Level sets of a linear numerator over a radial quadratic ($f(x)=(a\cdot x)/(1+\Vert x{\Vert }^2)$)

For a fixed nonzero $a\in {\mathbb{R}}^n$, a nonzero level set $f(x)=h$ is the sphereThis follows by completing the square after multiplying by the positive denominator. Nonzero levels exist only for $|h|\le \Vert a\Vert /2$. The zero level is the hyperplane $a\cdot x=0$; the extreme levels shrink to points $x=\pm a/\Vert a\Vert$. The Cauchy-Schwarz inequality gives $|f(x)|\le \Vert a\Vert \Vert x\Vert /(1+\Vert x{\Vert }^2)\le \Vert a\Vert /2$, so these extrema are global. At them the Hessian matrix is respectively $-\Vert a\Vert I/2$ and $+\Vert a\Vert I/2$. In two dimensions the spheres are circles whose centres all lie on the axis spanned by $a$.