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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?)

Thanks ;)

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1 Answer 1

If $f$ is convex and $F$ is its Legendre transform then $f(q)+F(p)\ge qp$, with equality in the case when $p$ and $q$ correspond to each other (i.e., $p=\nabla f(q)$, $q=\nabla F(p)$). This shows that the graph of $F$ lies above the tangent plane at any point, so that $F$ is indeed convex.

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