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Abstract
The Hahn–Banach theorem implies the axiom of choice for families of closed convex sets in normed reflexive spaces and for more general families of convex sets in locally convex spaces. It is, in fact, equivalent to several forms of coherent choice in inversely directed families of convex sets and affine continuous transformations. This is a consequence of results about convex compactness and barycenters of finitely additive measures. Two characterizations of reflexive normed spaces in terms of these last concepts follow from Hahn–Banach.
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References
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