This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article relies largely or entirely on a single source. Please help improve this article by introducing citations to additional sources. Find sources: "Extreme set" – news · newspapers · books · scholar · JSTOR (December 2024) This article has no lead section. Please improve this article by adding one in your own words. (December 2024) (Learn how and when to remove this message) This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (December 2024) (Learn how and when to remove this message) (Learn how and when to remove this message)

In mathematics, most commonly in convex geometry, an extreme set or face of a set ${\displaystyle C\subseteq V}$ in a vector space ${\displaystyle V}$ is a subset ${\displaystyle F\subseteq C}$ with the property that if for any two points ${\displaystyle x,y\in C}$ some in-between point ${\displaystyle z=\theta x+(1-\theta )y,\theta \in [0,1]}$ lies in ${\displaystyle F}$, then we must have had ${\displaystyle x,y\in F}$.^[1]

An extreme point of ${\displaystyle C}$ is a point ${\displaystyle p\in C}$ for which ${\displaystyle \{p\}}$ is a face.^[1]

An exposed face of ${\displaystyle C}$ is the subset of points of ${\displaystyle C}$ where a linear functional achieves its minimum on ${\displaystyle C}$. Thus, if ${\displaystyle f}$ is a linear functional on ${\displaystyle V}$ and ${\displaystyle \alpha =\inf\{f(c)\ \colon c\in C\}>-\infty }$, then ${\displaystyle \{c\in C\ \colon f(c)=\alpha \}}$ is an exposed face of ${\displaystyle C}$.

An exposed point of ${\displaystyle C}$ is a point ${\displaystyle p\in C}$ such that ${\displaystyle \{p\}}$ is an exposed face. That is, ${\displaystyle f(p)>f(c)}$ for all ${\displaystyle c\in C\setminus \{p\}}$.

An exposed face is a face, but the converse is not true (see the figure). An exposed face of ${\displaystyle C}$ is convex if ${\displaystyle C}$ is convex. If ${\displaystyle F}$ is a face of ${\displaystyle C\subseteq V}$, then ${\displaystyle E\subseteq F}$ is a face of ${\displaystyle F}$ if and only if ${\displaystyle E}$ is a face of ${\displaystyle C}$.

Some authors do not include ${\displaystyle C}$ and/or ${\displaystyle \varnothing }$ among the (exposed) faces. Some authors require ${\displaystyle F}$ and/or ${\displaystyle C}$ 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 ${\displaystyle f}$ 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