Epigraph (mathematics)

A function (in black) is convex if and only if the region above its graph (in green) is a convex set: This region is the function's epigraph.

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 same definitions are valid for a function that takes values in R ∪ ∞. In this case, the epigraph is empty if and only if f is identically equal to infinity.

Similarly, the set of points on or below the function is its hypograph.

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.