Hypograph (mathematics)

In mathematics, the hypograph or subgraph of a function $$f:\R^{n}\rightarrow \R$$ is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or above the function's graph.

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

Definition
The definition of the hypograph was inspired by that of the graph of a function, where the of $$f : X \to Y$$ is defined to be the set


 * $$\operatorname{graph} f := \left\{ (x, y) \in X \times Y ~:~ y = f(x) \right\}.$$

The or  of a function $$f : X \to [-\infty, \infty]$$ valued in the extended real numbers $$[-\infty, \infty] = \mathbb{R} \cup \{ \pm \infty \}$$ is the set



\begin{alignat}{4} \operatorname{hyp} f &= \left\{ (x, r) \in X \times \mathbb{R} ~:~ r \leq f(x) \right\} \\ &= \left[ f^{-1}(\infty) \times \mathbb{R} \right] \cup \bigcup_{x \in f^{-1}(\mathbb{R})} (\{ x \} \times (-\infty, f(x)]). \end{alignat} $$

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

The is the hypograph with the graph removed:



\begin{alignat}{4} \operatorname{hyp}_S f &= \left\{ (x, r) \in X \times \mathbb{R} ~:~ r < f(x) \right\} \\ &= \operatorname{hyp} f \setminus \operatorname{graph} f \\ &= \bigcup_{x \in X} (\{ x \} \times (-\infty, f(x))). \end{alignat} $$

Despite the fact that $$f$$ might take one (or both) of $$\pm \infty$$ as a value (in which case its graph would be a subset of $$X \times \mathbb{R}$$), the hypograph of $$f$$ is nevertheless defined to be a subset of $$X \times \mathbb{R}$$ rather than of $$X \times [-\infty, \infty].$$

Properties
The hypograph of a function $$f$$ is empty if and only if $$f$$ is identically equal to negative infinity.

A function is concave if and only if its hypograph is a convex set. The hypograph of a real affine function $$g : \mathbb{R}^n \to \mathbb{R}$$ is a halfspace in $$\mathbb{R}^{n+1}.$$

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