I know that the Legendre transform $F(p)$ of a given function $f(q)$ is well defined only if $f(q)$ has a definite convexity. Furthermore I know that I can take the Legendre transform twice to recover the original function $f(q)$.
So my guess is that also $F(p)$ should have a definite convexity (because I can take the Legendre transform twice), anyone can tell me if is it true or not?
If this is true, the convexity of $F(p)$ is the same of $f(q)$ or not? (The Legendre transform of a convex function is still convex or is concave or can be either convex or concave?)