yeah I did that, I was just curious why this didn’t work. However, Joaquim just showed me that indeed when v is not a variable, the constraint uses a matrix that is Positive-Semidefinite instead of Positive-Definite.
v'Σ*v = (x + y)'Σ*(x + y) = x'Σ*(x + y) + y'Σ*(x + y) = x'Σ*x + x'Σ*y + y'Σ*x + y'Σ*y
Since x'Σ*y == y'Σ*x, then:
v'Σ*v = (x + y)'Σ*(x + y) = x'Σ*(x + y) + y'Σ*(x + y) = x'Σ*x + x'Σ*y + y'Σ*x + y'Σ*y = x'Σ*x + 2x'Σ*y + y'Σ*y
And:
x'Σ*x + 2x'Σ*y + y'Σ*y == [x; y]' [Σ Σ;Σ Σ] [x; y]
At last:
[Σ Σ;Σ Σ] >= 0 and not > 0