In mathematics, the hypograph or subgraph of a function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {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 R n {\displaystyle \mathbb {R} ^{n}} .
Definition The definition of the hypograph was inspired by that of the graph of a function, where the graph of f : X → Y {\displaystyle f:X\to Y} is defined to be the set
graph f := { ( x , y ) ∈ X × Y : y = f ( x ) } . {\displaystyle \operatorname {graph} f:=\left\{(x,y)\in X\times Y~:~y=f(x)\right\}.}
The hypograph or subgraph of a function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} valued in the extended real numbers [ − ∞ , ∞ ] = R ∪ { ± ∞ } {\displaystyle [-\infty ,\infty ]=\mathbb {R} \cup \{\pm \infty \}} is the set
hyp f = { ( x , r ) ∈ X × R : r ≤ f ( x ) } = [ f − 1 ( ∞ ) × R ] ∪ ⋃ x ∈ f − 1 ( R ) ( { x } × ( − ∞ , f ( x ) ] ) . {\displaystyle {\begin{alignedat}{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{alignedat}}}
Similarly, the set of points on or above the function is its epigraph.
The strict hypograph is the hypograph with the graph removed:
hyp S f = { ( x , r ) ∈ X × R : r < f ( x ) } = hyp f ∖ graph f = ⋃ x ∈ X ( { x } × ( − ∞ , f ( x ) ) ) . {\displaystyle {\begin{alignedat}{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{alignedat}}}
Despite the fact that f {\displaystyle f} might take one (or both) of ± ∞ {\displaystyle \pm \infty } as a value (in which case its graph would not be a subset of X × R {\displaystyle X\times \mathbb {R} } ), the hypograph of f {\displaystyle f} is nevertheless defined to be a subset of X × R {\displaystyle X\times \mathbb {R} } rather than of X × [ − ∞ , ∞ ] . {\displaystyle X\times [-\infty ,\infty ].}
Properties The hypograph of a function f {\displaystyle f} is empty if and only if f {\displaystyle 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 : R n → R {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} } is a halfspace in R n + 1 . {\displaystyle \mathbb {R} ^{n+1}.} A function is upper semicontinuous if and only if its hypograph is closed.
See also Effective domain Epigraph (mathematics) – Region above a graph Proper convex function – Concept in convex analysis
Citations
References Rockafellar, R. Tyrrell; Wets, Roger J.-B. (26 June 2009). Variational Analysis. Grundlehren der mathematischen Wissenschaften. Vol. 317. Berlin New York: Springer Science & Business Media. ISBN 9783642024313. OCLC 883392544.