On the axiomatization of convex subsets of Banach spaces
Valerio Capraro, T. A. Fritz · Proceedings of the American Mathematical Society · 2013
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.