Convex Real Function is Pointwise Supremum of Affine Functions

Theorem
Let $f : \R \to \R$ be a convex real function.

Then there exists a set $\mathcal S \subseteq \R^2$ such that:


 * $\ds \map f x = \sup_{\tuple {a, b} \in \mathcal S} \paren {a x + b}$

for each $x \in \R$.

Proof
Let $x \in \R$.

We construct an affine function $\phi_x : \R \to \R$ such that:


 * $\map f y \ge \map {\phi_x} y$

and $\map {\phi_x} x = \map f x$.

We will then argue that:


 * $\map f x = \sup \set {\map {\phi_y} x : y \in \R}$

Define a function $g_x : \R \setminus \set x \to \R$ by:


 * $\ds \map {g_x} y = \frac {\map f y - \map f x} {y - x}$

for each $y \in \R \setminus \set x$.

From definition 3 of a convex real function, we have that $g_x$ is increasing.

Then from Limit of Increasing Function, we have:


 * $\sup \set {\map {g_x} y : y < x} = c_x < \infty$

and from Limit of Increasing Function: Corollary:


 * $\map {g_x} y \ge c_x$

for $y > x$.

That is, for $y - x > 0$, we have:


 * $\ds \frac {\map f y - \map f x} {y - x} \ge c_x$

and for $y - x < 0$, we have:


 * $\ds \frac {\map f y - \map f x} {y - x} \le c_x$

So for all $x, y \in \R$ with $x \ne y$ we have:


 * $\map f y - \map f x \ge c_x \paren {y - x}$

giving:


 * $\map f y \ge \map f x + c_x \paren {y - x}$

Clearly this inequality also holds when $x = y$.

Now set:


 * $\map {\phi_x} y = \map f x + c_x \paren {y - x}$

for each $y \in \R$.

Then we clearly have:


 * $\map {\phi_x} x = \map f x$

while:


 * $\map f x \ge \map {\phi_y} x$

for all $x, y \in \R$.

The latter gives that:


 * $\sup \set {\map {\phi_y} x : y \in \R} \le \map f x$

while the former assures that:


 * $\map f x \le \sup \set {\map {\phi_y} x : y \in \R}$

So we have:


 * $\map f x = \sup \set {\map {\phi_y} x : y \in \R}$

In particular, setting:


 * $\mathcal S = \set {\tuple {c_y, \map f y - c_y y} : y \in \R}$

we obtain:


 * $\ds \map f x = \sup_{\tuple {a, b} \in \mathcal S} \paren {a x + b}$

for each $x \in \R$.