Extreme set

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.