The Krein-Milman Theorem states that in a locally convex topological vector space, a compact convex set equals the closed convex hull of its own extreme points, the points that cannot be written as a genuine average of two other points in the set. Named for Mark Krein and David Milman, it is a foundational result of functional analysis guaranteeing that a compact convex set is fully determined by its extreme points alone, with applications throughout convex analysis and optimization.
Facts
StatementA compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of its extreme points. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Krein-Milman theorem, Wikipedia
Sources
1. Krein-Milman theorem, Wikipedia
Statement section
A compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of its extreme points.
History section
The original statement proved by Mark Krein and David Milman (1940) was somewhat less general than the form stated here.
- In Branch: Functional Analysis, Lead sentence
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