Consider the polynomial of degree $4$ in variable $r$

$$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$

The discriminant of this polynomial in $r$ is the following expression (obtained using Mathematica)

$$ \Delta := 16 x^2 y^2 \left( -x^6 + x^8 - 27 x^2 y^2 + 33 x^4 y^2 - 4 x^6 y^2 + 33 x^2 y^4 + 6 x^4 y^4 - y^6 - 4 x^2 y^6 + y^8 \right) $$

Consider the subset $D$ of $\mathbb{R}^2$ defined by the polynomial inequality $\frac{\Delta}{16 x^2 y^2} \le 0$

$$ D : -x^6 + x^8 - 27 x^2 y^2 - 4 x^6 y^2 - 4 x^2 y^6 + 33 x^4 y^2 + 33 x^2 y^4 + 6 x^4 y^4 - y^6 + y^8 \le 0 $$

I would like to prove that $D$ is convex. Any help would be appreciated. Thanks!