Exposed face ($F=\{x\in K:f(x)={\sup }_{K}f\}$)

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.