the Krein-Milman theorem
/ KRINE MIL-mun /
Look at a diamond: its whole solid is the convex hull of its sharp corners — fill in everything between the corners and you recover the gem. The Krein-Milman theorem says this is no accident: a compact convex set is completely rebuilt from its sharpest points, the extreme points, the ones that are not the midpoint (or any inside-average) of two other points of the set. The corners of a polygon, the vertices of a cube, every point of a sphere's surface — these are the extreme points, and the body is exactly their convex hull.
An extreme point of a convex set K is a point that does not lie strictly between two distinct points of K: if e = t*x + (1 - t)*y with x, y in K and 0 < t < 1 forces x = y = e. The theorem states that a nonempty compact convex set K in R^n (more generally in a locally convex topological vector space) equals the closed convex hull of its extreme points: K = conv(ext K), closure included. So the extreme points carry all the information; everything else is a weighted average of them. The infinite-dimensional version is the powerful one — it guarantees extreme points even when you cannot see them, provided K is compact in a suitable (often weak-star) topology.
This is the abstract engine behind many existence theorems: that linear programs attain their optimum at a vertex, that probability measures are mixtures of point masses (the extreme points of the simplex of measures), that the unit ball of L-infinity has extreme points (functions of modulus 1) while the unit ball of the non-dual space c_0 has NONE — which by Krein-Milman proves c_0 is not a dual space. Two honest caveats. First, you genuinely need compactness: the open ball has no extreme points at all, and an unbounded convex set like a half-plane is not the hull of its extremes. Second, Krein-Milman gives the closed convex hull; recovering K as the hull of extreme points without taking closure is the stronger, separate fact (true for polytopes and finite dimensions, subtler in general — the sharper Choquet theory refines it).
The extreme points of the solid square [0,1] x [0,1] are exactly its four corners (0,0), (1,0), (1,1), (0,1). Every other point — an edge midpoint, the center — is an average of two square points, so it is not extreme. And indeed the square is the convex hull of those four corners. By contrast, the closed disk's extreme points are its entire boundary circle: no boundary point is an inside average of two others.
A square is the hull of its 4 corners; a disk's extreme points fill its whole boundary circle.
Extreme points are not the same as exposed points (those cut out by a single supporting hyperplane touching only there). Every exposed point is extreme, but not conversely — a stadium shape (rectangle with semicircular caps) has extreme points at the cap-rectangle junctions that are not exposed. Krein-Milman needs only extreme points.