Hypograph (mathematics)

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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 .

The domain (rather than the co-domain) of the function is not particularly important for this definition; it can be an arbitrary set[1] instead of .

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.

See also[edit]


  1. ^ Charalambos D. Aliprantis; Kim C. Border (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer Science & Business Media. pp. 8–9. ISBN 978-3-540-32696-0.