I am given the following set
$$\Omega_{1} = \{ (x,y) \in \mathbf{R}^{n} \times \mathbf{R}^{n} | \Vert y \Vert_{2}^{2} \leq 10 + x^{T}y - \Vert x \Vert^{2}_{2} \}$$
and would like to determine whether it is convex. I know that the set
$$\Omega = \{ x \in \mathbf{R}^{n} | \Vert x \Vert_{2}^{2} \leq 10 \}$$
is convex. Can I use this fact to simplify the problem since the intersection of half spaces is a convex set? Otherwise, do I need to consider for example
$$ z = \lambda t + (1- \lambda)s$$ $$ w = \lambda p + (1-\lambda)q$$
where $t,s,p,q \in \Omega$ and show that $z, w$ are also in $\Omega_{1}$?
$\Vert y \Vert_{2}^{2} \leq 10 + x^{T}y - \Vert x \Vert^{2}_{2}$
Is the same as
$\Vert y-x/2\Vert^2+\frac{3}{4}\Vert x\Vert^2\leq 10$
Change variable $u=y-x/2$, $u=\sqrt{\frac{3}{4}}x$.
$\Vert z\Vert^2+\Vert v\Vert^2=\Vert(u,v)\Vert^2\leq 10$
If you know how to show that $\Omega$ is convex, then $\Omega_1$ is the same, just in a space of higher dimension and after a linear change of coordinates.