Hypograph (mathematics)

In mathematics, the hypograph of a function "f" : R^"n" → R is the set of points lying on or below its graph:

: $mbox\{hyp\}\; f\; =\; \{\; (x,\; mu)\; ,\; :\; ,\; x\; in\; mathbb\{R\}^n,,\; mu\; in\; mathbb\{R\},,\; mu\; le\; f(x)\; \}\; subseteq\; mathbb\{R\}^\{n+1\}$

and the strict hypograph of the function is:

:

The set is empty if $f\; equiv\; -infty$ .

Similarly, the set of points on or above the function's graph is its epigraph.

When referring to relations, such as preference relations in economics, a similarly defined set is generally called a lower contour set.

Properties

A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function "g" : R^"n" → R is a halfspace in R^"n"+1.

A function is upper semicontinuous if and only if its hypograph is closed.