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 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.