= Level sets of a linear numerator over a radial quadratic
{title2=$f(x)=(a\cdot x)/(1+\|x\|^2)$}
For a fixed nonzero $a\in\mathbb R^n$, a nonzero <level set> $f(x)=h$ is the sphere
$$
\left\|x-\frac a{2h}\right\|^2=\frac{\|a\|^2}{4h^2}-1.
$$
This follows by completing the square after multiplying by the positive denominator. Nonzero levels exist only for $|h|\leq\|a\|/2$. The zero level is the <hyperplane> $a\cdot x=0$; the extreme levels shrink to points $x=\pm a/\|a\|$. The <Cauchy-Schwarz inequality> gives $|f(x)|\leq\|a\|\|x\|/(1+\|x\|^2)\leq\|a\|/2$, so these extrema are global. At them the <Hessian matrix> is respectively $-\|a\|I/2$ and $+\|a\|I/2$. In two dimensions the spheres are circles whose centres all lie on the axis spanned by $a$.
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