Convex function via epigraph

A function is convex if and only if its epigraph is a convex set

Convex function via epigraph

Let $X$ be a vector space and let $f:X\to \mathbb{R}$ be an extended-real-valued function .

The function $f$ is convex if its epigraph (see epigraph ) is a convex set in $X\times\mathbb{R}$ .

Context. This geometric definition is equivalent to analytic inequalities such as Jensen’s inequality; see equivalent characterizations of convexity .

Examples:

On a normed space, $x\mapsto \|x\|$ is convex (uses the triangle inequality; see norm ).
The indicator function of a set $\Omega$ is convex iff $\Omega$ is convex.
The distance function to a convex set is convex (in normed spaces).