Convex Geometry - Main Theorem
Every compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points:
For polytopes (finite-dimensional compact convex sets), this simplifies: every polytope is the convex hull of its vertices (extreme points).
This fundamental result shows that convex sets are determined by their "corners"βpoints that cannot be expressed as combinations of others. It provides both theoretical insight (convex sets are built from extremal elements) and computational tools (represent sets via extreme points).
Game theory: Mixed strategy equilibria are extreme points of strategy simplices. Nash equilibria can be found by examining extreme points.
Economics: Efficient allocations in production economies lie on the boundary as extreme points of feasible sets.
Optimization: Linear programs achieve optima at vertices (extreme points) of feasible polyhedra, enabling the simplex algorithm.
The theorem extends to infinite dimensions with appropriate topological conditions. The Choquet theory further analyzes how general points decompose as "integrals" over extreme points, generalizing convex combinations to measure-theoretic settings.
A subset is a face if whenever a convex combination of points in lies in , those points themselves lie in .
Faces form a partially ordered structure under inclusion. Extreme points are 0-dimensional faces; edges are 1-dimensional faces; facets are -dimensional faces.
The Krein-Milman theorem has deep connections to functional analysis. In the dual space setting, extreme points of the unit ball characterize pure states in quantum mechanics and extremal functionals in optimization theory.
For polytopes, the number and structure of faces satisfy Euler's formula and its generalizations (Euler characteristic, -vectors), connecting convex geometry with combinatorics and algebraic topology.