Inmathematics, theepigraph orsupergraph^[1] of afunction${\displaystyle f:X\to [-\mathrm{\infty },\mathrm{\infty }]}$ valued in theextended real numbers${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]=\mathbb{R}\cup \{\pm \mathrm{\infty }\}}$ is theset$${\displaystyle \mathrm{epi}f=\{(x,r)\in X\times \mathbb{R}:r\ge f(x)\}}$$consisting of all points in theCartesian product${\displaystyle X\times \mathbb{R}}$ lying on or above the function'sgraph.^[2] Similarly, thestrict epigraph${\displaystyle {\mathrm{epi}}_{S}f}$ is the set of points in${\displaystyle X\times \mathbb{R}}$ lying strictly above its graph.

Importantly, unlike the graph of${\displaystyle f,}$ the epigraphalways consistsentirely 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${\displaystyle \pm \mathrm{\infty }}$ as a value then${\displaystyle \mathrm{graph}f}$ willnot be a subset of its epigraph${\displaystyle \mathrm{epi}f.}$ For example, if${\displaystyle f\left(x_0\right)=\mathrm{\infty }}$ then the point${\displaystyle \left(x_0,f\left(x_0\right)\right)=\left(x_0,\mathrm{\infty }\right)}$ will belong to${\displaystyle \mathrm{graph}f}$ but not to${\displaystyle \mathrm{epi}f.}$ These two sets are nevertheless closely related because the graph can always be reconstructed from the epigraph, and vice versa.

The study ofcontinuousreal-valued functions inreal analysis has traditionally been closely associated with the study of theirgraphs, which are sets that provide geometric information (and intuition) about these functions.^[2] Epigraphs serve this same purpose in the fields ofconvex analysis andvariational analysis, in which the primary focus is onconvex functions valued in${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]}$ 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 aconvex function's properties, to help formulate or prove hypotheses, or to aid in constructingcounterexamples.

The definition of the epigraph was inspired by that of thegraph of a function, where thegraph of${\displaystyle f:X\to Y}$ is defined to be the set$${\displaystyle \mathrm{graph}f:=\{(x,y)\in X\times Y:y=f(x)\}.}$$

Theepigraph orsupergraph of a function${\displaystyle f:X\to [-\mathrm{\infty },\mathrm{\infty }]}$ valued in theextended real numbers${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]=\mathbb{R}\cup \{\pm \mathrm{\infty }\}}$ is the set^[2]where all sets being unioned in the last line are pairwise disjoint.

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${\displaystyle \{x\}\times [f(x),\mathrm{\infty })}$ 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 itshypograph.

Thestrict epigraph is the epigraph with the graph removed:where all sets being unioned in the last line are pairwise disjoint, and some may be empty.

Despite the fact that${\displaystyle f}$ might take one (or both) of${\displaystyle \pm \mathrm{\infty }}$ as a value (in which case its graph wouldnot 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${\displaystyle X\times [-\mathrm{\infty },\mathrm{\infty }].}$ This is intentional because when${\displaystyle X}$ is avector space then so is${\displaystyle X\times \mathbb{R}}$ but${\displaystyle X\times [-\mathrm{\infty },\mathrm{\infty }]}$ isnever a vector space^[2] (since theextended real number line${\displaystyle [-\mathrm{\infty },\mathrm{\infty }]}$ is not a vector space). This deficiency in${\displaystyle X\times [-\mathrm{\infty },\mathrm{\infty }]}$ remains even if instead of being a vector space,${\displaystyle X}$ is merely a non-empty subset of some vector space. The epigraph being a subset of a vector space allows for tools related toreal analysis andfunctional analysis (and other fields) to be more readily applied.

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

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

The epigraph of a function${\displaystyle f:X\to [-\mathrm{\infty },\mathrm{\infty }]}$ is related to its graph and strict epigraph by$${\displaystyle \mathrm{epi}f\subseteq {\mathrm{epi}}_{S}f\cup \mathrm{graph}f}$$where set equality holds if and only if${\displaystyle f}$ is real-valued. However,$${\displaystyle \mathrm{epi}f=\left[{\mathrm{epi}}_{S}f\cup \mathrm{graph}f\right]\cap [X\times \mathbb{R}]}$$ always holds.

The epigraph isempty 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${\displaystyle E:=\mathrm{epi}f}$ (even when${\displaystyle f}$ takes on${\displaystyle \pm \mathrm{\infty }}$ as a value). Given${\displaystyle x\in X,}$ the value${\displaystyle f(x)}$ can be reconstructed from the intersection${\displaystyle E\cap (\{x\}\times \mathbb{R})}$ of${\displaystyle E}$ with the "vertical line"${\displaystyle \{x\}\times \mathbb{R}}$ passing through${\displaystyle x}$ as follows:

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

The above observations can be combined to give a single formula for${\displaystyle f(x)}$ in terms of${\displaystyle E:=\mathrm{epi}f.}$ Specifically, for any${\displaystyle x\in X,}$$${\displaystyle f(x)=\underset{}{inf}\{r\in \mathbb{R}:(x,r)\in E\}}$$where by definition,${\displaystyle \underset{}{inf}\varnothing :=\mathrm{\infty }.}$ This same formula can also be used to reconstruct${\displaystyle f}$ from its strict epigraph${\displaystyle E:={\mathrm{epi}}_{S}f.}$

A function isconvex if and only if its epigraph is aconvex set. The epigraph of a realaffine function${\displaystyle g:{\mathbb{R}}^n\to \mathbb{R}}$ is ahalfspace in${\displaystyle {\mathbb{R}}^{n+1}.}$

A function islower semicontinuous if and only if its epigraph isclosed.

• Effective domain
• Hypograph (mathematics) – Region underneath a graph
• Proper convex function

Wikimedia Commons has media related toepigraphs und hypographs.

• 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.

• Convex analysis
• Mathematical analysis
• Variational analysis

• Articles with short description
• Short description is different from Wikidata
• Commons category link is locally defined