Epigraph (mathematics)

Relationships with other sets

Despite the fact that ${\displaystyle f}$ might take one (or both) of ${\displaystyle \pm \mathrm{\infty }}$ 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 ${\displaystyle X\times [-\mathrm{\infty },\mathrm{\infty }].}$ This is intentional because when ${\displaystyle X}$ is a vector space then so is ${\displaystyle X\times \mathbb{R}}$ but ${\displaystyle X\times [-\mathrm{\infty },\mathrm{\infty }]}$ is never a vector space (since the extended 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 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 or even an arbitrary set 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.

Reconstructing functions from epigraphs

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 ${\displaystyle E:=\mathrm{epi}f}$ (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 the infimum of the interval.

The above observations can be combined to give a single formula for ${\displaystyle f(x)}$ in terms of ${\displaystyle E:=\mathrm{epi}f.}$ 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 ${\displaystyle E:={\mathrm{epi}}_{S}f.}$

Relationships between properties of functions and their epigraphs

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.