What is the convex hull of this function?

Question

Find the convex hull of this function:

y(x) = $\frac{1}{1+x^2} $

I'm attempting to understand the idea of a convex hull and attempted one of these problems. I'm a little confused because isn't the convex hull of this function just the function itself since contains every point and all points joining said points? Do I have the concept wrong and is there something more to it?

Answer 1

Hint: The convex hull of a function is derived from the convex hull of the epigraph of this function. More precise

The convex hull (or convex envelope) of a function $f:\mathbb{R}\to\mathbb{R}$ is defined as \begin{align*} g(x)=\inf\{t|(x,t)\in\mathrm{conv}\{\mathrm{epi}\ f\}\} \end{align*}

See for instance this post. Can you continue?