Krein–Milman theorem

In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about convex sets in topological vector spaces. A particular case of this theorem, which can be easily visualized, states that given a convex polygon, one only needs the corners of the polygon to recover the polygon shape. The statement of the theorem is false if the polygon is not convex, as then there can be many ways of drawing a polygon having given points as corners.

Formally, let ${\displaystyle X}$ be a locally convex topological vector space (assumed to be Hausdorff), and let ${\displaystyle K}$ be a compact convex subset of ${\displaystyle X}$. Then, the theorem states that ${\displaystyle K}$ is the closed convex hull of its extreme points.

...
Wikipedia