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Let ${\displaystyle C\subseteq V}$, where ${\displaystyle V}$ is a vector space.

A extreme set or face or of ${\displaystyle C}$ is a set ${\displaystyle F\subseteq C}$ such that ${\displaystyle x,y\in C\ \&\ 0<\theta <1\ \&\ \theta x+(1-\theta )y\in F\ \Rightarrow \ x,y\in F}$. That is, if a point ${\displaystyle p\in F}$ lies between some points ${\displaystyle x,y\in C}$, then ${\displaystyle x,y\in F}$.

An extreme point of ${\displaystyle C}$ is a point ${\displaystyle p\in C}$ such that ${\displaystyle \{p\}}$ is a face of ${\displaystyle C}$. That is, if ${\displaystyle p}$ lies between some points ${\displaystyle x,y\in C}$, then ${\displaystyle x=y=p}$.

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 ${\displaystyle \alpha =\inf\{fc\ \colon c\in C\}>-\infty }$, then ${\displaystyle \{c\in C\ \colon fc=\alpha \}}$ 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 of ${\displaystyle C}$. That is, ${\displaystyle fp>fc}$ for all ${\displaystyle c\in C\setminus \{p\}}$.

Competing definitions

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.

Facts

An exposed face is clearly a face. An exposed face of ${\displaystyle C}$ is clearly 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 ${\displaystyle F}$ iff ${\displaystyle E}$ is a face of ${\displaystyle C}$.

See also

• Face (geometry)

References

1. ^ Narici & Beckenstein 2011, pp. 275–339.

Bibliography

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

External links

• 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

• Convex geometry
• Convex hulls
• Functional analysis
• Mathematical analysis