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Extreme set

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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 $z=\theta x+(1-\theta )y,\theta \in [0,1]$ lies in $F$ , then we must have had $x,y\in F$ .

An extreme point of $C$ is a point $p\in C$ for which $\{p\}$ is a face.

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 $\alpha =\inf\{f(c)\ \colon c\in C\}>-\infty$ , then $\{c\in C\ \colon f(c)=\alpha \}$ is an exposed face of $C$ .

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

An exposed face is a face, but the converse is not true (see the figure). An exposed face of $C$ 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 $C$ .

01Competing 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.

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Sources and credits

This article is adapted from the Wikipedia article “Extreme set”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Extremenotexposed.png by Loxley, CC BY-SA 4.0

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Extreme set

01Competing definitions

Sources and credits

Related topics

Face (geometry)