Closed convex function

A closed convex function is an extended-real-valued function $f:\mathbb{R}^n\to(-\infty,+\infty]$ that is convex and has a closed epigraph

\mathrm{epi}(f)\;=\;\{(x,t)\in\mathbb{R}^n\times\mathbb{R}:\ t\ge f(x)\},

so that $\mathrm{epi}(f)$ is a closed convex set in $\mathbb{R}^n\times\mathbb{R}$ .

Equivalent characterizations

Equivalently, $f$ is closed convex iff it is convex and lower semicontinuous, meaning $\liminf_{x\to x_0} f(x)\ge f(x_0)$ for every $x_0$ .

Remarks

Closedness is the regularity condition that makes conjugation behave well: for proper functions, closed convexity is exactly the condition for equality $f=f^{**}$ in the biconjugate construction (see Fenchel–Moreau theorem).

Examples

The norm $f(x)=\|x\|_2$ is closed and convex on $\mathbb{R}^n$ .
If $C$ is a closed convex set, then the indicator $\delta_C$ is a closed convex function; if $C$ is convex but not closed (e.g. an open ball), then $\delta_C$ is convex but not closed.