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.
See also
Epigraph (mathematics)
References
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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