Convex Functions

Why

We generalize convex functions to arbitrary vector spaces.

Suppose $X$ is a vector space over $\R $, $D \subset X$ and $f: D \to \Rbar$. As before, define

\[ \epi f = \Set{(x, \alpha ) \in X \times \R }{f(x) \leq \alpha } \]

$f$ is convex if $\epi f$ is convex. It is straightforward that $f$ is convex if and only if

\[ f(tx + (1-t)y) \leq tf(x) + (1-t)f(y) \]

for all $t \in [0,1]$ and $x,y \in D$.

Any norm on $X$ is a convex function.