Epigraph (mathematics)

In mathematics, the epigraph or supergraph[1] of a function f : Rⁿ→R is the set of points lying on or above its graph:

${\displaystyle {\mbox{epi}}f=\{(x,\mu )\,:\,x\in \mathbb {R} ^{n},\,\mu \in \mathbb {R} ,\,\mu \geq f(x)\}\subseteq \mathbb {R} ^{n+1}.}$

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.

The strict epigraph is the epigraph with the graph itself removed:

${\displaystyle {\mbox{epi}}_{S}f=\{(x,\mu )\,:\,x\in \mathbb {R} ^{n},\,\mu \in \mathbb {R} ,\,\mu >f(x)\}\subseteq \mathbb {R} ^{n+1}.}$

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.

The domain (rather than the co-domain) of the function is not particularly important for this definition; it can be any linear space[1] or even an arbitrary set[2] instead of ${\displaystyle \mathbb {R} ^{n}}$.

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

The epigraph can often be used to give geometrical interpretations of the properties of convex functions or to prove these properties.