Face of a convex set

Summary

A face of a convex set $C$ is a convex subset $F$ of $C$ with the following property. Any segment of $C$ whose inside meets $F$ lies entirely in $F$. In other words, a face is a convex extreme set of $C$. For a polytope, the nonempty faces are the vertices, edges and higher-dimensional faces, together with the polytope itself. The zero-dimensional faces of $C$ are exactly its extreme points.