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Find the convex hull of this function:

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

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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?

M.N
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  • That function is not convex and so not a convex hull. You need to start with the definition of the convex hull of a function – Henry Oct 17 '22 at 21:35
  • Okay, Ive been putting more thought into it and realized the function can't possibly be the convex hull, but I'm thinking would a line y = 1 be a convex hull? Thats the smallest possible answer which encompasses this function – M.N Oct 17 '22 at 21:44
  • Could you give the definition of the convex hull of a function? – Giorgos Giapitzakis Oct 17 '22 at 21:48

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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?

Markus Scheuer
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