In mathematics, the epigraph of a function f : Rn→R is the set of points lying on or above its graph:
and the strict epigraph of the function is:
The set is empty if 
.
Similarly, the set of points on or below the function is its hypograph.
When referring to relations, such as preference relations in economics, a similarly defined set is generally called an upper contour set.
Properties
A function is convex if and only if its epigraph is a convex set. The epigraph of a real affine function g : Rn→R is a halfspace in Rn+1.
A function is lower semicontinuous if and only if its epigraph is closed.
References
- Rockafellar, Ralph Tyrell (1996), Convex Analysis, Princeton University Press, Princeton, NJ. ISBN 0-691-01586-4.
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