Epigraph (mathematics)

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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},\, f(x)\le \mu \} \subseteq \mathbb{R}^{n+1}.

[edit] 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.

[edit] References

  • Rockafellar, Ralph Tyrell (1996), Convex Analysis, Princeton University Press, Princeton, NJ. ISBN 0-691-01586-4.
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