Definition:Legendre Transform

Definition
Let $f \left({x}\right)$ be a strictly convex real function.

Let $p= f' \left({x}\right)$.

Let $f^* \left({p}\right) = -f \left({x \left({p}\right)}\right) + p x \left({p} \right)$.

The Legendre Transform on $x$ and $f$ is the mapping of the variable and function pair:
 * $\left({x, f \left({x} \right)} \right) \to \left({p, f^* \left({p}\right)}\right)$