Legendre Transform of Strictly Convex Real Function is Strictly Convex

Theorem
Let $\map f x$ be a strictly convex real function.

Then the function $\map {f^*} p$ acquired through the Legendre Transform is also strictly convex.

Proof
Since $\map f x$ is real and strictly convex, $\map {f'} x$ is strictly increasing.

Then:


 * $\map {f''} x > 0$

Therefore, the first derivative of $f^*$ is strictly increasing.

By Real Function is Strictly Convex iff Derivative is Strictly Increasing, $f^*$ is strictly convex.