Convex envelope of a function

Question

I am trying to calculate the convex envelope of a function, which is calculated by doing the Legendre transform twice (right?). Therefore, I was trying to calculare the convex envelope of $f(x)=(x^2-1)^2$ but I have some difficulties about it because even the Legendre transform of this function leads to problems so is this the method or should I do something else? Thanks

Answer 1

The convex envelope of $(x^2-1)^2$ is

$$\begin{align}|x|\ge1&\to(x^2-1)^2,\\|x|\le1&\to0.\end{align}$$

You can find it by considering the Legendre transform at every point of the curve and checking if it meets the curve elsewhere. If this is the case, the segment joining the intersections is also part of the convex envelope. In the case of smooth curves, the outline will be delimited by bitangent segments.