Epigraph

$ \newcommand{\epi}{\mathop{\mathrm{epi}}} $

The epigraph of a function \(f: \mathbb{R}^n \rightarrow \mathbb{R}\) is the set of points lying on or above its graph:

\[ \text{epi} f = \{ (x,\mu) : x\in \mathbb{R}^n, \mu \in \mathbb{R}, \mu \ge f(x) \} \subset \mathbb{R}^{n+1}\]. 

Properties:

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

A function is lower semicontinuous if and only if its epigraph is closed.