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 be a locally convex topological vector space (assumed to be Hausdorff), and let be a compact convex subset of . Then, the theorem states that is the closed convex hull of its extreme points.
The closed convex hull above is defined as the intersection of all closed convex subsets of that contain This turns out to be the same as the closure of the convex hull in the topological vector space. One direction in the theorem is easy; the main burden is to show that there are 'enough' extreme points.
Hermann Minkowski had already proved that if is 3-dimensional then equals the convex hull of the set of its extreme points. This assertion was expanded to the case of any finite dimension by Ernst Steinitz. The Krein–Milman theorem generalizes this to arbitrary locally convex , with a caveat: the closure may be needed.
Relation to the axiom of choice
The axiom of choice, or some weaker version of it, is needed to prove this theorem in Zermelo–Fraenkel set theory. Conversely, this theorem together with the Boolean prime ideal theorem can prove the axiom of choice.
Under the previous assumptions on if is a subset of and the closed convex hull of is all of , then every extreme point of belongs to the closure of This result is known as Milman's (partial) converse to the Krein–Milman theorem.
More general settings
The linearity is also needed, because the statement fails for weakly compact convex sets in CAT(0) spaces, as proved in 2016 by Nicolas Monod. However, Theo Buehler proved in 2006 that the Krein–Milman theorem does hold for metrically compact CAT(0) spaces.
- Carathéodory's theorem (convex hull)
- Shapley–Folkman lemma
- Helly's theorem
- Radon's theorem
- Choquet theory
- Banach–Alaoglu theorem
- Bell, J.; Fremlin, David (1972). "A geometric form of the axiom of choice" (PDF). Fundamenta Mathematicae. 77 (2): 167–170. Retrieved 11 June 2018.
Theorem 1.2. BPI [the Boolean Prime Ideal Theorem] & KM [Krein-Milman] ⇒ (*) [the unit ball of the dual of a normed vector space has an extreme point]…. Theorem 2.1. (*) ⇒ AC [the Axiom of Choice].
- Buehler, Theo (2006). "The Krein-Mil'man Theorem for Metric Spaces with a Convex Bicombing". arXiv:math/0604187.
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