Hypograph (mathematics)

Hypograph of a function

In mathematics, the hypograph or subgraph of a function $f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ 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 instead of ${\mathbb{ℝ}}^n$.

Definition

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

$\mathrm{graph}f:=\{(x,y)\in X\times Y:y=f(x)\}.$

The hypograph or subgraph of a function $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ valued in the extended real numbers $[-\mathrm{\infty },\mathrm{\infty }]=\mathbb{ℝ}\cup \{\pm \mathrm{\infty }\}$ is the set

$\begin{array}{rl}\mathrm{hyp}f& =\{(x,r)\in X\times \mathbb{ℝ}:r\le f(x)\}\\ & =[f^{-1}(\mathrm{\infty })\times \mathbb{ℝ}]\cup \bigcup _{x\in f^{-1}(\mathbb{ℝ})}\{x\}\times (-\mathrm{\infty },f(x)].\end{array}$

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

$\begin{array}{rl}{\mathrm{hyp}}_{S}f& =\{(x,r)\in X\times \mathbb{ℝ}:r<f(x)\}\\ & =\mathrm{hyp}f\setminus \mathrm{graph}f\\ & =\bigcup _{x\in X}\{x\}\times (-\mathrm{\infty },f(x)).\end{array}$

Despite the fact that $f$ might take one (or both) of $\pm \mathrm{\infty }$ as a value (in which case its graph would not be a subset of $X\times \mathbb{ℝ}$), the hypograph of $f$ is nevertheless defined to be a subset of $X\times \mathbb{ℝ}$ rather than of $X\times [-\mathrm{\infty },\mathrm{\infty }].$

Properties

The hypograph of a function $f$ is empty if and only if $f$ 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 $g:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ is a halfspace in ${\mathbb{ℝ}}^{n+1}.$

A function is upper semicontinuous if and only if its hypograph is closed.

See also

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

References

• Template:Rockafellar Wets Variational Analysis 2009 Springer