Extreme set

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

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

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

An exposed point of ${\displaystyle C}${\displaystyle C} is a point ${\displaystyle p\in C}${\displaystyle p\in C} such that ${\displaystyle \{p\}}${\displaystyle \{p\}} is an exposed face. That is, ${\displaystyle f(p)>f(c)}${\displaystyle f(p)>f(c)} for all ${\displaystyle c\in C\setminus \{p\}}${\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 ${\displaystyle C}${\displaystyle C} is convex if ${\displaystyle C}${\displaystyle C} is convex. If ${\displaystyle F}${\displaystyle F} is a face of ${\displaystyle C\subseteq V}${\displaystyle C\subseteq V}, then ${\displaystyle E\subseteq F}${\displaystyle E\subseteq F} is a face of ${\displaystyle F}${\displaystyle F} if and only if ${\displaystyle E}${\displaystyle E} is a face of ${\displaystyle C}${\displaystyle C}.

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

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

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

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