Supergraph
From Encyclopedia of Mathematics
of a function
Let be a function defined on some set
with values in the extended real line
. Then the supergraph of
is the subset of the Cartesian product
consisting of all points
lying "above" the graph of
. The supergraph is sometimes called the epigraph and is denoted by
:
![]() |
The projection of the supergraph onto is known as the effective domain of
, denoted by
:
![]() |
The function is said to be proper if
![]() |
A function defined on a real vector space
is convex if and only if
is a convex subset of
. A function
defined on a topological space
is upper semi-continuous if and only if
is a closed set.
Comments
References
[a1] | R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970) pp. 23; 307 |
How to Cite This Entry:
Supergraph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supergraph&oldid=17155
Supergraph. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Supergraph&oldid=17155
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article