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:

and the strict hypograph of the function is:

The set is empty if .

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 .

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.

gollark: I am not particularly angry. I just think you were being wrong.
gollark: Some offense, but if you were being precautionary you should probably have realised you were 50% over what you said was the maximum safe dose.
gollark: You sure did clear some thing in the past 15 minutes.
gollark: Also, just switch really fast.
gollark: Why not?

See also

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.


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