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:

$\mbox{hyp} f = \{ (x, \mu) \, : \, x \in \mathbb{R}^n,\, \mu \in \mathbb{R},\, \mu \le f(x) \} \subseteq \mathbb{R}^{n+1}$

and the strict hypograph of the function is:

$\mbox{hyp}_S f = \{ (x, \mu) \, : \, x \in \mathbb{R}^n,\, \mu \in \mathbb{R},\, \mu < f(x) \} \subseteq \mathbb{R}^{n+1}.$

The set is empty if $f \equiv -\infty$.

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

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

Properties

A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function g : Rⁿ → R is a halfspace in Rⁿ⁺¹.

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

References

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.