- Epigraph (mathematics)
In
mathematics , the epigraph of a function "f" : Rn→R is the set of points lying on or above its graph:: mbox{epi} f = { (x, mu) , : , x in mathbb{R}^n,, mu in mathbb{R},, mu ge f(x) } subseteq mathbb{R}^{n+1},
and the strict epigraph of the function is:
: mbox{epi}_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 .
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 realaffine function "g" : Rn→R is ahalfspace 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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