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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.
Contents
Formally, let
The closed convex hull above is defined as the intersection of all closed convex subsets of
The original statement proved by Mark Krein and David Milman was somewhat less general than this.
Hermann Minkowski had already proved that if
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. This theorem together with the Boolean prime ideal theorem, though, can prove the axiom of choice.
Related results
Under the previous assumptions on
The Choquet–Bishop–de Leeuw theorem states that every point in
More general settings
The assumption of local convexity for the ambient space is necessary, because James Roberts constructed a counter-example in 1977 for more general spaces.
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.