Hypograph (mathematics)

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

and the strict hypograph of the function is:

The set is empty if ${\displaystyle f\equiv -\infty }$.

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

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

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

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

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