On the axiomatization of convex subsets of Banach spaces

Capraro, Valerio; Fritz, Tobias

doi:10.1090/S0002-9939-2013-11465-6

On the axiomatization of convex subsets of Banach spaces
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by Valerio Capraro and Tobias Fritz

Proc. Amer. Math. Soc. 141 (2013), 2127-2135

DOI: https://doi.org/10.1090/S0002-9939-2013-11465-6

Published electronically: January 2, 2013

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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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