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In mathematics, most commonly in convex geometry, an extreme set or face of a set C ⊆ ⊆ V {\displaystyle C\subseteq V} in a vector space V {\displaystyle V} is a subset F ⊆ ⊆ C {\displaystyle F\subseteq C} with the property that if for any two points x , y ∈ ∈ C {\displaystyle x,y\in C} some in-between point z = θ θ x + ( 1 − − θ θ ) y , θ θ ∈ ∈ [ 0 , 1 ] {\displaystyle z=\theta x+(1-\theta )y,\theta \in [0,1]} lies in F {\displaystyle F} , then we must have had x , y ∈ ∈ F {\displaystyle x,y\in F} .cite-ref-footnotenaricibeckenstein2011275-339-1-0[1]

An extreme point of C {\displaystyle C} is a point p ∈ ∈ C {\displaystyle p\in C} for which { p } {\displaystyle \{p\}} is a face.cite-ref-footnotenaricibeckenstein2011275-339-1-1[1]

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

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

• Competing definitions

• See also

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

• Face (geometry)

cite-note-footnotenaricibeckenstein2011275-339-11. ↑ Narici & Beckenstein 2011, pp. 275–339.

• citerefnaricibeckenstein2011Narici, 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