Characterize convex sets that are the convex hull of their extreme points

Question

The famous Krein-Milman theorem states that every compact convex set in a topological vector space is the convex hull of its extreme points. Note, however, that many convex sets exist in $\mathbb R^n$ which are not bounded but still the convex hull of their extreme points (e.g. the closed epigraph of a parabola), so I was wondering whether a true characterization (not a sufficient condition) for this property exists in finite-dimensional spaces: which closed convex sets are the convex hulls of their extreme points? All my search results lead to the Krein-Milman theorem which doesn't interest me. Due to the huge literature on convex sets I can't imagine that this hasn't been attempted before.