Characterizations of Convex Functions

Why

Convex functions satisfy an interpolation property.

Discussion

By definition, given $S \subset \R ^n$, $f: S \to \R $ is convex means that the point

\[ (1 - \lambda )(x \mu ) + \lambda (x, \nu ) = ((1-\lambda )x + \lambda y, (1-\lambda )\mu + \lambda \nu \]

belongs to $\epi f$ whenever $(x, \mu )$ and $(y, \nu )$ belong to $\epi f$ and $0 \leq \lambda \leq 1$. Said differently, we have $(1-\lambda )x + \lambda y \in S$, and

\[ f((1-\lambda )x + \lambda y) \leq (1-\lambda )\mu + \lambda \nu \]

whenever $x \in S$, $y \in S$, $f(x) \leq \mu \in \R $, and $f(y) \leq \nu \in \R $.

Concave functions have a similar property, with the inequalities flipped, and affine functions satisfy with qualities. This shows that the functions $f: \R ^n \to \R $ we are calling affine coincide exactly with the affine transformations from $\R ^n$ to $\R $.

Visualization