# 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

 $\displaystyle \frac {\d f^*} {\d p}$ $=$ $\displaystyle -\frac {\d \map f {\map x p} } {\d p} + \frac {\map \d {p \map x p} } {\d p}$ Definition of Legendre Transform $\displaystyle$ $=$ $\displaystyle -f' \frac {\d x} {\d p} + x + p \frac {\d x} {\d p}$ Product Rule for Derivatives $\displaystyle$ $=$ $\displaystyle -p \frac {\d x} {\d p} + x + p \frac {\d x} {\d p}$ Definition of $p$ $\displaystyle$ $=$ $\displaystyle x$
 $\displaystyle \frac {\d^2 f^*} {\d p^2}$ $=$ $\displaystyle \map {x'} p$ $\displaystyle$ $=$ $\displaystyle \frac 1 {\map {p'} x}$ Derivative of Inverse Function $\displaystyle$ $=$ $\displaystyle \frac 1 {\map {f''} x}$ Definition of $p$

We have that $\map f x$ is real and strictly convex.

Hence, by Real Function is Strictly Convex iff Derivative is Strictly Increasing, $\map {f'} x$ is strictly increasing.

Then:

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

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

$\blacksquare$