= Exposed face
{title2=$F=\{x\in K:f(x)=\sup_K f\}$}
An exposed face of a <convex set> $K$ is the nonempty set on which a supporting real <linear functional> attains its maximum. It is a <face of a convex set>: if $x=ty+(1-t)z$ with $0<t<1$ attains the maximum and $y,z\in K$, each of $f(y),f(z)$ is at most that maximum, so both equal it. On a complex <vector space>, take the real part of a complex <linear functional>. An <exposed point> is an exposed face consisting of one point.
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