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 $x \in \R$.
Corollary
There exists a countable 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 $x \in \R$.
$\blacksquare$