Extreme set

In mathematics, most commonly in convex geometry, an extreme set or face of a set in a vector space is a subset with the property that if for any two points some in-between point lies in, then we must have had.
An extreme point of is a point for which is a face.
An exposed face of is the subset of points of where a linear functional achieves its minimum on. Thus, if is a linear functional on and, then is an exposed face of.
An exposed point of is a point such that is an exposed face. That is, for all.
An exposed face is a face, but the converse is not true. An exposed face of is convex if is convex.
If is a face of, then is a face of if and only if is a face of.

Competing definitions

Some authors do not include and/or among the faces. Some authors require and/or to be convex or closed. Some authors require the functional to be continuous in a given vector topology.