epigraph

definition

A epigraph or supergraph of a function $f: X \to [-\infty, \infty]$ valued in the extended real numbers $[-\infty, \infty]=\mathbb{R} \cup \{\pm \infty\}$ is the following set:

$$\text{epi} f = \{(x,r) \in X \times \mathbb{R} : r \ge f(x)\}$$

which consists of all points in the Cartesian product $X \times \mathbb{R}$ lying on or above the function’s graph.

A strict epigraph $\text{epi}_S f$ is the set of points in $X \times \mathbb{R}$ lying strictly above its graph.