Extreme point and face

Let $K$ be a convex subset of a real or complex vector space, where convex combinations always use real coefficients as in Local convexity, convex and balanced sets, and the continuous dual. A point $x\in K$ is an extreme point of $K$ if

$$x=(1-t)y+tz,y,z\in K,0<t<1,$$

implies $y=z=x$. The set of extreme points is denoted $\mathrm{ext}K$.

A face of $K$ is a nonempty convex subset $F\subseteq K$ such that

$$(1-t)y+tz\in F,y,z\in K,0<t<1,$$

implies $y,z\in F$. Thus $x$ is extreme exactly when the singleton $\{x\}$ is a face. Neither definition requires a topology. A face need not be exposed by a continuous linear functional; “face” below always means the intrinsic endpoint condition just stated.

For $K=\mathrm{\varnothing }$ there are no extreme points and no faces. If $K=\{x\}$, then $x$ is extreme and $K$ is its unique face. The strict restriction $0<t<1$ is essential: at $t=0$ or $t=1$ the displayed equality contains no information about the unused endpoint.