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On the axiomatization of convex subsets of Banach spaces

By Valerio Capraro and Tobias Fritz

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

Topics: Mathematics - Metric Geometry, Mathematics - Functional Analysis, Mathematics - Operator Algebras, (Primary) 52A01 (Secondary) 46L36
Year: 2015
DOI identifier: 10.1090/S0002-9939-2013-11465-6
OAI identifier: oai:arXiv.org:1105.1270
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