Epigraph (mathematics) Information & Epigraph (mathematics) Links at HealthHaven.com

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.

[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.

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