π1 +π3 −R13,π2 +π3 −R23,
π1 +π2 +π3 −R12 −R13 −R23 } ,
ν ≥ max { ν1, ν2, ν3,
r12 −π1,r13 −π1,r12 −π2,r23 −π2,r13 −π3,r23 −π3, ν1 −π2 +r23 −R12,ν1 −π3 +r23 −R13,
ν2 −π1 +r13 −R12,ν2 −π3 +r13 −R23,
ν3 −π1 +r12 −R13,ν3 −π2 +r12 −R23,
ν1 +ν2 −R12,ν1 +ν3 −R13,ν2 +ν3 −R23,
ν1 +ν2 +ν3 −R12 −R13 −R23 } ,
π ≤ min { π1 +n2 +n3,n1 +π2 +n3,n1 +n2 +π3,
π1 +π2 +n3 +R12,π1 +n2 +π3 +R13,n1 +π2 +π3 +R23, π1 +π2 +π3 +R12 +R13 +R23 } ,
ν ≤ min { ν1 +n2 +n3,n1 +ν2 +n3,n1 +n2 +ν3,
ν1 +ν2 +n3 +R12,ν1 +n2 +ν3 +R13,n1 +ν2 +ν3 +R23, ν1 +ν2 +ν3 +R12 +R13 +R23 } ,
π+ν ≥max{π1 +ν1 +π2 +ν2 −R12,π1 +ν1 +π3 +ν3 −R13, π2 +ν2 +π3 +ν3 −R23,
π1 +ν1 +π2 +ν2 +π3 +ν3 −R12 −R13 −R23,
r12 +r13 −R23,r12 +r23 −R13,r13 +r23 −R12,
π1 +ν1 −π2 −ν2 +2r23 −R12, π1 +ν1 −π3 −ν3 +2r23 −R13, π2 +ν2 −π1 −ν1 +2r13 −R12, π2 +ν2 −π3 −ν3 +2r13 −R23, π3+ν3−π1−ν1+2r12−R13,π3+ν3−π2−ν2+2r12−R23 } ,