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 Proof Method Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Krein-Milman theorem, Wikipedia
Proved By
Source Krein-Milman theorem, Wikipedia
Sources
1. Krein-Milman theorem, Wikipedia
History
The original statement proved by Mark Krein and David Milman (1940) was somewhat less general than the form stated here.
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
Proved By: Mark Krein, Article text
The original statement proved by Mark Krein and David Milman (1940) was somewhat less general than the form stated here.
View the Source2. ProofWiki: Krein-Milman Theorem
ProofWikiProof sectionQuote, Proof section
Aiming for a contradiction, suppose suppose that $K' \ne K$.
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