In mathematics, most commonly in convex geometry, an extreme set or face of a set
in a vector space
with the property that if for any two points
some in-between point
[1] An extreme point of
[1] An exposed face of
is the subset of points of
where a linear functional achieves its minimum on
is a linear functional on
α = inf { f ( c )
is an exposed face of
An exposed point of
is an exposed face.
An exposed face is a face, but the converse is not true (see the figure).
An exposed face of
Some authors do not include
among the (exposed) faces.
Some authors require
to be convex (else the boundary of a disc is a face of the disc, as well as any subset of the boundary) or closed.
Some authors require the functional
to be continuous in a given vector topology.