Hypograph (mathematics)

In mathematics, the hypograph or subgraph of a function $f:\R^\rightarrow \R$ 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 $\mathbb^n$.

Definition

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

:$\operatorname f := \left\.$

The or of a function f : X \to \infty, \infty/math> valued in the extended real numbers \infty, \infty= \mathbb \cup \ is the set

:
\begin
\operatorname f
&= \left\ \\
&= \left f^(\infty) \times \mathbb \right\cup \bigcup_ (\ \times (-\infty, f(x)]).
\end

Similarly, the set of points on or above the function is its epigraph.

The is the hypograph with the graph removed:

:$\begin \operatorname_S f &= \left\ \\ &= \operatorname f \setminus \operatorname f \\ &= \bigcup_ (\ \times (-\infty, f(x))). \end$

Despite the fact that $f$ might take one (or both) of $\pm \infty$ as a value (in which case its graph would be a subset of $X \times \mathbb$), the hypograph of $f$ is nevertheless defined to be a subset of $X \times \mathbb$ rather than of X \times \infty, \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 $g : \mathbb^n \to \mathbb$ is a halfspace in $\mathbb^.$

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

See also

*
*
*

Citations

References

*

{{mathanalysis-stub

Mathematical analysis
Convex analysis