On the axiomatization of convex subsets of Banach spaces

Abstract

We prove that any convex-like structure in the sense of Nate Brown is affinely and isometrically isomorphic to a closed convex subset of a Banach space. This answers an open question of Brown. As an intermediate step, we identify Brown's algebraic axioms as equivalent to certain well-known axioms of abstract convexity. We conclude with a new characterization of convex subsets of Banach spaces.

8 pages, 1 figure. v3: added post-publication note on missing reference with partly overlapping material

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