Extreme set

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

An extreme point of

C

is a point

p \in C

for which

{p}

is a face.^[1]

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

α = inf {f (c) : c \in C} > - \infty

, then

{c \in C : f (c) = α}

is an exposed face of

C

.

{c \in V : f (c) = α}

is called a supporting hyperplane of

C

.

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

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

Competing definitions

Some authors do not include $C$ and/or $\emptyset$ 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.

References

1 2 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

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

[FOOTNOTENariciBeckenstein2011275–339-1] 1 2 Narici & Beckenstein 2011, pp. 275–339.

[1]

Competing definitions

See also

References

Bibliography

External links