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 .^[1]

An extreme point of is a point for which is a face.^[1]

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 (see the figure). 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 .

Some authors do not include and/or among the (exposed) faces. Some authors require and/or 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.

• Face (geometry)

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.

• VECTOR SPACES AND CONTINUOUS LINEAR FUNCTIONALS, Chapter III of FUNCTIONAL ANALYSIS, Lawrence Baggett, University of Colorado Boulder.
• Analysis, Peter Philip, Ludwig-Maximilians-universität München, 2024