Convex function via epigraph

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

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

The function $f$ is convex if its epigraph

\operatorname{epi}(f)=\{(x,r)\in X\times\mathbb R:f(x)\le r\}

is a convex set in $X\times\mathbb{R}$ .

Equivalent characterizations

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).