Epigraph (mathematics)

The definition of the epigraph was inspired by that of the graph of a function, where the graph of ${\displaystyle f:X\to Y}$ is defined to be the set

${\displaystyle \operatorname {graph} f:=\left\{(x,y)\in X\times Y~:~y=f(x)\right\}.}$

The epigraph or supergraph 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[2]

${\displaystyle {\begin{alignedat}{4}\operatorname {epi} f&=\left\{(x,r)\in X\times \mathbb {R} ~:~r\geq f(x)\right\}\\&=\left[f^{-1}(-\infty )\times \mathbb {R} \right]\cup \bigcup _{x\in f^{-1}(\mathbb {R} )}\{x\}\times [f(x),\infty )~~~{\text{ (all sets being unioned are pairwise disjoint). }}\end{alignedat}}}$

In the union over ${\displaystyle x\in f^{-1}(\mathbb {R} )}$ that appears above on the right hand side of the last line, the set ${\displaystyle \{x\}\times [f(x),\infty )}$ may be interpreted as being a "vertical ray" consisting of ${\displaystyle (x,f(x))}$ and all points in ${\displaystyle X\times \mathbb {R} }$ "directly above" it. Similarly, the set of points on or below the graph of a function is its hypograph.

The strict epigraph is the epigraph with the graph removed:

${\displaystyle {\begin{alignedat}{4}\operatorname {epi} _{S}f&=\left\{(x,r)\in X\times \mathbb {R} ~:~r>f(x)\right\}\\&=\operatorname {epi} f\setminus \operatorname {graph} f\\&=\bigcup _{x\in X}\{x\}\times (f(x),\infty )~~~{\text{ (all sets being unioned are pairwise disjoint, some may be empty). }}\end{alignedat}}}$

Despite the fact that ${\displaystyle f}$ might take one (or both) of ${\displaystyle \pm \infty }$ as a value (in which case its graph would not be a subset of ${\displaystyle X\times \mathbb {R} }$), the epigraph of ${\displaystyle f}$ is nevertheless defined to be a subset of ${\displaystyle X\times \mathbb {R} }$ rather than of ${\displaystyle X\times [-\infty ,\infty ].}$ This is intentional because when ${\displaystyle X}$ is a vector space then so is ${\displaystyle X\times \mathbb {R} }$ but ${\displaystyle X\times [-\infty ,\infty ]}$ is never a vector space[2] (since the extended real number line ${\displaystyle [-\infty ,\infty ]}$ is not a vector space). More generally, if ${\displaystyle X}$ is only a non-empty subset of some vector space then ${\displaystyle X\times [-\infty ,\infty ]}$ is never even a subset of any vector space. The epigraph being a subset of a vector space allows for tools related to real analysis and functional analysis (and other fields) to be more readily applied.

The domain (rather than the codomain) of the function is not particularly important for this definition; it can be any linear space[1] or even an arbitrary set[3] instead of ${\displaystyle \mathbb {R} ^{n}}$.

The strict epigraph ${\displaystyle \operatorname {epi} _{S}f}$ and the graph ${\displaystyle \operatorname {graph} f}$ are always disjoint.

The epigraph of a function ${\displaystyle f:X\to [-\infty ,\infty ]}$ is related to its graph and strict epigraph by

${\displaystyle \,\operatorname {epi} f\,\subseteq \,\operatorname {epi} _{S}f\,\cup \,\operatorname {graph} f}$

where set equality holds if and only if ${\displaystyle f}$ is real-valued. However,

${\displaystyle \operatorname {epi} f=\left[\operatorname {epi} _{S}f\,\cup \,\operatorname {graph} f\right]\,\cap \,\left[X\times \mathbb {R} \right]}$

always holds.

The epigraph is empty if and only if the function is identically equal to infinity.

Just as any function can be reconstructed from its graph, so too can any extended real-valued function ${\displaystyle f}$ on ${\displaystyle X}$ be reconstructed from its epigraph ${\displaystyle E:=\operatorname {epi} f}$ (even when ${\displaystyle f}$ takes on ${\displaystyle \pm \infty }$ as a value). Given ${\displaystyle x\in X,}$ the value ${\displaystyle f(x)}$ can be reconstructed from the intersection ${\displaystyle E\cap \left(\{x\}\times \mathbb {R} \right)}$ of ${\displaystyle E}$ with the "vertical line" ${\displaystyle \{x\}\times \mathbb {R} }$ passing through ${\displaystyle x}$ as follows:

• case 1: ${\displaystyle E\cap \left(\{x\}\times \mathbb {R} \right)=\varnothing }$ if and only if ${\displaystyle f(x)=\infty ,}$
• case 2: ${\displaystyle E\cap \left(\{x\}\times \mathbb {R} \right)=\{x\}\times \mathbb {R} }$ if and only if ${\displaystyle f(x)=-\infty ,}$
• case 3: otherwise, ${\displaystyle E\cap \left(\{x\}\times \mathbb {R} \right)}$ is necessarily of the form ${\displaystyle \{x\}\times [f(x),\infty ),}$ from which the value of ${\displaystyle f(x)}$ can be obtained by taking the infimum of the interval.

The above observations can be combined to give a single formula for ${\displaystyle f(x)}$ in terms of ${\displaystyle E:=\operatorname {epi} f.}$ Specifically, for any ${\displaystyle x\in X,}$

${\displaystyle f(x)=\inf _{}\{r\in \mathbb {R} ~:~(x,r)\in E\}}$

where by definition, ${\displaystyle \inf _{}\varnothing$ :=\infty .} This same formula can also be used to reconstruct ${\displaystyle f}$ from its strict epigraph ${\displaystyle E:=\operatorname {epi} _{S}f.}$

A function is convex if and only if its epigraph is a convex set. The epigraph of a real affine function ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} }$ is a halfspace in ${\displaystyle \mathbb {R} ^{n+1}.}$

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

• Effective domain
• Hypograph (mathematics) – Mathematical analysis termPages displaying wikidata descriptions as a fallback
• Proper convex function

• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.
• Rockafellar, Ralph Tyrell (1996), Convex Analysis, Princeton University Press, Princeton, NJ. ISBN 0-691-01586-4.