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.
Capraro, V., Fritz, T. (2013). On the axiomatization of convex subsets of a Banach space. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 141(6), 2127-2135 [10.1090/S0002-9939-2013-11465-6].
On the axiomatization of convex subsets of a Banach space
Capraro V;
2013
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.
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