Hypograph (mathematics)

In mathematics, the hypograph or subgraph of a function f : Rⁿ → R is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph.

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

Definition

The definition of the hypograph 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 hypograph or subgraph 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]

${\displaystyle {\begin{alignedat}{4}\operatorname {hyp} f&=\left\{(x,r)\in X\times \mathbb {R} ~:~r\leq f(x)\right\}\\&=\left[f^{-1}(\infty )\times \mathbb {R} \right]\cup \bigcup _{x\in f^{-1}(\mathbb {R} )}\{x\}\times (-\infty ,f(x)].\end{alignedat}}}$

Similarly, the set of points on or below the function is its epigraph. The strict hypograph is the hypograph with the graph removed:

${\displaystyle {\begin{alignedat}{4}\operatorname {hyp} _{S}f&=\left\{(x,r)\in X\times \mathbb {R} ~:~r<f(x)\right\}\\&=\operatorname {hyp} f\setminus \operatorname {graph} f\\&=\bigcup _{x\in X}\{x\}\times (-\infty ,f(x)).\end{alignedat}}}$

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 ${\displaystyle X\times [-\infty ,\infty ].}$

Properties

The epigraph of a function ${\displaystyle f}$ is empty if and only if ${\displaystyle f}$ is identically equal to negative infinity.

A function is concave if and only if its hypograph is a convex set. The hypograph 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 upper semicontinuous if and only if its hypograph is closed.

See also

• Effective domain
• Epigraph (mathematics)
• Proper convex function

Citations

1. Charalambos D. Aliprantis; Kim C. Border (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer Science & Business Media. pp. 8–9. ISBN 978-3-540-32696-0.
2. Rockafellar & Wets 2009, pp. 1-37. sfn error: no target: CITEREFRockafellarWets2009 (help)

References

• Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.CS1 maint: date and year (link)