Extreme set

The two distinguished points are examples of extreme points of a convex set that are not exposed points. Therefore, not every convex face of a convex set is an exposed face.

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

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

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

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

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

Competing definitions

Some authors do not include $C$ and/or $\varnothing$ among the (exposed) faces. Some authors require $F$ and/or $C$ 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 $f$ to be continuous in a given vector topology.

Facts

An exposed face is clearly a face. An exposed face of $C$ is clearly convex if $C$ is convex.

If $F$ is a face of $C\subseteq V$, then $E\subseteq F$ is a face of $F$ iff $E$ is a face of $C$.

See also

• Face (geometry)

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