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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 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 Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

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