Real Function Epigraphs

Definition

Let $f: D \to R$ be a multivariate real-valued function where $D \subset \R ^d$. The graph of $f$ is the set in $\R ^{d + 1}$ defined by

\[ \Set{(x, f(x)) \in \R ^d \times \R }{x \in D} \]

Of course, in these sheets, the graph is the same object as the function $f$ (see discussion in Functions). The epigraph of $f$ is the set in $\R ^{d + 1}$ defined by

\[ \Set{(x, \alpha ) \in D \times \R }{f(x) \leq \alpha }. \]

The prefix “epi” is Greek, meaning “upon” or “above”. It is merited (see the visualization below) by the fact that $f \neq \epi f$.

Visualization of epigraph

Notation

We denote the epigraph of a function $f$ by $\epi f$.

Extension to extended real numbers

We can extend this concept in the natural way to extended real value function $f: D \to \Rbar$.

\[ \epi f \Set{(x, \alpha ) \in D \times \R }{f(x) \leq \alpha } \]

Caution, in this case $f \not\subset \epi f$.