Extreme set

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

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

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

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

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

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