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:

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

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.