Epigraph (mathematics)

In mathematics, the epigraph or supergraph^[1] of a function ${\displaystyle f:X\to [-\infty ,\infty ]}$ valued in the extended real numbers ${\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}}$ is the set $${\displaystyle \operatorname {epi} f=\{(x,r)\in X\times \mathbb {R} ~:~r\geq f(x)\}}$$ consisting of all points in the Cartesian product ${\displaystyle X\times \mathbb {R} }$ lying on or above the function's graph.^[2] Similarly, the strict epigraph ${\displaystyle \operatorname {epi} _{S}f}$ is the set of points in ${\displaystyle X\times \mathbb {R} }$ lying strictly above its graph.

Importantly, unlike the graph of ${\displaystyle f,}$ the epigraph always consists entirely of points in ${\displaystyle X\times \mathbb {R} }$ (this is true of the graph only when ${\displaystyle f}$ is real-valued). If the function takes ${\displaystyle \pm \infty }$ as a value then ${\displaystyle \operatorname {graph} f}$ will not be a subset of its epigraph ${\displaystyle \operatorname {epi} f.}$ For example, if ${\displaystyle f\left(x_{0}\right)=\infty }$ then the point ${\displaystyle \left(x_{0},f\left(x_{0}\right)\right)=\left(x_{0},\infty \right)}$ will belong to ${\displaystyle \operatorname {graph} f}$ but not to ${\displaystyle \operatorname {epi} f.}$ These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.

The study of continuous real-valued functions in real analysis has traditionally been closely associated with the study of their graphs, which are sets that provide geometric information (and intuition) about these functions.^[2] Epigraphs serve this same purpose in the fields of convex analysis and variational analysis, in which the primary focus is on convex functions valued in ${\displaystyle [-\infty ,\infty ]}$ instead of continuous functions valued in a vector space (such as ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {R} ^{2}}$).^[2] This is because in general, for such functions, geometric intuition is more readily obtained from a function's epigraph than from its graph.^[2] Similarly to how graphs are used in real analysis, the epigraph can often be used to give geometrical interpretations of a convex function's properties, to help formulate or prove hypotheses, or to aid in constructing counterexamples.